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