\[ \int (e x)^m \left (a+b x^n\right )^3 \left (A+B x^n\right ) \left (c+d x^n\right )^3 \, dx \]
Optimal antiderivative \[ \frac {a^{2} c^{2} \left (a B c +3 A \left (a d +b c \right )\right ) x^{1+n} \left (e x \right )^{m}}{1+m +n}+\frac {3 a c \left (a B c \left (a d +b c \right )+A \left (a^{2} d^{2}+3 a b c d +b^{2} c^{2}\right )\right ) x^{1+2 n} \left (e x \right )^{m}}{1+m +2 n}+\frac {\left (3 a B c \left (a^{2} d^{2}+3 a b c d +b^{2} c^{2}\right )+A \left (a^{3} d^{3}+9 a^{2} b c \,d^{2}+9 a \,b^{2} c^{2} d +b^{3} c^{3}\right )\right ) x^{1+3 n} \left (e x \right )^{m}}{1+m +3 n}+\frac {\left (a^{3} B \,d^{3}+9 a \,b^{2} c d \left (A d +B c \right )+3 a^{2} b \,d^{2} \left (A d +3 B c \right )+b^{3} c^{2} \left (3 A d +B c \right )\right ) x^{1+4 n} \left (e x \right )^{m}}{1+m +4 n}+\frac {3 b d \left (a^{2} B \,d^{2}+b^{2} c \left (A d +B c \right )+a b d \left (A d +3 B c \right )\right ) x^{1+5 n} \left (e x \right )^{m}}{1+m +5 n}+\frac {b^{2} d^{2} \left (A b d +3 a B d +3 b B c \right ) x^{1+6 n} \left (e x \right )^{m}}{1+m +6 n}+\frac {b^{3} B \,d^{3} x^{1+7 n} \left (e x \right )^{m}}{1+m +7 n}+\frac {a^{3} A \,c^{3} \left (e x \right )^{1+m}}{e \left (1+m \right )} \]
command
integrate((e*x)^m*(a+b*x^n)^3*(A+B*x^n)*(c+d*x^n)^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Timed out} \]_______________________________________________________