\[ \int \frac {(c+d x)^2}{a+b \left (F^{g (e+f x)}\right )^n} \, dx \]
Optimal antiderivative \[ \frac {\left (d x +c \right )^{3}}{3 a d}-\frac {\left (d x +c \right )^{2} \ln \left (1+\frac {b \left (F^{g \left (f x +e \right )}\right )^{n}}{a}\right )}{a f g n \ln \left (F \right )}-\frac {2 d \left (d x +c \right ) \polylog \left (2, -\frac {b \left (F^{g \left (f x +e \right )}\right )^{n}}{a}\right )}{a \,f^{2} g^{2} n^{2} \ln \left (F \right )^{2}}+\frac {2 d^{2} \polylog \left (3, -\frac {b \left (F^{g \left (f x +e \right )}\right )^{n}}{a}\right )}{a \,f^{3} g^{3} n^{3} \ln \left (F \right )^{3}} \]
command
integrate((d*x+c)^2/(a+b*(F^(g*(f*x+e)))^n),x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ c^{2} {\left (\frac {f g n x + g n e}{a f g n} - \frac {\log \left (F^{f g n x + g n e} b + a\right )}{a f g n \log \left (F\right )}\right )} - \frac {2 \, {\left (f g n x \log \left (\frac {F^{f g n x} F^{g n e} b}{a} + 1\right ) \log \left (F\right ) + {\rm Li}_2\left (-\frac {F^{f g n x} F^{g n e} b}{a}\right )\right )} c d}{a f^{2} g^{2} n^{2} \log \left (F\right )^{2}} - \frac {{\left (f^{2} g^{2} n^{2} x^{2} \log \left (\frac {F^{f g n x} F^{g n e} b}{a} + 1\right ) \log \left (F\right )^{2} + 2 \, f g n x {\rm Li}_2\left (-\frac {F^{f g n x} F^{g n e} b}{a}\right ) \log \left (F\right ) - 2 \, {\rm Li}_{3}(-\frac {F^{f g n x} F^{g n e} b}{a})\right )} d^{2}}{a f^{3} g^{3} n^{3} \log \left (F\right )^{3}} + \frac {d^{2} f^{3} g^{3} n^{3} x^{3} \log \left (F\right )^{3} + 3 \, c d f^{3} g^{3} n^{3} x^{2} \log \left (F\right )^{3}}{3 \, a f^{3} g^{3} n^{3} \log \left (F\right )^{3}} \]
Maxima 5.44 via sagemath 9.3 output
\[ -c^{2} {\left (\frac {\log \left ({\left (F^{f g x + e g}\right )}^{n} b + a\right )}{a f g n \log \left (F\right )} - \frac {\log \left ({\left (F^{f g x + e g}\right )}^{n}\right )}{a f g n \log \left (F\right )}\right )} + \int \frac {d^{2} x^{2} + 2 \, c d x}{{\left (F^{f g x}\right )}^{n} {\left (F^{e g}\right )}^{n} b + a}\,{d x} \]________________________________________________________________________________________