\[ \int \frac {(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{9/2}} \, dx \]
Optimal antiderivative \[ -\frac {a^{3} \tan \left (f x +e \right )}{f \left (c -c \sec \left (f x +e \right )\right )^{\frac {9}{2}} \sqrt {a +a \sec \left (f x +e \right )}}-\frac {a^{3} \tan \left (f x +e \right )}{2 c^{2} f \left (c -c \sec \left (f x +e \right )\right )^{\frac {5}{2}} \sqrt {a +a \sec \left (f x +e \right )}}-\frac {a^{3} \tan \left (f x +e \right )}{c^{3} f \left (c -c \sec \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {a +a \sec \left (f x +e \right )}}+\frac {a^{3} \ln \left (1-\cos \left (f x +e \right )\right ) \tan \left (f x +e \right )}{c^{4} f \sqrt {a +a \sec \left (f x +e \right )}\, \sqrt {c -c \sec \left (f x +e \right )}} \]
command
integrate((a+a*sec(f*x+e))^(5/2)/(c-c*sec(f*x+e))^(9/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\frac {48 \, \sqrt {-a c} a^{3} \log \left ({\left | a \right |} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}\right )}{c^{5} {\left | a \right |}} - \frac {48 \, \sqrt {-a c} a^{3} \log \left ({\left | -a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a \right |}\right )}{c^{5} {\left | a \right |}} - \frac {100 \, {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a\right )}^{4} \sqrt {-a c} a^{3} + 352 \, {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a\right )}^{3} \sqrt {-a c} a^{4} + 480 \, {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a\right )}^{2} \sqrt {-a c} a^{5} + 292 \, {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a\right )} \sqrt {-a c} a^{6} + 67 \, \sqrt {-a c} a^{7}}{a^{4} c^{5} {\left | a \right |} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8}}}{48 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________