\[ \int \frac {(c+d x)^3}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx \]
Optimal antiderivative \[ \frac {\left (d x +c \right )^{4}}{4 a^{2} d}-\frac {\left (d x +c \right )^{3}}{a^{2} f g n \ln \left (F \right )}+\frac {\left (d x +c \right )^{3}}{a f \left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right ) g n \ln \left (F \right )}+\frac {3 d \left (d x +c \right )^{2} \ln \left (1+\frac {b \left (F^{g \left (f x +e \right )}\right )^{n}}{a}\right )}{a^{2} f^{2} g^{2} n^{2} \ln \left (F \right )^{2}}-\frac {\left (d x +c \right )^{3} \ln \left (1+\frac {b \left (F^{g \left (f x +e \right )}\right )^{n}}{a}\right )}{a^{2} f g n \ln \left (F \right )}+\frac {6 d^{2} \left (d x +c \right ) \polylog \left (2, -\frac {b \left (F^{g \left (f x +e \right )}\right )^{n}}{a}\right )}{a^{2} f^{3} g^{3} n^{3} \ln \left (F \right )^{3}}-\frac {3 d \left (d x +c \right )^{2} \polylog \left (2, -\frac {b \left (F^{g \left (f x +e \right )}\right )^{n}}{a}\right )}{a^{2} f^{2} g^{2} n^{2} \ln \left (F \right )^{2}}-\frac {6 d^{3} \polylog \left (3, -\frac {b \left (F^{g \left (f x +e \right )}\right )^{n}}{a}\right )}{a^{2} f^{4} g^{4} n^{4} \ln \left (F \right )^{4}}+\frac {6 d^{2} \left (d x +c \right ) \polylog \left (3, -\frac {b \left (F^{g \left (f x +e \right )}\right )^{n}}{a}\right )}{a^{2} f^{3} g^{3} n^{3} \ln \left (F \right )^{3}}-\frac {6 d^{3} \polylog \left (4, -\frac {b \left (F^{g \left (f x +e \right )}\right )^{n}}{a}\right )}{a^{2} f^{4} g^{4} n^{4} \ln \left (F \right )^{4}} \]
command
integrate((d*x+c)^3/(a+b*(F^(g*(f*x+e)))^n)^2,x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ c^{3} {\left (\frac {f g n x + g n e}{a^{2} f g n} + \frac {1}{{\left (F^{f g n x + g n e} a b + a^{2}\right )} f g n \log \left (F\right )} - \frac {\log \left (F^{f g n x + g n e} b + a\right )}{a^{2} f g n \log \left (F\right )}\right )} + \frac {d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x}{F^{f g n x} F^{g n e} a b f g n \log \left (F\right ) + a^{2} f g n \log \left (F\right )} - \frac {3 \, c^{2} d x}{a^{2} f g n \log \left (F\right )} + \frac {3 \, c^{2} d \log \left (F^{f g n x} F^{g n e} b + a\right )}{a^{2} f^{2} g^{2} n^{2} \log \left (F\right )^{2}} - \frac {3 \, {\left (c^{2} d f g n \log \left (F\right ) - 2 \, c d^{2}\right )} {\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 )}}{a^{2} f^{3} g^{3} n^{3} \log \left (F\right )^{3}} - \frac {{\left (f^{3} g^{3} n^{3} x^{3} \log \left (\frac {F^{f g n x} F^{g n e} b}{a} + 1\right ) \log \left (F\right )^{3} + 3 \, f^{2} g^{2} n^{2} x^{2} {\rm Li}_2\left (-\frac {F^{f g n x} F^{g n e} b}{a}\right ) \log \left (F\right )^{2} - 6 \, f g n x \log \left (F\right ) {\rm Li}_{3}(-\frac {F^{f g n x} F^{g n e} b}{a}) + 6 \, {\rm Li}_{4}(-\frac {F^{f g n x} F^{g n e} b}{a})\right )} d^{3}}{a^{2} f^{4} g^{4} n^{4} \log \left (F\right )^{4}} - \frac {3 \, {\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 )} {\left (c d^{2} f g n \log \left (F\right ) - d^{3}\right )}}{a^{2} f^{4} g^{4} n^{4} \log \left (F\right )^{4}} + \frac {d^{3} f^{4} g^{4} n^{4} x^{4} \log \left (F\right )^{4} + 4 \, {\left (c d^{2} f g n \log \left (F\right ) - d^{3}\right )} f^{3} g^{3} n^{3} x^{3} \log \left (F\right )^{3} + 6 \, {\left (c^{2} d f^{2} g^{2} n^{2} \log \left (F\right )^{2} - 2 \, c d^{2} f g n \log \left (F\right )\right )} f^{2} g^{2} n^{2} x^{2} \log \left (F\right )^{2}}{4 \, a^{2} f^{4} g^{4} n^{4} \log \left (F\right )^{4}} \]
Maxima 5.44 via sagemath 9.3 output
\[ c^{3} {\left (\frac {1}{{\left ({\left (F^{f g x + e g}\right )}^{n} a b n + a^{2} n\right )} f g \log \left (F\right )} + \frac {\log \left (F^{f g x + e g}\right )}{a^{2} f g \log \left (F\right )} - \frac {\log \left (\frac {{\left (F^{f g x + e g}\right )}^{n} b + a}{b}\right )}{a^{2} f g n \log \left (F\right )}\right )} + \frac {d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x}{{\left (F^{f g x}\right )}^{n} {\left (F^{e g}\right )}^{n} a b f g n \log \left (F\right ) + a^{2} f g n \log \left (F\right )} + \int \frac {d^{3} f g n x^{3} \log \left (F\right ) - 3 \, c^{2} d + 3 \, {\left (c d^{2} f g n \log \left (F\right ) - d^{3}\right )} x^{2} + 3 \, {\left (c^{2} d f g n \log \left (F\right ) - 2 \, c d^{2}\right )} x}{{\left (F^{f g x}\right )}^{n} {\left (F^{e g}\right )}^{n} a b f g n \log \left (F\right ) + a^{2} f g n \log \left (F\right )}\,{d x} \]________________________________________________________________________________________