\[ \int \sec (e+f x) (a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{7/2} \, dx \]
Optimal antiderivative \[ \frac {a \left (a +a \sec \left (f x +e \right )\right )^{\frac {3}{2}} \left (c -c \sec \left (f x +e \right )\right )^{\frac {7}{2}} \tan \left (f x +e \right )}{6 f}+\frac {a^{3} \left (c -c \sec \left (f x +e \right )\right )^{\frac {7}{2}} \tan \left (f x +e \right )}{15 f \sqrt {a +a \sec \left (f x +e \right )}}+\frac {2 a^{2} \left (c -c \sec \left (f x +e \right )\right )^{\frac {7}{2}} \sqrt {a +a \sec \left (f x +e \right )}\, \tan \left (f x +e \right )}{15 f} \]
command
integrate(sec(f*x+e)*(a+a*sec(f*x+e))^(5/2)*(c-c*sec(f*x+e))^(7/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {16 \, {\left (20 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{3} c^{4} + 45 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{2} c^{5} + 36 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )} c^{6} + 10 \, c^{7}\right )} \sqrt {-a c} a^{2} c {\left | c \right |} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{15 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{6} f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________