\[ \int \frac {F^{f \left (a+b \log ^2\left (c (d+e x)^n\right )\right )}}{d g+e g x} \, dx \]
Optimal antiderivative \[ \frac {F^{a f} \erfi \left (\ln \left (c \left (e x +d \right )^{n}\right ) \sqrt {b}\, \sqrt {f}\, \sqrt {\ln \left (F \right )}\right ) \sqrt {\pi }}{2 e g n \sqrt {b}\, \sqrt {f}\, \sqrt {\ln \left (F \right )}} \]
command
int(F^(f*(a+b*ln(c*(e*x+d)^n)^2))/(e*g*x+d*g),x)
Maple 2022.1 output
\[\frac {\sqrt {\pi }\, F^{f \left (-i b \ln \left (c \right ) \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right )+b \,\pi ^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )-b \,\pi ^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right )+b \,\pi ^{2} \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right )+i b \ln \left (c \right ) \pi \,\mathrm {csgn}\left (i c \right )-i b \ln \left (c \right ) \pi \,\mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )+i b \ln \left (c \right ) \pi \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right )-b \,\pi ^{2}+b \ln \left (c \right )^{2}+a \right )} F^{-\frac {f b \left (i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right )-i \pi \,\mathrm {csgn}\left (i c \right )+i \pi \,\mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )-i \pi \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right )-2 \ln \left (c \right )\right )^{2}}{4}} \erf \left (\sqrt {-\ln \left (F \right ) b f}\, \ln \left (\left (e x +d \right )^{n}\right )-\frac {f b \left (2 \ln \left (c \right )-i \pi \,\mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right ) \left (-\mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )+\mathrm {csgn}\left (i c \right )\right ) \left (-\mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )+\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right )\right )\right ) \ln \left (F \right )}{2 \sqrt {-\ln \left (F \right ) b f}}\right )}{2 g e n \sqrt {-\ln \left (F \right ) b f}}\]
Maple 2021.1 output
\[ \int \frac {F^{\left (b \ln \left (c \left (e x +d \right )^{n}\right )^{2}+a \right ) f}}{e g x +d g}\, dx \]________________________________________________________________________________________