\[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{7/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (a +a \sec \left (f x +e \right )\right )^{\frac {3}{2}} \tan \left (f x +e \right )}{6 f \left (c -c \sec \left (f x +e \right )\right )^{\frac {7}{2}}}-\frac {\left (a +a \sec \left (f x +e \right )\right )^{\frac {3}{2}} \tan \left (f x +e \right )}{24 c f \left (c -c \sec \left (f x +e \right )\right )^{\frac {5}{2}}} \]
command
integrate(sec(f*x+e)*(a+a*sec(f*x+e))^(3/2)/(c-c*sec(f*x+e))^(7/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {{\left (a - \frac {3 \, {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a\right )} a^{3} + a^{4}}{a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6}}\right )} a^{2}}{24 \, \sqrt {-a c} c^{3} f {\left | a \right |} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________