\[ \int (b+2 c x) (d+e x)^m \left (a+b x+c x^2\right )^3 \, dx \]
Optimal antiderivative \[ -\frac {\left (-b e +2 c d \right ) \left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \left (e x +d \right )^{1+m}}{e^{8} \left (1+m \right )}+\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (14 c^{2} d^{2}+3 b^{2} e^{2}-2 c e \left (-a e +7 b d \right )\right ) \left (e x +d \right )^{2+m}}{e^{8} \left (2+m \right )}-\frac {3 \left (-b e +2 c d \right ) \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (7 c^{2} d^{2}+b^{2} e^{2}-c e \left (-3 a e +7 b d \right )\right ) \left (e x +d \right )^{3+m}}{e^{8} \left (3+m \right )}+\frac {\left (70 c^{4} d^{4}+b^{4} e^{4}-4 b^{2} c \,e^{3} \left (-3 a e +5 b d \right )-20 c^{3} d^{2} e \left (-3 a e +7 b d \right )+6 c^{2} e^{2} \left (a^{2} e^{2}-10 a b d e +15 b^{2} d^{2}\right )\right ) \left (e x +d \right )^{4+m}}{e^{8} \left (4+m \right )}-\frac {5 c \left (-b e +2 c d \right ) \left (7 c^{2} d^{2}+b^{2} e^{2}-c e \left (-3 a e +7 b d \right )\right ) \left (e x +d \right )^{5+m}}{e^{8} \left (5+m \right )}+\frac {3 c^{2} \left (14 c^{2} d^{2}+3 b^{2} e^{2}-2 c e \left (-a e +7 b d \right )\right ) \left (e x +d \right )^{6+m}}{e^{8} \left (6+m \right )}-\frac {7 c^{3} \left (-b e +2 c d \right ) \left (e x +d \right )^{7+m}}{e^{8} \left (7+m \right )}+\frac {2 c^{4} \left (e x +d \right )^{8+m}}{e^{8} \left (8+m \right )} \]
command
integrate((2*c*x+b)*(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 \[ \text {Timed out} \]_____________________________________________________