\[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))}{(c-c \sec (e+f x))^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {a \arctan \left (\frac {\sqrt {c}\, \tan \left (f x +e \right ) \sqrt {2}}{2 \sqrt {c -c \sec \left (f x +e \right )}}\right ) \sqrt {2}}{16 c^{\frac {5}{2}} f}-\frac {a \tan \left (f x +e \right )}{2 f \left (c -c \sec \left (f x +e \right )\right )^{\frac {5}{2}}}+\frac {a \tan \left (f x +e \right )}{8 c f \left (c -c \sec \left (f x +e \right )\right )^{\frac {3}{2}}} \]
command
integrate(sec(f*x+e)*(a+a*sec(f*x+e))/(c-c*sec(f*x+e))^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\sqrt {2} {\left (a \sqrt {c} \arctan \left (\frac {\sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c}}{\sqrt {c}}\right ) + \frac {{\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {3}{2}} a c - \sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c} a c^{2}}{c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4}}\right )}}{16 \, c^{3} f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________