List of integrals of rational functions explained

The following is a list of integrals (antiderivative functions) of rational functions. Any rational function can be integrated by partial fraction decomposition of the function into a sum of functions of the form:which can then be integrated term by term.

For other types of functions, see lists of integrals.

Miscellaneous integrands

\int (f'(x))/(f(x)) dx = ln \left| f(x)\right| + C \int (1)/(x ^{2+a} ^{2}) dx = (1)/(\arctan a) (x)/(a) + C \int (1)/(x ^{2-a} ^{2}) dx = (1)/(ln \left| 2 a) (x - a)/(x + a) \right| + C = \begin{cases} \displaystyle - (1)/(\operatorname{artanh} a) (x)/(a) + C = (1)/(ln 2 a) (a - x)/(a + x) + C & (for |x| < |a| ) \\[ 12 pt] \displaystyle - (1)/(\operatorname{arcoth} a) (x)/(a) + C = (1)/(ln 2 a) (x - a)/(x + a) + C & (for |x| > |a| ) \end{cases} \int (1)/(a ^{2-x} ^{2}) dx = (1)/(ln \left| 2 a) (a + x)/(a - x) \right| + C = \begin{cases} \displaystyle (1)/(\operatorname{artanh} a) (x)/(a) + C = (1)/(ln 2 a) (a + x)/(a - x) + C & (for |x| < |a| ) \\[ 12 pt] \displaystyle (1)/(\operatorname{arcoth} a) (x)/(a) + C = (1)/(ln 2 a) (x + a)/(x - a) + C & (for |x| > |a| ) \end{cases} \int (dx)/(x ^{2 ^{n}} + 1) = (1)/(2 ^{n-1}) \sum ^{2 ^{n-1}} _{k =1} \sin \left( (2 k -1)/(2 ^{n}) \pi\right) \arctan\left[\left(x - \cos \left( (2 k -1)/(2 ^{n}) \pi \right) \right ) \csc \left( (2 k -1)/(2 ^{n}) \pi \right) \right] - (1)/(2) \cos \left( (2 k -1)/(2 ^{n}) \pi \right) ln \left | x ^{2} - 2 x \cos \left( (2 k -1)/(2 ^{n}) \pi \right) + 1 \right | + C

Integrands of the form x^m(a x + b)ⁿ

Many of the following antiderivatives have a term of the form ln |ax + b|. Because this is undefined when x = −b / a, the most general form of the antiderivative replaces the constant of integration with a locally constant function.^[1] However, it is conventional to omit this from the notation. For example,is usually abbreviated as$$\backslash int\backslash frac\; \backslash ,\; dx=\; \backslash frac\backslash ln\backslash left|ax\; +\; b\backslash right|\; +\; C,$$where C is to be understood as notation for a locally constant function of x. This convention will be adhered to in the following.

\int (ax + b) ^{n} dx = ((ax + b) ^{n+1})/(a(n +1 )) + C (for n ≠ -1 ) (Cavalieri's quadrature formula)\int (x)/(ax + b) dx = (x)/(a) - (b)/(a ^{2}) ln \left|ax + b\right| + C \int (mx + n)/(ax + b) dx = (m)/(a) x + (an - bm)/(a ^{2}) ln \left|ax + b\right| + C \int (x)/((ax + b) ^{2}) dx = (b)/(a ^{2(ax} + b)) + (1)/(a ^{2}) ln \left|ax + b\right| + C \int (x)/((ax + b) ^{n}) dx = (a( 1- n)x - b)/(a ^{2(n} -1 )(n -2 )(ax + b) ^{n-1}) + C (for n\not\in \{ 1 , 2 \} ) \int x(ax + b) ^{n} dx = (a(n +1 )x - b)/(a ^{2(n} +1 )(n +2 )) (ax + b) ^{n+1} + C (for n \not\in \{ -1 , -2 \} ) \int (x ^{2})/(ax + b) dx = (b ^{2ln(\left|ax} + b\right|))/(+ a ^{3}) (ax ^{2} -2 bx)/(2 a ^{2}) + C \int (x ^{2})/((ax + b) ^{2}) dx = (1)/(a ^{3}) \left(ax - 2 b ln \left|ax + b\right| - (b ^{2})/(ax + b) \right) + C \int (x ^{2})/((ax + b) ^{3}) dx = (1)/(a ^{3}) \left( ln \left|ax + b\right| + (2 b)/(ax + b) - (b ^{2})/(2 (ax + b) ^{2}) \right) + C \int (x ^{2})/((ax + b) ^{n}) dx = (1)/(\left( - a ^{3}) ((ax + b) ^{3-n})/((n -3 )) + (2 b (ax + b) ^{2-n})/((n -2 )) - (b ^{2} (ax + b) ^{1-n})/((n -1 )) \right) + C (for n\not\in \{ 1 , 2 , 3 \} ) \int (1)/(x(ax + b)) dx =- (1)/(ln \left| b) (ax + b)/(x) \right| + C \int (1)/(x ^{2(ax+b)}) dx =- (1)/(bx) + (a)/(ln \left| b ^{2}) (ax + b)/(x) \right| + C \int (1)/(x ^{2(ax+b)} ^{2}) dx =- a\left( (1)/(b ^{2(ax+b)}) + (1)/(ab ^{2x}) - (2)/(ln \left| b ^{3}) (ax + b)/(x) \right|\right) + C

Integrands of the form x^m / (a x² + b x + c)ⁿ

For

a ≠ 0 : \int (1)/(ax ^{2+bx+c}) dx = \begin{cases} \displaystyle (2)/(\sqrt{ 4 ac - b ^{2}) }\arctan\frac + C & \text4ac-b^2>0\mbox \\[12pt]\displaystyle \frac\ln\left|\frac\right| + C =\begin\displaystyle -\frac\,\operatorname\frac + C &\text|2ax+b|<\sqrt\mbox \\[6pt]\displaystyle -\frac\,\operatorname\frac + C &\text\end & \text4ac-b^2<0\mbox \\[12pt]\displaystyle -\frac + C & \text4ac-b^2=0\mbox\end\int (x)/(ax ^{2+bx+c}) dx = (1)/(2 a) ln \left|ax ^{2+ bx + c\right| - (b)/(2 a)} \int (dx)/(ax ^{2+bx+c}) + C \int (mx + n)/(ax ^{2+bx+c}) dx = \begin{cases} \displaystyle (m)/(2 a) ln \left|ax ^{2+ bx + c\right| + (2 an - bm)/(a\sqrt{ 4 ac - b ^{2})} }\arctan\frac + C &\text4ac-b^2>0\mbox \\[12pt] \displaystyle \frac\ln\left|ax^2+bx+c\right|+\frac\ln\left|\frac\right| + C =\begin\displaystyle \frac\ln\left|ax^2+bx+c\right|-\frac\,\operatorname\frac + C &\text|2ax+b|<\sqrt\mbox \\[6pt]\displaystyle \frac\ln\left|ax^2+bx+c\right|-\frac\,\operatorname\frac + C &\text\end & \text4ac-b^2<0\mbox \\[12pt]\displaystyle \frac\ln\left|ax^2+bx+c\right|-\frac + C = \frac\ln\left|x+\frac\right|-\frac + C &\text4ac-b^2=0\mbox\end\int (1)/((ax ^{2+bx+c)} ^{n}) dx = (2 ax + b)/(+ (n -1 )( 4 ac - b ^{2)(ax} ^{2+bx+c)} ^{n-1}) (( 2 n -3 ) 2 a)/(\int (n -1 )( 4 ac - b ^{2)}) (1)/((ax ^{2+bx+c)} ^{n-1}) dx + C \int (x)/((ax ^{2+bx+c)} ^{n}) dx =- (bx +2 c)/(- (n -1 )( 4 ac - b ^{2)(ax} ^{2+bx+c)} ^{n-1}) (b( 2 n -3 ))/(\int (n -1 )( 4 ac - b ^{2)}) (1)/((ax ^{2+bx+c)} ^{n-1}) dx + C \int (1)/(x(ax ^{2+bx+c)}) dx = (1)/(ln \left| 2 c) (x ^{2})/(\right| - ax ^{2+bx+c}) (b)/(\int 2 c) (1)/(ax ^{2+bx+c}) dx + C

Integrands of the form x^m (a + b xⁿ)^p

• The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m and p toward 0.
• These reduction formulas can be used for integrands having integer and/or fractional exponents.

\int x ^{m} \left(a + bx ^{n\right)} ^{p} dx = (x ^{m+1} \left(a + bx ^{n\right)} ^{p})/(m + np +1) + (anp)/(m + np +1) \int x ^{m} \left(a + bx ^{n\right)} ^{p-1} dx \int x ^{m} \left(a + bx ^{n\right)} ^{p} dx =- (x ^{m+1} \left(a + bx ^{n\right)} ^{p+1})/(an (p +1 )) + (m + n (p +1 ) +1)/(an (p +1 )) \int x ^{m} \left(a + bx ^{n\right)} ^{p+1} dx \int x ^{m} \left(a + bx ^{n\right)} ^{p} dx = (x ^{m+1} \left(a + bx ^{n\right)} ^{p})/(m +1) - (bnp)/(m +1) \int x ^{m+n} \left(a + bx ^{n\right)} ^{p-1} dx \int x ^{m} \left(a + bx ^{n\right)} ^{p} dx = (x ^{m-n+1} \left(a + bx ^{n\right)} ^{p+1})/(bn (p +1 )) - (m - n +1)/(bn (p +1 )) \int x ^{m-n} \left(a + bx ^{n\right)} ^{p+1} dx \int x ^{m} \left(a + bx ^{n\right)} ^{p} dx = (x ^{m-n+1} \left(a + bx ^{n\right)} ^{p+1})/(b (m + np +1 )) - (a (m - n +1 ))/(b (m + np +1 )) \int x ^{m-n} \left(a + bx ^{n\right)} ^{pdx} \int x ^{m} \left(a + bx ^{n\right)} ^{p} dx = (x ^{m+1} \left(a + bx ^{n\right)} ^{p+1})/(a (m +1 )) - (b (m + n (p +1 ) +1 ))/(a (m +1 )) \int x ^{m+n} \left(a + bx ^{n\right)} ^{pdx}

Integrands of the form (A + B x) (a + b x)^m (c + d x)ⁿ (e + f x)^p

• The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m, n and p toward 0.
• These reduction formulas can be used for integrands having integer and/or fractional exponents.
• Special cases of these reductions formulas can be used for integrands of the form

(a + bx) ^{m} (c + dx) ^{n} (e + fx) ^{p} by setting B to 0.\begin{align} &\int (A + Bx) (a + bx) ^{m} (c + dx) ^{n} (e + fx) ^{p} dx = - ((Ab - aB)(a + bx) ^{m+1} (c + dx) ^{n(e+fx)} ^{p+1})/(b (m +1 ) (af - be)) + (1)/(b (m +1 ) (af - be)) ⋅ \\ & \int (bc(m +1 ) (Af - Be) + (Ab - aB) (nde + cf(p +1 )) + d(b(m +1 ) (Af - Be) + f(n + p +1 ) (Ab - aB))x)(a + bx) ^{m+1} (c + dx) ^{n-1} (e + fx) ^{p} dx \end{align} \begin{align} &\int (A + Bx) (a + bx) ^{m} (c + dx) ^{n} (e + fx) ^{p} dx = (B(a + bx) ^{m} (c + dx) ^{n+1} (e + fx) ^{p+1})/(df(m + n + p +2 )) + (1)/(df(m + n + p +2 )) ⋅ \\ & \int (Aadf(m + n + p +2 ) - B (bcem + a(de(n +1 ) + cf(p +1 ))) + (Abdf(m + n + p +2 ) + B (adfm - b(de(m + n +1 ) + cf(m + p +1 )))) x)(a + bx) ^{m-1} (c + dx) ^{n(e+fx)} ^{p} dx \end{align} \begin{align} &\int (A + Bx) (a + bx) ^{m} (c + dx) ^{n} (e + fx) ^{p} dx = ((Ab - aB)(a + bx) ^{m+1} (c + dx) ^{n+1} (e + fx) ^{p+1})/((m +1 )(ad - bc)(af - be)) + (1)/((m +1 )(ad - bc)(af - be)) ⋅ \\ & \int ((m +1 ) (A (adf - b(cf + de)) + Bbce) - (Ab - aB) (de(n +1 ) + cf(p +1 )) - df(m + n + p +3 ) (Ab - aB)x)(a + bx) ^{m+1} (c + dx) ^{n(e+fx)} ^{p} dx \end{align}

Integrands of the form x^m (A + B xⁿ) (a + b xⁿ)^p (c + d xⁿ)^q

• The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m, p and q toward 0.
• These reduction formulas can be used for integrands having integer and/or fractional exponents.
• Special cases of these reductions formulas can be used for integrands of the form

\left(a + bx ^{n\right)} ^{p\left(c+dx} ^{n\right)} ^{q} and x ^{m\left(a+bx} ^{n\right)} ^{p\left(c+dx} ^{n\right)} ^{q} by setting m and/or B to 0.\begin{align} &\int x ^{m\left(A+Bx} ^{n\right)\left(a+bx} ^{n\right)} ^{p\left(c+dx} ^{n\right)} ^{qdx=} - ((Ab - aB) x ^{m+1} \left(a + bx ^{n\right)} ^{p+1} \left(c + dx ^{n\right)} ^{q})/(abn (p +1 )) + (1)/(abn (p +1 )) ⋅ \\ & \int x ^{m\left(c} (Abn (p +1 ) + (Ab - aB) (m +1 )) + d (Abn (p +1 ) + (Ab - aB) (m + nq +1 )) x ^{n\right)\left(a+bx} ^{n\right)} ^{p+1} \left(c + dx ^{n\right)} ^{q-1} dx \end{align} \begin{align} &\int x ^{m\left(A+Bx} ^{n\right)\left(a+bx} ^{n\right)} ^{p\left(c+dx} ^{n\right)} ^{qdx=} (Bx ^{m+1} \left(a + bx ^{n\right)} ^{p+1} \left(c + dx ^{n\right)} ^{q})/(b (m + n (p + q +1 ) +1 )) + (1)/(b (m + n (p + q +1 ) +1 )) ⋅ \\ & \int x ^{m\left(c} ((Ab - aB) ( 1+ m) + Abn ( 1+ p + q)) + (d(Ab - aB) ( 1+ m) + Bnq(bc - ad) + Abdn ( 1+ p + q))x ^{n\right)\left(a+bx} ^{n\right)} ^{p\left(c+dx} ^{n\right)} ^{q-1} dx \end{align} \begin{align} &\int x ^{m\left(A+Bx} ^{n\right)\left(a+bx} ^{n\right)} ^{p\left(c+dx} ^{n\right)} ^{qdx=} - ((Ab - aB) x ^{m+1} \left(a + bx ^{n\right)} ^{p+1} \left(c + dx ^{n\right)} ^{q+1})/(an (bc - ad) (p +1 )) + (1)/(an(bc - ad)(p +1 )) ⋅ \\ & \int x ^{m\left(c(Ab-aB)(m+1)+An} (bc - ad)(p +1 ) + d(Ab - aB) (m + n (p + q +2 ) +1 ) x ^{n\right)\left(a+bx} ^{n\right)} ^{p+1} \left(c + dx ^{n\right)} ^{qdx} \end{align} \begin{align} &\int x ^{m\left(A+Bx} ^{n\right)\left(a+bx} ^{n\right)} ^{p\left(c+dx} ^{n\right)} ^{qdx=} (Bx ^{m-n+1} \left(a + bx ^{n\right)} ^{p+1} \left(c + dx ^{n\right)} ^{q+1})/(bd (m + n (p + q +1 ) +1 )) - (1)/(bd (m + n (p + q +1 ) +1 )) ⋅ \\ & \int x ^{m-n} \left(aBc (m - n +1 ) + (aBd (m + nq +1 ) - b ( - Bc (m + np +1 ) + Ad (m + n (p + q +1 ) +1 ))) x ^{n\right)\left(a+bx} ^{n\right)} ^{p\left(c+dx} ^{n\right)} ^{qdx} \end{align} \begin{align} &\int x ^{m\left(A+Bx} ^{n\right)\left(a+bx} ^{n\right)} ^{p\left(c+dx} ^{n\right)} ^{qdx=} (Ax ^{m+1} \left(a + bx ^{n\right)} ^{p+1} \left(c + dx ^{n\right)} ^{q+1})/(ac (m +1 )) + (1)/(ac (m +1 )) ⋅ \\ & \int x ^{m+n} \left(aBc (m +1 ) - A (bc + ad) (m + n +1 ) - An (bcp + adq) - Abd (m + n (p + q +2 ) +1 ) x ^{n\right)\left(a+bx} ^{n\right)} ^{p\left(c+dx} ^{n\right)} ^{qdx} \end{align} \begin{align} &\int x ^{m\left(A+Bx} ^{n\right)\left(a+bx} ^{n\right)} ^{p\left(c+dx} ^{n\right)} ^{qdx=} (Ax ^{m+1} \left(a + bx ^{n\right)} ^{p+1} \left(c + dx ^{n\right)} ^{q})/(a (m +1 )) - (1)/(a (m +1 )) ⋅ \\ & \int x ^{m+n} \left(c(Ab - aB)(m +1 ) + An (bc (p +1 ) + adq) + d ((Ab - aB) (m +1 ) + Abn (p + q +1 )) x ^{n\right)\left(a+bx} ^{n\right)} ^{p\left(c+dx} ^{n\right)} ^{q-1} dx \end{align} \begin{align} &\int x ^{m\left(A+Bx} ^{n\right)\left(a+bx} ^{n\right)} ^{p\left(c+dx} ^{n\right)} ^{qdx=} ((Ab - aB) x ^{m-n+1} \left(a + bx ^{n\right)} ^{p+1} \left(c + dx ^{n\right)} ^{q+1})/(bn (bc - ad) (p +1 )) - (1)/(bn(bc - ad)(p +1 )) ⋅ \\ & \int x ^{m-n} \left(c(Ab - aB)(m - n +1 ) + (d(Ab - aB)(m + nq +1 ) - bn(Bc - Ad)(p +1 )) x ^{n\right)\left(a+bx} ^{n\right)} ^{p+1} \left(c + dx ^{n\right)} ^{qdx} \end{align}

Integrands of the form (d + e x)^m (a + b x + c x²)^p when b² − 4 a c = 0

• The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m and p toward 0.
• These reduction formulas can be used for integrands having integer and/or fractional exponents.
• Special cases of these reductions formulas can be used for integrands of the form

\left(a + bx + cx ^{2\right)} ^{p} when b ^{2-4ac=0} by setting m to 0.\int (d + ex) ^{m} \left(a + bx + cx ^{2\right)} ^{pdx=} ((d + ex) ^{m+1} \left(a + bx + cx ^{2\right)} ^{p})/(e(m +1 )) - (p (d + ex) ^{m+2} (b +2 cx) \left(a + bx + cx ^{2\right)} ^{p-1})/(e ^{2(m+1)(m+2} p +1 )) + (p( 2 p -1 )( 2 cd - be))/(e ^{2(m+1)(m+2} p +1 )) \int (d + ex) ^{m+1} \left(a + bx + cx ^{2\right)} ^{p-1} dx \int (d + ex) ^{m} \left(a + bx + cx ^{2\right)} ^{pdx=} ((d + ex) ^{m+1} \left(a + bx + cx ^{2\right)} ^{p})/(e(m +1 )) - (p (d + ex) ^{m+2} (b +2 cx)\left(a + bx + cx ^{2\right)} ^{p-1})/(e ^{2(m+1)(m+2)}) + (2 cp( 2 p -1 ))/(e ^{2(m+1)(m+2)}) \int (d + ex) ^{m+2} \left(a + bx + cx ^{2\right)} ^{p-1} dx \int (d + ex) ^{m\left(a+bx+cx} ^{2\right)} ^{pdx=} - (e(m +2 p +2 )(d + ex) ^{m} \left(a + bx + cx ^{2\right)} ^{p+1})/((p +1 )( 2 p +1 )( 2 cd - be)) + ((d + ex) ^{m+1} (b +2 cx) \left(a + bx + cx ^{2\right)} ^{p})/(( 2 p +1 )( 2 cd - be)) + (e ^{2m(m+2} p +2 ))/((p +1 )( 2 p +1 )( 2 cd - be)) \int (d + ex) ^{m-1} \left(a + bx + cx ^{2\right)} ^{p+1} dx \int (d + ex) ^{m} \left(a + bx + cx ^{2\right)} ^{pdx=} - (em(d + ex) ^{m-1} \left(a + bx + cx ^{2\right)} ^{p+1})/(2 c (p +1 ) ( 2 p +1 )) + ((d + ex) ^{m(b+2} cx)\left(a + bx + cx ^{2\right)} ^{p})/(2 c ( 2 p +1 )) + (e ^{2m(m-1)})/(2 c (p +1 ) ( 2 p +1 )) \int (d + ex) ^{m-2} \left(a + bx + cx ^{2\right)} ^{p+1} dx \int (d + ex) ^{m} \left(a + bx + cx ^{2\right)} ^{pdx=} ((d + ex) ^{m+1} \left(a + bx + cx ^{2\right)} ^{p})/(e(m +2 p +1 )) - (p( 2 cd - be)(d + ex) ^{m+1} (b +2 cx)\left(a + bx + cx ^{2\right)} ^{p-1})/(2 ce ^{2(m+2} p)(m +2 p +1 )) + (p ( 2 p -1 )( 2 cd - be) ^{2})/(2 ce ^{2(m+2} p)(m +2 p +1 )) \int (d + ex) ^{m} \left(a + bx + cx ^{2\right)} ^{p-1} dx \int (d + ex) ^{m} \left(a + bx + cx ^{2\right)} ^{pdx=} - (2 ce(m +2 p +2 )(d + ex) ^{m+1} \left(a + bx + cx ^{2\right)} ^{p+1})/((p +1 ) ( 2 p +1 )( 2 cd - be) ^{2}) + ((d + ex) ^{m+1} (b +2 cx)\left(a + bx + cx ^{2\right)} ^{p})/(( 2 p +1 )( 2 cd - be)) + (2 ce ^{2(m+2p+2)(m+2} p +3 ))/((p +1 ) ( 2 p +1 )( 2 cd - be) ^{2}) \int (d + ex) ^{m} \left(a + bx + cx ^{2\right)} ^{p+1} dx \int (d + ex) ^{m} \left(a + bx + cx ^{2\right)} ^{pdx=} ((d + ex) ^{m} (b +2 cx)\left(a + bx + cx ^{2\right)} ^{p})/(2 c (m +2 p +1 )) + (m( 2 cd - be))/(2 c (m +2 p +1 )) \int (d + ex) ^{m-1} \left(a + bx + cx ^{2\right)} ^{pdx} \int (d + ex) ^{m\left(a+bx+cx} ^{2\right)} ^{pdx=} - ((d + ex) ^{m+1} (b +2 cx)\left(a + bx + cx ^{2\right)} ^{p})/((m +1 )( 2 cd - be)) + (2 c (m +2 p +2 ))/((m +1 )( 2 cd - be)) \int (d + ex) ^{m+1} \left(a + bx + cx ^{2\right)} ^{pdx}

Integrands of the form (d + e x)^m (A + B x) (a + b x + c x²)^p

• The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m and p toward 0.
• These reduction formulas can be used for integrands having integer and/or fractional exponents.
• Special cases of these reductions formulas can be used for integrands of the form

\left(a + bx + cx ^{2\right)} ^{p} and (d + ex) ^{m} \left(a + bx + cx ^{2\right)} ^{p} by setting m and/or B to 0.\begin{align} &\int (d + ex) ^{m} (A + Bx) \left(a + bx + cx ^{2\right)} ^{pdx=} ((d + ex) ^{m+1} (Ae (m +2 p +2 ) - Bd ( 2 p +1 ) + eB (m +1 ) x) \left(a + bx + cx ^{2\right)} ^{p})/(e ^{2(m+1)} (m +2 p +2 )) + (1)/(e ^{2(m+1)} (m +2 p +2 )) p ⋅ \\ & \int (d + ex) ^{m+1} (B (bd +2 ae +2 aem +2 bdp) - Abe (m +2 p +2 ) + (B ( 2 cd + be + be m +4 cdp) -2 Ace (m +2 p +2 ))x)\left(a + bx + cx ^{2\right)} ^{p-1} dx \end{align} \begin{align} &\int (d + ex) ^{m} (A + Bx) \left(a + bx + cx ^{2\right)} ^{pdx=} ((d + ex) ^{m} (Ab -2 aB - (bB -2 Ac) x)\left(a + bx + cx ^{2\right)} ^{p+1})/((p +1 )\left(b ^{2-4} ac\right)) + (1)/((p +1 )\left(b ^{2-4} ac\right)) ⋅ \\ & \int (d + ex) ^{m-1} (B ( 2 aem + bd ( 2 p +3 )) - A (bem +2 cd ( 2 p +3 )) + e(bB -2 Ac) (m +2 p +3 ) x)\left(a + bx + cx ^{2\right)} ^{p+1} dx \end{align} \begin{align} &\int (d + ex) ^{m} (A + Bx) \left(a + bx + cx ^{2\right)} ^{pdx=} ((d + ex) ^{m+1} (Ace (m +2 p +2 ) - B (cd +2 cdp - bep) + Bce(m +2 p +1 ) x)\left(a + bx + cx ^{2\right)} ^{p})/(ce ^{2(m+2} p +1 ) (m +2 p +2 )) - (p)/(ce ^{2(m+2} p +1 ) (m +2 p +2 )) ⋅ \\ & \int (d + ex) ^{m} (Ace (bd -2 ae) (m +2 p +2 ) + B (ae (be -2 cdm + bem) + bd (bep - cd -2 cdp)) + \\ & \left(Ace ( 2 cd - be) (m +2 p +2 ) - B \left( - b ^{2} e ^{2} (m + p +1 ) +2 c ^{2} d ^{2} ( 1+2 p) + ce (bd (m -2 p) +2 ae (m +2 p +1 ))\right)\right) x)\left(a + bx + cx ^{2\right)} ^{p-1} dx \end{align} \begin{align} &\int (d + ex) ^{m} (A + Bx) \left(a + bx + cx ^{2\right)} ^{pdx=} ((d + ex) ^{m+1} \left(A \left(bcd - b ^{2} e +2 ace\right) - aB ( 2 cd - be) + c (A ( 2 cd - be) - B (bd -2 ae)) x\right)\left(a + bx + cx ^{2\right)} ^{p+1})/((p +1 )\left(b ^{2-4} ac\right) \left(cd ^{2-bde+ae} ^{2\right)}) + \\ & (1)/((p +1 )\left(b ^{2-4} ac\right) \left(cd ^{2-bde+ae} ^{2\right)}) ⋅ \\ & \int (d + ex) ^{m} (A \left(bcde ( 2 p - m +2 ) + b ^{2} e ^{2} (m + p +2 ) -2 c ^{2} d ^{2} ( 3+2 p) -2 ace ^{2} (m +2 p +3 )\right) - \\ & B (ae (be -2 cd m + bem) + bd ( -3 cd + be -2 cdp + bep)) + ce(B (bd -2 ae) - A ( 2 cd - be)) (m +2 p +4 ) x)\left(a + bx + cx ^{2\right)} ^{p+1} dx \end{align} \begin{align} &\int (d + ex) ^{m} (A + Bx) \left(a + bx + cx ^{2\right)} ^{pdx=} (B(d + ex) ^{m\left(a+bx+cx} ^{2\right)} ^{p+1})/(c(m +2 p +2 )) + (1)/(c(m +2 p +2 )) ⋅ \\ & \int (d + ex) ^{m-1} (m(Acd - aBe) - d(bB -2 Ac)(p +1 ) + ((Bcd - bBe + Ace) m - e(bB -2 Ac)(p +1 ))x) \left(a + bx + cx ^{2\right)} ^{pdx} \end{align} \begin{align} &\int (d + ex) ^{m} (A + Bx) \left(a + bx + cx ^{2\right)} ^{pdx=} - ((Bd - Ae) (d + ex) ^{m+1} \left(a + bx + cx ^{2\right)} ^{p+1})/((m +1 )\left(cd ^{2-bde+ae} ^{2\right)}) + (1)/((m +1 )\left(cd ^{2-bde+ae} ^{2\right)}) ⋅ \\ & \int (d + ex) ^{m+1} ((Acd - Abe + aBe) (m +1 ) + b (Bd - Ae) (p +1 ) + c (Bd - Ae) (m +2 p +3 ) x)\left(a + bx + cx ^{2\right)} ^{pdx} \end{align}

Integrands of the form x^m (a + b xⁿ + c x²ⁿ)^p when b² − 4 a c = 0

• The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m and p toward 0.
• These reduction formulas can be used for integrands having integer and/or fractional exponents.
• Special cases of these reductions formulas can be used for integrands of the form

\left(a + bx ^{n+cx} ^{2} \right) ^{p} when b ^{2-4ac=0} by setting m to 0.\int x ^{m} \left(a + bx ^{n+cx} ^{2} \right) ^{p} dx = (x ^{m+1} \left(a + bx ^{n+cx} ^{2} \right) ^{p})/(m +2 np +1) + (npx ^{m+1} \left( 2 a + bx ^{n\right)\left(a+bx} ^{n+cx} ^{2} \right) ^{p-1})/((m +1 )(m +2 np +1 )) - (bn ^{2} p ( 2 p -1 ))/((m +1 )(m +2 np +1 )) \int x ^{m+n} \left(a + bx ^{n+cx} ^{2} \right) ^{p-1} dx \int x ^{m} \left(a + bx ^{n+cx} ^{2} \right) ^{p} dx = ((m + n( 2 p -1 ) +1 ) x ^{m+1} \left(a + bx ^{n+cx} ^{2} \right) ^{p})/((m +1 )(m + n +1 )) + (npx ^{m+1} \left( 2 a + bx ^{n\right)\left(a+bx} ^{n+cx} ^{2} \right) ^{p-1})/((m +1 )(m + n +1 )) + (2 cpn ^{2(2} p -1 ))/((m +1 )(m + n +1 )) \int x ^{m+2n} \left(a + bx ^{n+cx} ^{2} \right) ^{p-1} dx \int x ^{m} \left(a + bx ^{n+cx} ^{2} \right) ^{p} dx = ((m + n( 2 p +1 ) +1 ) x ^{m-n+1} \left(a + bx ^{n+cx} ^{2} \right) ^{p+1})/(bn ^{2} (p +1 ) ( 2 p +1 )) - (x ^{m+1} \left(b +2 cx ^{n\right)\left(a+bx} ^{n+cx} ^{2} \right) ^{p})/(bn ( 2 p +1 )) - ((m - n +1 )(m + n( 2 p +1 ) +1 ))/(bn ^{2} (p +1 ) ( 2 p +1 )) \int x ^{m-n} \left(a + bx ^{n+cx} ^{2} \right) ^{p+1} dx \int x ^{m} \left(a + bx ^{n+cx} ^{2} \right) ^{p} dx = - ((m -3 n -2 np +1 ) x ^{m-2n+1} \left(a + bx ^{n+cx} ^{2} \right) ^{p+1})/(2 cn ^{2(p+1)(2p+1)}) - (x ^{m-2n+1} \left( 2 a + bx ^{n\right)\left(a+bx} ^{n+cx} ^{2} \right) ^{p})/(2 cn( 2 p +1 )) + ((m - n +1 )(m -2 n +1 ))/(2 cn ^{2(p+1)(2p+1)}) \int x ^{m-2n} \left(a + bx ^{n+cx} ^{2} \right) ^{p+1} dx \int x ^{m} \left(a + bx ^{n+cx} ^{2} \right) ^{p} dx = (x ^{m+1} \left(a + bx ^{n+cx} ^{2} \right) ^{p})/(m +2 np +1) + (npx ^{m+1} \left( 2 a + bx ^{n\right)\left(a+bx} ^{n+cx} ^{2} \right) ^{p-1})/((m +2 np +1 ) (m + n( 2 p -1 ) +1 )) + (2 an ^{2} p ( 2 p -1 ))/((m +2 np +1 ) (m + n( 2 p -1 ) +1 )) \int x ^{m} \left(a + bx ^{n+cx} ^{2} \right) ^{p-1} dx \int x ^{m} \left(a + bx ^{n+cx} ^{2} \right) ^{p} dx = - ((m + n +2 np +1 ) x ^{m+1} \left(a + bx ^{n+cx} ^{2} \right) ^{p+1})/(2 an ^{2} (p +1 ) ( 2 p +1 )) - (x ^{m+1} \left( 2 a + bx ^{n\right)\left(a+bx} ^{n+cx} ^{2} \right) ^{p})/(2 an( 2 p +1 )) + ((m + n( 2 p +1 ) +1 )(m +2 n (p +1 ) +1 ))/(2 an ^{2} (p +1 ) ( 2 p +1 )) \int x ^{m} \left(a + bx ^{n+cx} ^{2} \right) ^{p+1} dx \int x ^{m\left(a+bx} ^{n+cx} ^{2} \right) ^{p} dx = (x ^{m-n+1} \left(b +2 cx ^{n\right)\left(a+bx} ^{n+cx} ^{2} \right) ^{p})/(2 c (m +2 np +1 )) - (b (m - n +1 ))/(2 c (m +2 np +1 )) \int x ^{m-n} \left(a + bx ^{n+cx} ^{2} \right) ^{p} dx \int x ^{m\left(a+bx} ^{n+cx} ^{2} \right) ^{p} dx = (x ^{m+1} \left(b +2 cx ^{n\right)\left(a+bx} ^{n+cx} ^{2} \right) ^{p})/(b (m +1 )) - (2 c (m + n( 2 p +1 ) +1 ))/(b (m +1 )) \int x ^{m+n} \left(a + bx ^{n+cx} ^{2} \right) ^{p} dx

Integrands of the form x^m (A + B xⁿ) (a + b xⁿ + c x²ⁿ)^p

• The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m and p toward 0.
• These reduction formulas can be used for integrands having integer and/or fractional exponents.
• Special cases of these reductions formulas can be used for integrands of the form

\left(a + bx ^{n+cx} ^{2} \right) ^{p} and x ^{m} \left(a + bx ^{n+cx} ^{2} \right) ^{p} by setting m and/or B to 0.\begin{align} &\int x ^{m} \left(A + Bx ^{n\right)} \left(a + bx ^{n+cx} ^{2} \right) ^{pdx=} (x ^{m+1} \left(A (m + n ( 2 p +1 ) +1 ) + B (m +1 ) x ^{n\right)} \left(a + bx ^{n+cx} ^{2} \right) ^{p})/((m +1 ) (m + n ( 2 p +1 ) +1 )) + (np)/((m +1 ) (m + n ( 2 p +1 ) +1 )) ⋅ \\ & \int x ^{m+n} \left( 2 aB (m +1 ) - Ab (m + n ( 2 p +1 ) +1 ) + (bB (m +1 ) -2 Ac (m + n ( 2 p +1 ) +1 )) x ^{n\right)\left(a+bx} ^{n+cx} ^{2} \right) ^{p-1} dx \end{align} \begin{align} &\int x ^{m} \left(A + Bx ^{n\right)} \left(a + bx ^{n+cx} ^{2} \right) ^{pdx=} (x ^{m-n+1} \left(Ab -2 aB - (bB -2 Ac) x ^{n\right)\left(a+bx} ^{n+cx} ^{2} \right) ^{p+1})/(n(p +1 ) \left(b ^{2-4} ac\right)) + (1)/(n(p +1 ) \left(b ^{2-4} ac\right)) ⋅ \\ & \int x ^{m-n} \left((m - n +1 )( 2 aB - Ab) + (m +2 n (p +1 ) +1 ) (bB -2 Ac) x ^{n\right)\left(a+bx} ^{n+cx} ^{2} \right) ^{p+1} dx \end{align} \begin{align} &\int x ^{m} \left(A + Bx ^{n\right)} \left(a + bx ^{n+cx} ^{2} \right) ^{pdx=} (x ^{m+1} \left(bBnp + Ac (m + n ( 2 p +1 ) +1 ) + Bc (m +2 np +1 ) x ^{n\right)\left(a+bx} ^{n+cx} ^{2} \right) ^{p})/(c (m +2 np +1 ) (m + n ( 2 p +1 ) +1 )) + (np)/(c (m +2 np +1 ) (m + n ( 2 p +1 ) +1 )) ⋅ \\ & \int x ^{m} \left( 2 aAc (m + n ( 2 p +1 ) +1 ) - abB (m +1 ) + \left( 2 aBc (m +2 np +1 ) + Abc (m + n ( 2 p +1 ) +1 ) - b ^{2} B (m + np +1 )\right) x ^{n\right)\left(a+bx} ^{n+cx} ^{2} \right) ^{p-1} dx \end{align} \begin{align} &\int x ^{m} \left(A + Bx ^{n\right)} \left(a + bx ^{n+cx} ^{2} \right) ^{pdx=} - (x ^{m+1} \left(Ab ^{2-abB-2} aAc + (Ab -2 aB) cx ^{n\right)\left(a+bx} ^{n+cx} ^{2} \right) ^{p+1})/(an(p +1 ) \left(b ^{2-4} ac\right)) + (1)/(an(p +1 ) \left(b ^{2-4} ac\right)) ⋅ \\ & \int x ^{m} \left((m + n (p +1 ) +1 ) Ab ^{2-abB(m+1)-2(m+2n} (p +1 ) +1 )aAc + (m + n ( 2 p +3 ) +1 )(Ab -2 aB) cx ^{n\right)\left(a+bx} ^{n+cx} ^{2} \right) ^{p+1} dx \end{align} \begin{align} &\int x ^{m} \left(A + Bx ^{n\right)} \left(a + bx ^{n+cx} ^{2} \right) ^{pdx=} (Bx ^{m-n+1} \left(a + bx ^{n+cx} ^{2} \right) ^{p+1})/(c (m + n ( 2 p +1 ) +1 )) - (1)/(c (m + n ( 2 p +1 ) +1 )) ⋅ \\ & \int x ^{m-n} \left(aB (m - n +1 ) + (bB (m + np +1 ) - Ac (m + n ( 2 p +1 ) +1 )) x ^{n\right)} \left(a + bx ^{n+cx} ^{2} \right) ^{pdx} \end{align} \begin{align} &\int x ^{m} \left(A + Bx ^{n\right)} \left(a + bx ^{n+cx} ^{2} \right) ^{pdx=} (Ax ^{m+1} \left(a + bx ^{n+cx} ^{2} \right) ^{p+1})/(a(m +1 )) + (1)/(a(m +1 )) ⋅ \\ & \int x ^{m+n} \left(aB (m +1 ) - Ab (m + n (p +1 ) +1 ) - Ac (m +2 n(p +1 ) +1 ) x ^{n\right)\left(a+bx} ^{n+cx} ^{2} \right) ^{pdx} \end{align}

Notes and References

1. "Reader Survey: log|x| + C", Tom Leinster, The n-category Café, March 19, 2012