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