\[ \int (e x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \left (c+d x^2\right )^3 \, dx \]
Optimal antiderivative \[ \frac {a^{3} A \,c^{3} \left (e x \right )^{1+m}}{e \left (1+m \right )}+\frac {a^{2} c^{2} \left (a B c +3 A \left (a d +b c \right )\right ) \left (e x \right )^{3+m}}{e^{3} \left (3+m \right )}+\frac {3 a c \left (a B c \left (a d +b c \right )+A \left (a^{2} d^{2}+3 a b c d +b^{2} c^{2}\right )\right ) \left (e x \right )^{5+m}}{e^{5} \left (5+m \right )}+\frac {\left (3 a B c \left (a^{2} d^{2}+3 a b c d +b^{2} c^{2}\right )+A \left (a^{3} d^{3}+9 a^{2} b c \,d^{2}+9 a \,b^{2} c^{2} d +b^{3} c^{3}\right )\right ) \left (e x \right )^{7+m}}{e^{7} \left (7+m \right )}+\frac {\left (a^{3} B \,d^{3}+9 a \,b^{2} c d \left (A d +B c \right )+3 a^{2} b \,d^{2} \left (A d +3 B c \right )+b^{3} c^{2} \left (3 A d +B c \right )\right ) \left (e x \right )^{9+m}}{e^{9} \left (9+m \right )}+\frac {3 b d \left (a^{2} B \,d^{2}+b^{2} c \left (A d +B c \right )+a b d \left (A d +3 B c \right )\right ) \left (e x \right )^{11+m}}{e^{11} \left (11+m \right )}+\frac {b^{2} d^{2} \left (A b d +3 a B d +3 b B c \right ) \left (e x \right )^{13+m}}{e^{13} \left (13+m \right )}+\frac {b^{3} B \,d^{3} \left (e x \right )^{15+m}}{e^{15} \left (15+m \right )} \]
command
integrate((e*x)**m*(b*x**2+a)**3*(B*x**2+A)*(d*x**2+c)**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 \[ \text {Timed out} \]_____________________________________________________