\[ \int (d+e x)^3 (f+g x)^n \left (a+2 c d x+c e x^2\right ) \, dx \]
Optimal antiderivative \[ -\frac {\left (-d g +e f \right )^{3} \left (a \,g^{2}+c f \left (-2 d g +e f \right )\right ) \left (g x +f \right )^{1+n}}{g^{6} \left (1+n \right )}+\frac {\left (-d g +e f \right )^{2} \left (3 a e \,g^{2}+c \left (2 d^{2} g^{2}-10 d e f g +5 e^{2} f^{2}\right )\right ) \left (g x +f \right )^{2+n}}{g^{6} \left (2+n \right )}-\frac {e \left (-d g +e f \right ) \left (3 a e \,g^{2}+c \left (7 d^{2} g^{2}-20 d e f g +10 e^{2} f^{2}\right )\right ) \left (g x +f \right )^{3+n}}{g^{6} \left (3+n \right )}+\frac {e^{2} \left (a e \,g^{2}+c \left (9 d^{2} g^{2}-20 d e f g +10 e^{2} f^{2}\right )\right ) \left (g x +f \right )^{4+n}}{g^{6} \left (4+n \right )}-\frac {5 c \,e^{3} \left (-d g +e f \right ) \left (g x +f \right )^{5+n}}{g^{6} \left (5+n \right )}+\frac {c \,e^{4} \left (g x +f \right )^{6+n}}{g^{6} \left (6+n \right )} \]
command
integrate((e*x+d)**3*(g*x+f)**n*(c*e*x**2+2*c*d*x+a),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 \[ \text {Timed out} \]_____________________________________________________