11.10 Problem number 606

\[ \int \frac {F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}}{d g+e g x} \, dx \]

Optimal antiderivative \[ \frac {\erfi \left (a \sqrt {f}\, \sqrt {\ln \left (F \right )}+b \ln \left (c \left (e x +d \right )^{n}\right ) \sqrt {f}\, \sqrt {\ln \left (F \right )}\right ) \sqrt {\pi }}{2 b e g n \sqrt {f}\, \sqrt {\ln \left (F \right )}} \]

command

int(F^(f*(a+b*ln(c*(e*x+d)^n))^2)/(e*g*x+d*g),x)

Maple 2022.1 output

\[-\frac {\sqrt {\pi }\, \erf \left (-b \sqrt {-f \ln \left (F \right )}\, \ln \left (\left (e x +d \right )^{n}\right )+\frac {f \left (a +b \left (\ln \left (c \right )-\frac {i \pi \,\mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right ) \left (-\mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )+\mathrm {csgn}\left (i c \right )\right ) \left (-\mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )+\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right )\right )}{2}\right )\right ) \ln \left (F \right )}{\sqrt {-f \ln \left (F \right )}}\right )}{2 g e n b \sqrt {-f \ln \left (F \right )}}\]

Maple 2021.1 output

\[ \int \frac {F^{\left (b \ln \left (c \left (e x +d \right )^{n}\right )+a \right )^{2} f}}{e g x +d g}\, dx \]________________________________________________________________________________________