\[ \int \frac {(a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {a^{2} \tan \left (f x +e \right )}{f \left (c -c \sec \left (f x +e \right )\right )^{\frac {5}{2}} \sqrt {a +a \sec \left (f x +e \right )}}-\frac {a^{2} \tan \left (f x +e \right )}{c f \left (c -c \sec \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {a +a \sec \left (f x +e \right )}}+\frac {a^{2} \ln \left (1-\cos \left (f x +e \right )\right ) \tan \left (f x +e \right )}{c^{2} f \sqrt {a +a \sec \left (f x +e \right )}\, \sqrt {c -c \sec \left (f x +e \right )}} \]
command
integrate((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
\[ -\frac {\frac {4 \, \sqrt {-a c} a^{2} \log \left ({\left | a \right |} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}\right )}{c^{3} {\left | a \right |}} - \frac {4 \, \sqrt {-a c} a^{2} \log \left ({\left | -a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a \right |}\right )}{c^{3} {\left | a \right |}} - \frac {6 \, {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a\right )}^{2} \sqrt {-a c} a^{2} + 8 \, {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a\right )} \sqrt {-a c} a^{3} + 3 \, \sqrt {-a c} a^{4}}{a^{2} c^{3} {\left | a \right |} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4}}}{4 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________