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