∫ x m ( A + B x n ) ( a + b x n + c x 2 n ) p d x = x m + 1 ( A ( m + n ( 2 p + 1 ) + 1 ) + B ( m + 1 ) x n ) ( a + b x n + c x 2 n ) p ( m + 1 ) ( m + n ( 2 p + 1 ) + 1 ) + n p ( m + 1 ) ( m + n ( 2 p + 1 ) + 1 ) ⋅ {\displaystyle \int x^{m}\left(A+B\,x^{n}\right)\left(a+b\,x^{n}+c\,x^{2n}\right)^{p}dx={\frac {x^{m+1}\left(A(m+n(2p+1)+1)+B(m+1)x^{n}\right)\left(a+b\,x^{n}+c\,x^{2n}\right)^{p}}{(m+1)(m+n(2p+1)+1)}}\,+\,{\frac {n\,p}{(m+1)(m+n(2p+1)+1)}}\,\cdot } ∫ x m + n ( 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 ) ( a + b x n + c x 2 n ) p − 1 d x {\displaystyle \int x^{m+n}\left(2a\,B(m+1)-A\,b(m+n(2p+1)+1)+(b\,B(m+1)-2\,A\,c(m+n(2p+1)+1))x^{n}\right)\left(a+b\,x^{n}+c\,x^{2n}\right)^{p-1}dx} ∫ x m ( A + B x n ) ( a + b x n + c x 2 n ) p d x = x m − n + 1 ( A b − 2 a B − ( b B − 2 A c ) x n ) ( a + b x n + c x 2 n ) p + 1 n ( p + 1 ) ( b 2 − 4 a c ) + 1 n ( p + 1 ) ( b 2 − 4 a c ) ⋅ {\displaystyle \int x^{m}\left(A+B\,x^{n}\right)\left(a+b\,x^{n}+c\,x^{2n}\right)^{p}dx={\frac {x^{m-n+1}\left(A\,b-2a\,B-(b\,B-2A\,c)x^{n}\right)\left(a+b\,x^{n}+c\,x^{2n}\right)^{p+1}}{n(p+1)\left(b^{2}-4a\,c\right)}}\,+\,{\frac {1}{n(p+1)\left(b^{2}-4a\,c\right)}}\,\cdot } ∫ x m − n ( ( m − n + 1 ) ( 2 a B − A b ) + ( m + 2 n ( p + 1 ) + 1 ) ( b B − 2 A c ) x n ) ( a + b x n + c x 2 n ) p + 1 d x {\displaystyle \int x^{m-n}\left((m-n+1)(2a\,B-A\,b)+(m+2n(p+1)+1)(b\,B-2A\,c)x^{n}\right)\left(a+b\,x^{n}+c\,x^{2n}\right)^{p+1}dx} ∫ x m ( A + B x n ) ( a + b x n + c x 2 n ) p d x = x m + 1 ( b B n p + A c ( m + n ( 2 p + 1 ) + 1 ) + B c ( m + 2 n p + 1 ) x n ) ( a + b x n + c x 2 n ) p c ( m + 2 n p + 1 ) ( m + n ( 2 p + 1 ) + 1 ) + n p c ( m + 2 n p + 1 ) ( m + n ( 2 p + 1 ) + 1 ) ⋅ {\displaystyle \int x^{m}\left(A+B\,x^{n}\right)\left(a+b\,x^{n}+c\,x^{2n}\right)^{p}dx={\frac {x^{m+1}\left(b\,B\,n\,p+A\,c(m+n(2p+1)+1)+B\,c(m+2n\,p+1)x^{n}\right)\left(a+b\,x^{n}+c\,x^{2n}\right)^{p}}{c(m+2n\,p+1)(m+n(2p+1)+1)}}\,+\,{\frac {n\,p}{c(m+2n\,p+1)(m+n(2p+1)+1)}}\,\cdot } ∫ x m ( 2 a A c ( m + n ( 2 p + 1 ) + 1 ) − a b B ( m + 1 ) + ( 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 ) ) x n ) ( a + b x n + c x 2 n ) p − 1 d x {\displaystyle \int x^{m}\left(2a\,A\,c(m+n(2p+1)+1)-a\,b\,B(m+1)+\left(2a\,B\,c(m+2n\,p+1)+A\,b\,c(m+n(2p+1)+1)-b^{2}B(m+n\,p+1)\right)x^{n}\right)\left(a+b\,x^{n}+c\,x^{2n}\right)^{p-1}dx} ∫ x m ( A + B x n ) ( a + b x n + c x 2 n ) p d x = − x m + 1 ( A b 2 − a b B − 2 a A c + ( A b − 2 a B ) c x n ) ( a + b x n + c x 2 n ) p + 1 a n ( p + 1 ) ( b 2 − 4 a c ) + 1 a n ( p + 1 ) ( b 2 − 4 a c ) ⋅ {\displaystyle \int x^{m}\left(A+B\,x^{n}\right)\left(a+b\,x^{n}+c\,x^{2n}\right)^{p}dx=-{\frac {x^{m+1}\left(A\,b^{2}-a\,b\,B-2a\,A\,c+(A\,b-2a\,B)c\,x^{n}\right)\left(a+b\,x^{n}+c\,x^{2n}\right)^{p+1}}{a\,n(p+1)\left(b^{2}-4a\,c\right)}}\,+\,{\frac {1}{a\,n(p+1)\left(b^{2}-4a\,c\right)}}\,\cdot } ∫ x m ( ( m + n ( p + 1 ) + 1 ) A b 2 − a b B ( m + 1 ) − 2 ( m + 2 n ( p + 1 ) + 1 ) a A c + ( m + n ( 2 p + 3 ) + 1 ) ( A b − 2 a B ) c x n ) ( a + b x n + c x 2 n ) p + 1 d x {\displaystyle \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-2a\,B)c\,x^{n}\right)\left(a+b\,x^{n}+c\,x^{2n}\right)^{p+1}dx} ∫ x m ( A + B x n ) ( a + b x n + c x 2 n ) p d x = B x m − n + 1 ( a + b x n + c x 2 n ) p + 1 c ( m + n ( 2 p + 1 ) + 1 ) − 1 c ( m + n ( 2 p + 1 ) + 1 ) ⋅ {\displaystyle \int x^{m}\left(A+B\,x^{n}\right)\left(a+b\,x^{n}+c\,x^{2n}\right)^{p}dx={\frac {B\,x^{m-n+1}\left(a+b\,x^{n}+c\,x^{2n}\right)^{p+1}}{c(m+n(2p+1)+1)}}\,-\,{\frac {1}{c(m+n(2p+1)+1)}}\,\cdot } ∫ x m − n ( a B ( m − n + 1 ) + ( b B ( m + n p + 1 ) − A c ( m + n ( 2 p + 1 ) + 1 ) ) x n ) ( a + b x n + c x 2 n ) p d x {\displaystyle \int x^{m-n}\left(a\,B(m-n+1)+(b\,B(m+n\,p+1)-A\,c(m+n(2p+1)+1))x^{n}\right)\left(a+b\,x^{n}+c\,x^{2n}\right)^{p}dx} ∫ x m ( A + B x n ) ( a + b x n + c x 2 n ) p d x = A x m + 1 ( a + b x n + c x 2 n ) p + 1 a ( m + 1 ) + 1 a ( m + 1 ) ⋅ {\displaystyle \int x^{m}\left(A+B\,x^{n}\right)\left(a+b\,x^{n}+c\,x^{2n}\right)^{p}dx={\frac {A\,x^{m+1}\left(a+b\,x^{n}+c\,x^{2n}\right)^{p+1}}{a(m+1)}}\,+\,{\frac {1}{a(m+1)}}\,\cdot } ∫ x m + n ( a B ( m + 1 ) − A b ( m + n ( p + 1 ) + 1 ) − A c ( m + 2 n ( p + 1 ) + 1 ) x n ) ( a + b x n + c x 2 n ) p d x {\displaystyle \int x^{m+n}\left(a\,B(m+1)-A\,b(m+n(p+1)+1)-A\,c(m+2n(p+1)+1)x^{n}\right)\left(a+b\,x^{n}+c\,x^{2n}\right)^{p}dx}