\[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{9/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (a +a \sec \left (f x +e \right )\right )^{\frac {5}{2}} \tan \left (f x +e \right )}{8 f \left (c -c \sec \left (f x +e \right )\right )^{\frac {9}{2}}}-\frac {\left (a +a \sec \left (f x +e \right )\right )^{\frac {5}{2}} \tan \left (f x +e \right )}{48 c f \left (c -c \sec \left (f x +e \right )\right )^{\frac {7}{2}}} \]
command
integrate(sec(f*x+e)*(a+a*sec(f*x+e))^(5/2)/(c-c*sec(f*x+e))^(9/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {{\left (a^{2} - \frac {4 \, {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a\right )} a^{5} + a^{6}}{a^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8}}\right )} a^{2}}{48 \, \sqrt {-a c} c^{4} f {\left | a \right |} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________