List of Integrals of Rational Functions - Integrands of The Form xm (A + B Xn) (a + b Xn

Integrands of The Form xm (A + B Xn) (a + b Xn

  • 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 and by setting m and/or B to 0.
\int x^m \left(A+B\,x^n\right) \left(a+b\,x^n+c\,x^{2 n}\right)^pdx= \frac{x^{m+1} \left(A (m+n (2 p+1)+1)+B (m+1) x^n\right) \left(a+b\,x^n+c\,x^{2 n}\right)^p}{(m+1) (m+n (2 p+1)+1)}\,+\, \frac{n\,p}{(m+1) (m+n (2 p+1)+1)}\,\cdot

\int x^{m+n} \left(2 a\,B (m+1)-A\,b (m+n (2 p+1)+1)+(b\,B (m+1)-2\,A\,c (m+n (2 p+1)+1)) x^n\right)\left(a+b\,x^n+c\,x^{2 n}\right)^{p-1}dx
\int x^m \left(A+B\,x^n\right) \left(a+b\,x^n+c\,x^{2 n}\right)^pdx= \frac{x^{m-n+1} \left(A\,b-2 a\,B-(b\,B-2 A\,c) x^n\right)\left(a+b\,x^n+c\,x^{2 n}\right)^{p+1}}{n(p+1) \left(b^2-4 a\,c\right)}\,+\, \frac{1}{n(p+1) \left(b^2-4 a\,c\right)}\,\cdot

\int x^{m-n}\left((m-n+1)(2 a\,B-A\,b)+(m+2n (p+1)+1) (b\,B-2 A\,c) x^n\right)\left(a+b\,x^n+c\,x^{2 n}\right)^{p+1}dx
\int x^m \left(A+B\,x^n\right) \left(a+b\,x^n+c\,x^{2 n}\right)^pdx= \frac{x^{m+1} \left(b\,B\,n\,p+A\,c (m+n (2 p+1)+1)+B\,c (m+2 n\,p+1) x^n\right)\left(a+b\,x^n+c\,x^{2 n}\right)^p}{c (m+2 n\,p+1) (m+n (2 p+1)+1)}\,+\, \frac{n\,p}{c (m+2 n\,p+1) (m+n (2 p+1)+1)}\,\cdot

\int x^m \left(2 a\,A\,c (m+n (2 p+1)+1)-a\,b\,B (m+1)+\left(2 a\,B\,c (m+2 n\,p+1)+A\,b\,c (m+n (2 p+1)+1)-b^2 B (m+n\,p+1)\right) x^n\right)\left(a+b\,x^n+c\,x^{2 n}\right)^{p-1}dx
\int x^m \left(A+B\,x^n\right) \left(a+b\,x^n+c\,x^{2 n}\right)^pdx= -\frac{x^{m+1} \left(A\,b^2-a\,b\,B-2 a\,A\,c+(A\,b-2 a\,B) c\,x^n\right)\left(a+b\,x^n+c\,x^{2 n}\right)^{p+1}}{a\,n(p+1) \left(b^2-4 a\,c\right)}\,+\, \frac{1}{a\,n(p+1) \left(b^2-4 a\,c\right)}\,\cdot

\int x^m \left((m+n (p+1)+1) A\,b^2-a\,b\,B(m+1)-2(m+2n (p+1)+1)a\,A\,c+(m+n (2p+3)+1)(A\,b-2 a\,B) c\,x^n\right)\left(a+b\,x^n+c\,x^{2 n}\right)^{p+1}dx
\int x^m \left(A+B\,x^n\right) \left(a+b\,x^n+c\,x^{2 n}\right)^pdx= \frac{B\,x^{m-n+1}\left(a+b\,x^n+c\,x^{2 n}\right)^{p+1}}{c (m+n (2 p+1)+1)}\,-\, \frac{1}{c (m+n (2 p+1)+1)}\,\cdot

\int x^{m-n} \left(a\,B (m-n+1)+(b\,B (m+n\,p+1)-A\,c (m+n (2 p+1)+1)) x^n\right) \left(a+b\,x^n+c\,x^{2 n}\right)^pdx
\int x^m \left(A+B\,x^n\right) \left(a+b\,x^n+c\,x^{2 n}\right)^pdx= \frac{A\,x^{m+1} \left(a+b\,x^n+c\,x^{2 n}\right)^{p+1}}{a(m+1)}\,+\, \frac{1}{a(m+1)}\,\cdot

\int x^{m+n} \left(a\,B (m+1)-A\,b (m+n (p+1)+1)-A\,c (m+2 n(p+1)+1) x^n\right)\left(a+b\,x^n+c\,x^{2 n}\right)^pdx

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