\[ \int (A+B x) (d+e x)^m \left (a+b x+c x^2\right )^3 \, dx \]
Optimal antiderivative \[ -\frac {\left (-A e +B d \right ) \left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \left (e x +d \right )^{1+m}}{e^{8} \left (1+m \right )}-\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (3 A e \left (-b e +2 c d \right )-B \left (7 c \,d^{2}-e \left (-a e +4 b d \right )\right )\right ) \left (e x +d \right )^{2+m}}{e^{8} \left (2+m \right )}-\frac {3 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (B \left (7 c^{2} d^{3}-c d e \left (-3 a e +8 b d \right )+b \,e^{2} \left (-a e +2 b d \right )\right )-A e \left (5 c^{2} d^{2}+b^{2} e^{2}-c e \left (-a e +5 b d \right )\right )\right ) \left (e x +d \right )^{3+m}}{e^{8} \left (3+m \right )}-\frac {\left (A e \left (-b e +2 c d \right ) \left (10 c^{2} d^{2}+b^{2} e^{2}-2 c e \left (-3 a e +5 b d \right )\right )-B \left (35 c^{3} d^{4}-b^{2} e^{3} \left (-3 a e +4 b d \right )-30 c^{2} d^{2} e \left (-a e +2 b d \right )+3 c \,e^{2} \left (a^{2} e^{2}-8 a b d e +10 b^{2} d^{2}\right )\right )\right ) \left (e x +d \right )^{4+m}}{e^{8} \left (4+m \right )}-\frac {\left (B \left (35 c^{3} d^{3}-b^{3} e^{3}+3 b c \,e^{2} \left (-2 a e +5 b d \right )-15 c^{2} d e \left (-a e +3 b d \right )\right )-3 A c e \left (5 c^{2} d^{2}+b^{2} e^{2}-c e \left (-a e +5 b d \right )\right )\right ) \left (e x +d \right )^{5+m}}{e^{8} \left (5+m \right )}-\frac {3 c \left (A c e \left (-b e +2 c d \right )-B \left (7 c^{2} d^{2}+b^{2} e^{2}-c e \left (-a e +6 b d \right )\right )\right ) \left (e x +d \right )^{6+m}}{e^{8} \left (6+m \right )}-\frac {c^{2} \left (-A c e -3 b B e +7 B c d \right ) \left (e x +d \right )^{7+m}}{e^{8} \left (7+m \right )}+\frac {B \,c^{3} \left (e x +d \right )^{8+m}}{e^{8} \left (8+m \right )} \]
command
integrate((B*x+A)*(e*x+d)**m*(c*x**2+b*x+a)**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 \[ \text {Timed out} \]_____________________________________________________