\[ \int (A+B x) (d+e x)^m \left (b x+c x^2\right )^3 \, dx \]
Optimal antiderivative \[ -\frac {d^{3} \left (-A e +B d \right ) \left (-b e +c d \right )^{3} \left (e x +d \right )^{1+m}}{e^{8} \left (1+m \right )}+\frac {d^{2} \left (-b e +c d \right )^{2} \left (B d \left (-4 b e +7 c d \right )-3 A e \left (-b e +2 c d \right )\right ) \left (e x +d \right )^{2+m}}{e^{8} \left (2+m \right )}+\frac {3 d \left (-b e +c d \right ) \left (A e \left (b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right )-B d \left (2 b^{2} e^{2}-8 b c d e +7 c^{2} d^{2}\right )\right ) \left (e x +d \right )^{3+m}}{e^{8} \left (3+m \right )}+\frac {\left (B d \left (-4 b^{3} e^{3}+30 b^{2} c d \,e^{2}-60 b \,c^{2} d^{2} e +35 c^{3} d^{3}\right )-A e \left (-b^{3} e^{3}+12 b^{2} c d \,e^{2}-30 b \,c^{2} d^{2} e +20 c^{3} d^{3}\right )\right ) \left (e x +d \right )^{4+m}}{e^{8} \left (4+m \right )}+\frac {\left (3 A c e \left (b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right )-B \left (-b^{3} e^{3}+15 b^{2} c d \,e^{2}-45 b \,c^{2} d^{2} e +35 c^{3} d^{3}\right )\right ) \left (e x +d \right )^{5+m}}{e^{8} \left (5+m \right )}-\frac {3 c \left (A c e \left (-b e +2 c d \right )-B \left (b^{2} e^{2}-6 b c d e +7 c^{2} d^{2}\right )\right ) \left (e x +d \right )^{6+m}}{e^{8} \left (6+m \right )}-\frac {c^{2} \left (-A c e -3 b B e +7 B c d \right ) \left (e x +d \right )^{7+m}}{e^{8} \left (7+m \right )}+\frac {B \,c^{3} \left (e x +d \right )^{8+m}}{e^{8} \left (8+m \right )} \]
command
integrate((B*x+A)*(e*x+d)**m*(c*x**2+b*x)**3,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {output too large to display} \]
Sympy 1.8 under Python 3.8.8 output \[ \text {Timed out} \]_____________________________________________________