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