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