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