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