\[ \int \frac {(c+d x)^3}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx \]
Optimal antiderivative \[ \frac {\left (d x +c \right )^{4}}{4 a^{3} d}+\frac {3 d \left (d x +c \right )^{2}}{2 a^{3} f^{2} g^{2} n^{2} \ln \left (F \right )^{2}}-\frac {3 d \left (d x +c \right )^{2}}{2 a^{2} f^{2} \left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right ) g^{2} n^{2} \ln \left (F \right )^{2}}-\frac {3 \left (d x +c \right )^{3}}{2 a^{3} f g n \ln \left (F \right )}+\frac {\left (d x +c \right )^{3}}{2 a f \left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right )^{2} g n \ln \left (F \right )}+\frac {\left (d x +c \right )^{3}}{a^{2} f \left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right ) g n \ln \left (F \right )}-\frac {3 d^{2} \left (d x +c \right ) \ln \left (1+\frac {b \left (F^{g \left (f x +e \right )}\right )^{n}}{a}\right )}{a^{3} f^{3} g^{3} n^{3} \ln \left (F \right )^{3}}+\frac {9 d \left (d x +c \right )^{2} \ln \left (1+\frac {b \left (F^{g \left (f x +e \right )}\right )^{n}}{a}\right )}{2 a^{3} 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^{3} f g n \ln \left (F \right )}-\frac {3 d^{3} \polylog \left (2, -\frac {b \left (F^{g \left (f x +e \right )}\right )^{n}}{a}\right )}{a^{3} f^{4} g^{4} n^{4} \ln \left (F \right )^{4}}+\frac {9 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^{3} 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^{3} f^{2} g^{2} n^{2} \ln \left (F \right )^{2}}-\frac {9 d^{3} \polylog \left (3, -\frac {b \left (F^{g \left (f x +e \right )}\right )^{n}}{a}\right )}{a^{3} 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^{3} 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^{3} 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)^3,x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ \frac {1}{2} \, c^{3} {\left (\frac {2 \, F^{f g n x + g n e} b + 3 \, a}{{\left (2 \, F^{f g n x + g n e} a^{3} b + F^{2 \, f g n x + 2 \, g n e} a^{2} b^{2} + a^{4}\right )} f g n \log \left (F\right )} + \frac {2 \, {\left (f g n x + g n e\right )}}{a^{3} f g n} - \frac {2 \, \log \left (F^{f g n x + g n e} b + a\right )}{a^{3} f g n \log \left (F\right )}\right )} + \frac {3 \, a d^{3} f g n x^{3} \log \left (F\right ) - 3 \, a c^{2} d + 3 \, {\left (3 \, a c d^{2} f g n \log \left (F\right ) - a d^{3}\right )} x^{2} + {\left (2 \, F^{g n e} b d^{3} f g n x^{3} \log \left (F\right ) - 3 \, F^{g n e} b c^{2} d + 3 \, {\left (2 \, F^{g n e} b c d^{2} f g n \log \left (F\right ) - F^{g n e} b d^{3}\right )} x^{2} + 6 \, {\left (F^{g n e} b c^{2} d f g n \log \left (F\right ) - F^{g n e} b c d^{2}\right )} x\right )} F^{f g n x} + 3 \, {\left (3 \, a c^{2} d f g n \log \left (F\right ) - 2 \, a c d^{2}\right )} x}{2 \, {\left (2 \, F^{f g n x} F^{g n e} a^{3} b f^{2} g^{2} n^{2} \log \left (F\right )^{2} + F^{2 \, f g n x} F^{2 \, g n e} a^{2} b^{2} f^{2} g^{2} n^{2} \log \left (F\right )^{2} + a^{4} f^{2} g^{2} n^{2} \log \left (F\right )^{2}\right )}} - \frac {3 \, {\left (3 \, c^{2} d f g n \log \left (F\right ) - 2 \, c d^{2}\right )} x}{2 \, a^{3} f^{2} g^{2} n^{2} \log \left (F\right )^{2}} + \frac {3 \, {\left (3 \, c^{2} d f g n \log \left (F\right ) - 2 \, c d^{2}\right )} \log \left (F^{f g n x} F^{g n e} b + a\right )}{2 \, a^{3} 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^{3} 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 (2 \, c d^{2} f g n \log \left (F\right ) - 3 \, d^{3}\right )}}{2 \, a^{3} f^{4} g^{4} n^{4} \log \left (F\right )^{4}} - \frac {3 \, {\left (c^{2} d f^{2} g^{2} n^{2} \log \left (F\right )^{2} - 3 \, c d^{2} f g n \log \left (F\right ) + d^{3}\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^{3} 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} + 2 \, {\left (2 \, c d^{2} f g n \log \left (F\right ) - 3 \, 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} - 3 \, c d^{2} f g n \log \left (F\right ) + d^{3}\right )} f^{2} g^{2} n^{2} x^{2} \log \left (F\right )^{2}}{4 \, a^{3} f^{4} g^{4} n^{4} \log \left (F\right )^{4}} \]
Maxima 5.44 via sagemath 9.3 output
\[ \frac {1}{2} \, c^{3} {\left (\frac {2 \, {\left (F^{f g x + e g}\right )}^{n} b + 3 \, a}{{\left (2 \, {\left (F^{f g x + e g}\right )}^{n} a^{3} b n + {\left (F^{f g x + e g}\right )}^{2 \, n} a^{2} b^{2} n + a^{4} n\right )} f g \log \left (F\right )} + \frac {2 \, \log \left (F^{f g x + e g}\right )}{a^{3} f g \log \left (F\right )} - \frac {2 \, \log \left (\frac {{\left (F^{f g x + e g}\right )}^{n} b + a}{b}\right )}{a^{3} f g n \log \left (F\right )}\right )} + \frac {3 \, a d^{3} f g n x^{3} \log \left (F\right ) - 3 \, a c^{2} d + 3 \, {\left (3 \, a c d^{2} f g n \log \left (F\right ) - a d^{3}\right )} x^{2} + {\left (2 \, {\left (F^{e g}\right )}^{n} b d^{3} f g n x^{3} \log \left (F\right ) - 3 \, {\left (F^{e g}\right )}^{n} b c^{2} d + 3 \, {\left (2 \, {\left (F^{e g}\right )}^{n} b c d^{2} f g n \log \left (F\right ) - {\left (F^{e g}\right )}^{n} b d^{3}\right )} x^{2} + 6 \, {\left ({\left (F^{e g}\right )}^{n} b c^{2} d f g n \log \left (F\right ) - {\left (F^{e g}\right )}^{n} b c d^{2}\right )} x\right )} {\left (F^{f g x}\right )}^{n} + 3 \, {\left (3 \, a c^{2} d f g n \log \left (F\right ) - 2 \, a c d^{2}\right )} x}{2 \, {\left (2 \, {\left (F^{f g x}\right )}^{n} {\left (F^{e g}\right )}^{n} a^{3} b f^{2} g^{2} n^{2} \log \left (F\right )^{2} + {\left (F^{f g x}\right )}^{2 \, n} {\left (F^{e g}\right )}^{2 \, n} a^{2} b^{2} f^{2} g^{2} n^{2} \log \left (F\right )^{2} + a^{4} f^{2} g^{2} n^{2} \log \left (F\right )^{2}\right )}} + \int \frac {2 \, d^{3} f^{2} g^{2} n^{2} x^{3} \log \left (F\right )^{2} - 9 \, c^{2} d f g n \log \left (F\right ) + 6 \, c d^{2} + 3 \, {\left (2 \, c d^{2} f^{2} g^{2} n^{2} \log \left (F\right )^{2} - 3 \, d^{3} f g n \log \left (F\right )\right )} x^{2} + 6 \, {\left (c^{2} d f^{2} g^{2} n^{2} \log \left (F\right )^{2} - 3 \, c d^{2} f g n \log \left (F\right ) + d^{3}\right )} x}{2 \, {\left ({\left (F^{f g x}\right )}^{n} {\left (F^{e g}\right )}^{n} a^{2} b f^{2} g^{2} n^{2} \log \left (F\right )^{2} + a^{3} f^{2} g^{2} n^{2} \log \left (F\right )^{2}\right )}}\,{d x} \]________________________________________________________________________________________