\[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)^3 \, dx \]
Optimal antiderivative \[ \frac {a^{2} \left (d x +c \right )^{4}}{4 d}-\frac {12 a b \,d^{3} \left (F^{f g x +e g}\right )^{n}}{f^{4} g^{4} n^{4} \ln \left (F \right )^{4}}-\frac {3 b^{2} d^{3} \left (F^{f g x +e g}\right )^{2 n}}{8 f^{4} g^{4} n^{4} \ln \left (F \right )^{4}}+\frac {12 a b \,d^{2} \left (F^{f g x +e g}\right )^{n} \left (d x +c \right )}{f^{3} g^{3} n^{3} \ln \left (F \right )^{3}}+\frac {3 b^{2} d^{2} \left (F^{f g x +e g}\right )^{2 n} \left (d x +c \right )}{4 f^{3} g^{3} n^{3} \ln \left (F \right )^{3}}-\frac {6 a b d \left (F^{f g x +e g}\right )^{n} \left (d x +c \right )^{2}}{f^{2} g^{2} n^{2} \ln \left (F \right )^{2}}-\frac {3 b^{2} d \left (F^{f g x +e g}\right )^{2 n} \left (d x +c \right )^{2}}{4 f^{2} g^{2} n^{2} \ln \left (F \right )^{2}}+\frac {2 a b \left (F^{f g x +e g}\right )^{n} \left (d x +c \right )^{3}}{f g n \ln \left (F \right )}+\frac {b^{2} \left (F^{f g x +e g}\right )^{2 n} \left (d x +c \right )^{3}}{2 f g n \ln \left (F \right )} \]
command
integrate((a+b*(F^(g*(f*x+e)))^n)^2*(d*x+c)^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 \[ \text {Timed out} \]_______________________________________________________