\[ \int \frac {\sqrt {c-c \sec (e+f x)}}{(a+a \sec (e+f x))^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {c \tan \left (f x +e \right )}{2 f \left (a +a \sec \left (f x +e \right )\right )^{\frac {5}{2}} \sqrt {c -c \sec \left (f x +e \right )}}-\frac {c \tan \left (f x +e \right )}{a f \left (a +a \sec \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {c -c \sec \left (f x +e \right )}}+\frac {c \ln \left (1+\cos \left (f x +e \right )\right ) \tan \left (f x +e \right )}{a^{2} f \sqrt {a +a \sec \left (f x +e \right )}\, \sqrt {c -c \sec \left (f x +e \right )}} \]
command
integrate((c-c*sec(f*x+e))^(1/2)/(a+a*sec(f*x+e))^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\sqrt {2} {\left (\frac {8 \, \sqrt {2} \sqrt {-a c} c \log \left ({\left | c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + c \right |}\right )}{a^{3} {\left | c \right |}} + \frac {\sqrt {2} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{2} \sqrt {-a c} a^{3} c {\left | c \right |} - 4 \, \sqrt {2} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )} \sqrt {-a c} a^{3} c^{2} {\left | c \right |}}{a^{6} c^{4}}\right )}}{16 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________