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