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