\[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {15 a^{3} \arctan \left (\frac {\sqrt {c}\, \tan \left (f x +e \right ) \sqrt {2}}{2 \sqrt {c -c \sec \left (f x +e \right )}}\right ) \sqrt {2}}{4 c^{\frac {5}{2}} f}-\frac {a \left (a +a \sec \left (f x +e \right )\right )^{2} \tan \left (f x +e \right )}{2 f \left (c -c \sec \left (f x +e \right )\right )^{\frac {5}{2}}}+\frac {5 \left (a^{3}+a^{3} \sec \left (f x +e \right )\right ) \tan \left (f x +e \right )}{4 c f \left (c -c \sec \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {15 a^{3} \tan \left (f x +e \right )}{4 c^{2} f \sqrt {c -c \sec \left (f x +e \right )}} \]
command
integrate(sec(f*x+e)*(a+a*sec(f*x+e))^3/(c-c*sec(f*x+e))^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {a^{3} {\left (\frac {15 \, \sqrt {2} \arctan \left (\frac {\sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c}}{\sqrt {c}}\right )}{c^{\frac {5}{2}}} + \frac {8 \, \sqrt {2}}{\sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c} c^{2}} + \frac {7 \, \sqrt {2} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {3}{2}} + 9 \, \sqrt {2} \sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c} c}{c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4}}\right )}}{4 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________