List of integrals of rational functions

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Template:Verifiability 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:

a(x−b)n, and ax+b((x−c)2+d2)n.

which can then be integrated term by term.

For other types of functions that can be integrated, see lists of integrals.

## Contents

* 1 Miscellaneous integrands
* 2 Integrands of the form xm(a x \+ b)n
* 3 Integrands of the form xm / (a x2 \+ b x \+ c)n
* 4 Integrands of the form xm (a \+ b xn)p
* 5 Integrands of the form (A \+ B x) (a \+ b x)m (c \+ d x)n (e \+ f x)p
* 6 Integrands of the form xm (A \+ B xn) (a \+ b xn)p (c \+ d xn)q
* 7 Integrands of the form (d \+ e x)m (a \+ b x \+ c x2)p when b2 − 4 a c = 0
* 8 Integrands of the form (d \+ e x)m (A \+ B x) (a \+ b x \+ c x2)p
* 9 Integrands of the form xm (a \+ b xn \+ c x2n)p when b2 − 4 a c = 0
* 10 Integrands of the form xm (A \+ B xn) (a \+ b xn \+ c x2n)p
* 11 References

## Miscellaneous integrands

* ∫f′(x)f(x)dx=ln⁡|f(x)|+C
* ∫1x2+a2dx=1aarctan⁡xa+C
* ∫1x2−a2dx=12aln⁡|x−ax+a|+C={−1aartanh⁡xa+C=12aln⁡a−xa+x+C(for |x|<|a|)−1aarcoth⁡xa+C=12aln⁡x−ax+a+C(for |x|>|a|)
* ∫1a2−x2dx=12aln⁡|a+xa−x|+C={1aartanh⁡xa+C=12aln⁡a+xa−x+C(for |x|<|a|)1aarcoth⁡xa+C=12aln⁡x+ax−a+C(for |x|>|a|)
* ∫1x2n+1dx=12n−1∑k=12n−1sin⁡(2k−12nπ)arctan⁡[(x−cos⁡(2k−12nπ))csc⁡(2k−12nπ)]−12cos⁡(2k−12nπ)ln⁡|x2−2xcos⁡(2k−12nπ)+1|+C

## Integrands of the form xm(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, ∫1ax+bdx={1aln⁡(−(ax+b))+C−ax+b<01aln⁡(ax+b)+C+ax+b>0 is usually abbreviated as ∫1ax+bdx=1aln⁡|ax+b|+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.

* ∫(ax+b)ndx=(ax+b)n+1a(n+1)+C(for n≠−1) (Cavalieri's quadrature formula)
* ∫xa−1(1−x)b−1dx=B(x;a,b)+C(for Re⁡(a),Re⁡(b)>0) (Incomplete beta function)
* ∫xax+bdx=xa−ba2ln⁡|ax+b|+C
* ∫mx+nax+bdx=max+an−bma2ln⁡|ax+b|+C
* ∫x(ax+b)2dx=ba2(ax+b)+1a2ln⁡|ax+b|+C
* ∫x(ax+b)ndx=a(1−n)x−ba2(n−1)(n−2)(ax+b)n−1+C(for n∉{1,2})
* ∫x(ax+b)ndx=a(n+1)x−ba2(n+1)(n+2)(ax+b)n+1+C(for n∉{−1,−2})
* ∫x2ax+bdx=b2ln⁡(|ax+b|)a3+ax2−2bx2a2+C
* ∫x2(ax+b)2dx=1a3(ax−2bln⁡|ax+b|−b2ax+b)+C
* ∫x2(ax+b)3dx=1a3(ln⁡|ax+b|+2bax+b−b22(ax+b)2)+C
* ∫x2(ax+b)ndx=1a3(−(ax+b)3−n(n−3)+2b(ax+b)2−n(n−2)−b2(ax+b)1−n(n−1))+C(for n∉{1,2,3})
* ∫1x(ax+b)dx=−1bln⁡|ax+bx|+C
* ∫1x2(ax+b)dx=−1bx+ab2ln⁡|ax+bx|+C
* ∫1x2(ax+b)2dx=−a(1b2(ax+b)+1ab2x−2b3ln⁡|ax+bx|)+C

## Integrands of the form xm / (a x2 \+ b x \+ c)n

For a≠0:

* ∫1ax2+bx+cdx={24ac−b2arctan⁡2ax+b4ac−b2+C(for 4ac−b2>0)1b2−4acln⁡|2ax+b−b2−4ac2ax+b+b2−4ac|+C={−2b2−4acartanh⁡2ax+bb2−4ac+C(for |2ax+b|<b2−4ac)−2b2−4acarcoth⁡2ax+bb2−4ac+C(else)(for 4ac−b2<0)−22ax+b+C(for 4ac−b2=0)
* ∫xax2+bx+cdx=12aln⁡|ax2+bx+c|−b2a∫dxax2+bx+c+C
* ∫mx+nax2+bx+cdx={m2aln⁡|ax2+bx+c|+2an−bma4ac−b2arctan⁡2ax+b4ac−b2+C(for 4ac−b2>0)m2aln⁡|ax2+bx+c|+2an−bm2ab2−4acln⁡|2ax+b−b2−4ac2ax+b+b2−4ac|+C={m2aln⁡|ax2+bx+c|−2an−bmab2−4acartanh⁡2ax+bb2−4ac+C(for |2ax+b|<b2−4ac)m2aln⁡|ax2+bx+c|−2an−bmab2−4acarcoth⁡2ax+bb2−4ac+C(else)(for 4ac−b2<0)m2aln⁡|ax2+bx+c|−2an−bma(2ax+b)+C=maln⁡|x+b2a|−2an−bma(2ax+b)+C(for 4ac−b2=0)
* ∫1(ax2+bx+c)ndx=2ax+b(n−1)(4ac−b2)(ax2+bx+c)n−1+(2n−3)2a(n−1)(4ac−b2)∫1(ax2+bx+c)n−1dx+C
* ∫x(ax2+bx+c)ndx=−bx+2c(n−1)(4ac−b2)(ax2+bx+c)n−1−b(2n−3)(n−1)(4ac−b2)∫1(ax2+bx+c)n−1dx+C
* ∫1x(ax2+bx+c)dx=12cln⁡|x2ax2+bx+c|−b2c∫1ax2+bx+cdx+C

## Integrands of the form xm (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.

* ∫xm(a+bxn)pdx=xm+1(a+bxn)pm+np+1+anpm+np+1∫xm(a+bxn)p−1dx
* ∫xm(a+bxn)pdx=−xm+1(a+bxn)p+1an(p+1)+m+n(p+1)+1an(p+1)∫xm(a+bxn)p+1dx
* ∫xm(a+bxn)pdx=xm+1(a+bxn)pm+1−bnpm+1∫xm+n(a+bxn)p−1dx
* ∫xm(a+bxn)pdx=xm−n+1(a+bxn)p+1bn(p+1)−m−n+1bn(p+1)∫xm−n(a+bxn)p+1dx
* ∫xm(a+bxn)pdx=xm−n+1(a+bxn)p+1b(m+np+1)−a(m−n+1)b(m+np+1)∫xm−n(a+bxn)pdx
* ∫xm(a+bxn)pdx=xm+1(a+bxn)p+1a(m+1)−b(m+n(p+1)+1)a(m+1)∫xm+n(a+bxn)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.

* ∫(A+Bx)(a+bx)m(c+dx)n(e+fx)pdx=−(Ab−aB)(a+bx)m+1(c+dx)n(e+fx)p+1b(m+1)(af−be)+1b(m+1)(af−be)⋅∫(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)pdx
* ∫(A+Bx)(a+bx)m(c+dx)n(e+fx)pdx=B(a+bx)m(c+dx)n+1(e+fx)p+1df(m+n+p+2)+1df(m+n+p+2)⋅∫(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)pdx
* ∫(A+Bx)(a+bx)m(c+dx)n(e+fx)pdx=(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)⋅∫((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)pdx

## Integrands of the form xm (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 (a+bxn)p(c+dxn)q and xm(a+bxn)p(c+dxn)q by setting m and/or B to 0.

* ∫xm(A+Bxn)(a+bxn)p(c+dxn)qdx=−(Ab−aB)xm+1(a+bxn)p+1(c+dxn)qabn(p+1)+1abn(p+1)⋅∫xm(c(Abn(p+1)+(Ab−aB)(m+1))+d(Abn(p+1)+(Ab−aB)(m+nq+1))xn)(a+bxn)p+1(c+dxn)q−1dx
* ∫xm(A+Bxn)(a+bxn)p(c+dxn)qdx=Bxm+1(a+bxn)p+1(c+dxn)qb(m+n(p+q+1)+1)+1b(m+n(p+q+1)+1)⋅∫xm(c((Ab−aB)(1+m)+Abn(1+p+q))+(d(Ab−aB)(1+m)+Bnq(bc−ad)+Abdn(1+p+q))xn)(a+bxn)p(c+dxn)q−1dx
* ∫xm(A+Bxn)(a+bxn)p(c+dxn)qdx=−(Ab−aB)xm+1(a+bxn)p+1(c+dxn)q+1an(bc−ad)(p+1)+1an(bc−ad)(p+1)⋅∫xm(c(Ab−aB)(m+1)+An(bc−ad)(p+1)+d(Ab−aB)(m+n(p+q+2)+1)xn)(a+bxn)p+1(c+dxn)qdx
* ∫xm(A+Bxn)(a+bxn)p(c+dxn)qdx=Bxm−n+1(a+bxn)p+1(c+dxn)q+1bd(m+n(p+q+1)+1)−1bd(m+n(p+q+1)+1)⋅∫xm−n(aBc(m−n+1)+(aBd(m+nq+1)−b(−Bc(m+np+1)+Ad(m+n(p+q+1)+1)))xn)(a+bxn)p(c+dxn)qdx
* ∫xm(A+Bxn)(a+bxn)p(c+dxn)qdx=Axm+1(a+bxn)p+1(c+dxn)q+1ac(m+1)+1ac(m+1)⋅∫xm+n(aBc(m+1)−A(bc+ad)(m+n+1)−An(bcp+adq)−Abd(m+n(p+q+2)+1)xn)(a+bxn)p(c+dxn)qdx
* ∫xm(A+Bxn)(a+bxn)p(c+dxn)qdx=Axm+1(a+bxn)p+1(c+dxn)qa(m+1)−1a(m+1)⋅∫xm+n(c(Ab−aB)(m+1)+An(bc(p+1)+adq)+d((Ab−aB)(m+1)+Abn(p+q+1))xn)(a+bxn)p(c+dxn)q−1dx
* ∫xm(A+Bxn)(a+bxn)p(c+dxn)qdx=(Ab−aB)xm−n+1(a+bxn)p+1(c+dxn)q+1bn(bc−ad)(p+1)−1bn(bc−ad)(p+1)⋅∫xm−n(c(Ab−aB)(m−n+1)+(d(Ab−aB)(m+nq+1)−bn(Bc−Ad)(p+1))xn)(a+bxn)p+1(c+dxn)qdx

## 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 (a+bx+cx2)p when b2−4ac=0 by setting m to 0.

* ∫(d+ex)m(a+bx+cx2)pdx=(d+ex)m+1(a+bx+cx2)pe(m+1)−p(d+ex)m+2(b+2cx)(a+bx+cx2)p−1e2(m+1)(m+2p+1)+p(2p−1)(2cd−be)e2(m+1)(m+2p+1)∫(d+ex)m+1(a+bx+cx2)p−1dx
* ∫(d+ex)m(a+bx+cx2)pdx=(d+ex)m+1(a+bx+cx2)pe(m+1)−p(d+ex)m+2(b+2cx)(a+bx+cx2)p−1e2(m+1)(m+2)+2cp(2p−1)e2(m+1)(m+2)∫(d+ex)m+2(a+bx+cx2)p−1dx
* ∫(d+ex)m(a+bx+cx2)pdx=−e(m+2p+2)(d+ex)m(a+bx+cx2)p+1(p+1)(2p+1)(2cd−be)+(d+ex)m+1(b+2cx)(a+bx+cx2)p(2p+1)(2cd−be)+e2m(m+2p+2)(p+1)(2p+1)(2cd−be)∫(d+ex)m−1(a+bx+cx2)p+1dx
* ∫(d+ex)m(a+bx+cx2)pdx=−em(d+ex)m−1(a+bx+cx2)p+12c(p+1)(2p+1)+(d+ex)m(b+2cx)(a+bx+cx2)p2c(2p+1)+e2m(m−1)2c(p+1)(2p+1)∫(d+ex)m−2(a+bx+cx2)p+1dx
* ∫(d+ex)m(a+bx+cx2)pdx=(d+ex)m+1(a+bx+cx2)pe(m+2p+1)−p(2cd−be)(d+ex)m+1(b+2cx)(a+bx+cx2)p−12ce2(m+2p)(m+2p+1)+p(2p−1)(2cd−be)22ce2(m+2p)(m+2p+1)∫(d+ex)m(a+bx+cx2)p−1dx
* ∫(d+ex)m(a+bx+cx2)pdx=−2ce(m+2p+2)(d+ex)m+1(a+bx+cx2)p+1(p+1)(2p+1)(2cd−be)2+(d+ex)m+1(b+2cx)(a+bx+cx2)p(2p+1)(2cd−be)+2ce2(m+2p+2)(m+2p+3)(p+1)(2p+1)(2cd−be)2∫(d+ex)m(a+bx+cx2)p+1dx
* ∫(d+ex)m(a+bx+cx2)pdx=(d+ex)m(b+2cx)(a+bx+cx2)p2c(m+2p+1)+m(2cd−be)2c(m+2p+1)∫(d+ex)m−1(a+bx+cx2)pdx
* ∫(d+ex)m(a+bx+cx2)pdx=−(d+ex)m+1(b+2cx)(a+bx+cx2)p(m+1)(2cd−be)+2c(m+2p+2)(m+1)(2cd−be)∫(d+ex)m+1(a+bx+cx2)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 (a+bx+cx2)p and (d+ex)m(a+bx+cx2)p by setting m and/or B to 0.

* ∫(d+ex)m(A+Bx)(a+bx+cx2)pdx=(d+ex)m+1(Ae(m+2p+2)−Bd(2p+1)+eB(m+1)x)(a+bx+cx2)pe2(m+1)(m+2p+2)+1e2(m+1)(m+2p+2)p⋅∫(d+ex)m+1(B(bd+2ae+2aem+2bdp)−Abe(m+2p+2)+(B(2cd+be+bem+4cdp)−2Ace(m+2p+2))x)(a+bx+cx2)p−1dx
* ∫(d+ex)m(A+Bx)(a+bx+cx2)pdx=(d+ex)m(Ab−2aB−(bB−2Ac)x)(a+bx+cx2)p+1(p+1)(b2−4ac)+1(p+1)(b2−4ac)⋅∫(d+ex)m−1(B(2aem+bd(2p+3))−A(bem+2cd(2p+3))+e(bB−2Ac)(m+2p+3)x)(a+bx+cx2)p+1dx
* ∫(d+ex)m(A+Bx)(a+bx+cx2)pdx=(d+ex)m+1(Ace(m+2p+2)−B(cd+2cdp−bep)+Bce(m+2p+1)x)(a+bx+cx2)pce2(m+2p+1)(m+2p+2)−pce2(m+2p+1)(m+2p+2)⋅∫(d+ex)m(Ace(bd−2ae)(m+2p+2)+B(ae(be−2cdm+bem)+bd(bep−cd−2cdp))+(Ace(2cd−be)(m+2p+2)−B(−b2e2(m+p+1)+2c2d2(1+2p)+ce(bd(m−2p)+2ae(m+2p+1))))x)(a+bx+cx2)p−1dx
* ∫(d+ex)m(A+Bx)(a+bx+cx2)pdx=(d+ex)m+1(A(bcd−b2e+2ace)−aB(2cd−be)+c(A(2cd−be)−B(bd−2ae))x)(a+bx+cx2)p+1(p+1)(b2−4ac)(cd2−bde+ae2)+1(p+1)(b2−4ac)(cd2−bde+ae2)⋅∫(d+ex)m(A(bcde(2p−m+2)+b2e2(m+p+2)−2c2d2(3+2p)−2ace2(m+2p+3))−B(ae(be−2cdm+bem)+bd(−3cd+be−2cdp+bep))+ce(B(bd−2ae)−A(2cd−be))(m+2p+4)x)(a+bx+cx2)p+1dx
* ∫(d+ex)m(A+Bx)(a+bx+cx2)pdx=B(d+ex)m(a+bx+cx2)p+1c(m+2p+2)+1c(m+2p+2)⋅∫(d+ex)m−1(m(Acd−aBe)−d(bB−2Ac)(p+1)+((Bcd−bBe+Ace)m−e(bB−2Ac)(p+1))x)(a+bx+cx2)pdx
* ∫(d+ex)m(A+Bx)(a+bx+cx2)pdx=−(Bd−Ae)(d+ex)m+1(a+bx+cx2)p+1(m+1)(cd2−bde+ae2)+1(m+1)(cd2−bde+ae2)⋅∫(d+ex)m+1((Acd−Abe+aBe)(m+1)+b(Bd−Ae)(p+1)+c(Bd−Ae)(m+2p+3)x)(a+bx+cx2)pdx

## Integrands of the form xm (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 (a+bxn+cx2n)p when b2−4ac=0 by setting m to 0.

* ∫xm(a+bxn+cx2n)pdx=xm+1(a+bxn+cx2n)pm+2np+1+npxm+1(2a+bxn)(a+bxn+cx2n)p−1(m+1)(m+2np+1)−bn2p(2p−1)(m+1)(m+2np+1)∫xm+n(a+bxn+cx2n)p−1dx
* ∫xm(a+bxn+cx2n)pdx=(m+n(2p−1)+1)xm+1(a+bxn+cx2n)p(m+1)(m+n+1)+npxm+1(2a+bxn)(a+bxn+cx2n)p−1(m+1)(m+n+1)+2cpn2(2p−1)(m+1)(m+n+1)∫xm+2n(a+bxn+cx2n)p−1dx
* ∫xm(a+bxn+cx2n)pdx=(m+n(2p+1)+1)xm−n+1(a+bxn+cx2n)p+1bn2(p+1)(2p+1)−xm+1(b+2cxn)(a+bxn+cx2n)pbn(2p+1)−(m−n+1)(m+n(2p+1)+1)bn2(p+1)(2p+1)∫xm−n(a+bxn+cx2n)p+1dx
* ∫xm(a+bxn+cx2n)pdx=−(m−3n−2np+1)xm−2n+1(a+bxn+cx2n)p+12cn2(p+1)(2p+1)−xm−2n+1(2a+bxn)(a+bxn+cx2n)p2cn(2p+1)+(m−n+1)(m−2n+1)2cn2(p+1)(2p+1)∫xm−2n(a+bxn+cx2n)p+1dx
* ∫xm(a+bxn+cx2n)pdx=xm+1(a+bxn+cx2n)pm+2np+1+npxm+1(2a+bxn)(a+bxn+cx2n)p−1(m+2np+1)(m+n(2p−1)+1)+2an2p(2p−1)(m+2np+1)(m+n(2p−1)+1)∫xm(a+bxn+cx2n)p−1dx
* ∫xm(a+bxn+cx2n)pdx=−(m+n+2np+1)xm+1(a+bxn+cx2n)p+12an2(p+1)(2p+1)−xm+1(2a+bxn)(a+bxn+cx2n)p2an(2p+1)+(m+n(2p+1)+1)(m+2n(p+1)+1)2an2(p+1)(2p+1)∫xm(a+bxn+cx2n)p+1dx
* ∫xm(a+bxn+cx2n)pdx=xm−n+1(b+2cxn)(a+bxn+cx2n)p2c(m+2np+1)−b(m−n+1)2c(m+2np+1)∫xm−n(a+bxn+cx2n)pdx
* ∫xm(a+bxn+cx2n)pdx=xm+1(b+2cxn)(a+bxn+cx2n)pb(m+1)−2c(m+n(2p+1)+1)b(m+1)∫xm+n(a+bxn+cx2n)pdx

## Integrands of the form xm (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 (a+bxn+cx2n)p and xm(a+bxn+cx2n)p by setting m and/or B to 0.

* ∫xm(A+Bxn)(a+bxn+cx2n)pdx=xm+1(A(m+n(2p+1)+1)+B(m+1)xn)(a+bxn+cx2n)p(m+1)(m+n(2p+1)+1)+np(m+1)(m+n(2p+1)+1)⋅∫xm+n(2aB(m+1)−Ab(m+n(2p+1)+1)+(bB(m+1)−2Ac(m+n(2p+1)+1))xn)(a+bxn+cx2n)p−1dx
* ∫xm(A+Bxn)(a+bxn+cx2n)pdx=xm−n+1(Ab−2aB−(bB−2Ac)xn)(a+bxn+cx2n)p+1n(p+1)(b2−4ac)+1n(p+1)(b2−4ac)⋅∫xm−n((m−n+1)(2aB−Ab)+(m+2n(p+1)+1)(bB−2Ac)xn)(a+bxn+cx2n)p+1dx
* ∫xm(A+Bxn)(a+bxn+cx2n)pdx=xm+1(bBnp+Ac(m+n(2p+1)+1)+Bc(m+2np+1)xn)(a+bxn+cx2n)pc(m+2np+1)(m+n(2p+1)+1)+npc(m+2np+1)(m+n(2p+1)+1)⋅∫xm(2aAc(m+n(2p+1)+1)−abB(m+1)+(2aBc(m+2np+1)+Abc(m+n(2p+1)+1)−b2B(m+np+1))xn)(a+bxn+cx2n)p−1dx
* ∫xm(A+Bxn)(a+bxn+cx2n)pdx=−xm+1(Ab2−abB−2aAc+(Ab−2aB)cxn)(a+bxn+cx2n)p+1an(p+1)(b2−4ac)+1an(p+1)(b2−4ac)⋅∫xm((m+n(p+1)+1)Ab2−abB(m+1)−2(m+2n(p+1)+1)aAc+(m+n(2p+3)+1)(Ab−2aB)cxn)(a+bxn+cx2n)p+1dx
* ∫xm(A+Bxn)(a+bxn+cx2n)pdx=Bxm−n+1(a+bxn+cx2n)p+1c(m+n(2p+1)+1)−1c(m+n(2p+1)+1)⋅∫xm−n(aB(m−n+1)+(bB(m+np+1)−Ac(m+n(2p+1)+1))xn)(a+bxn+cx2n)pdx
* ∫xm(A+Bxn)(a+bxn+cx2n)pdx=Axm+1(a+bxn+cx2n)p+1a(m+1)+1a(m+1)⋅∫xm+n(aB(m+1)−Ab(m+n(p+1)+1)−Ac(m+2n(p+1)+1)xn)(a+bxn+cx2n)pdx

## References

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

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Lists of integrals

* Rational functions
* Irrational functions
* Trigonometric functions
* Inverse trigonometric functions
* Hyperbolic functions
* Inverse hyperbolic functions
* Exponential functions
* Logarithmic functions
* Gaussian functions

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