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