\[ \int \frac {\sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {a \cos \left (f x +e \right )}{2 f \left (c -c \sin \left (f x +e \right )\right )^{\frac {5}{2}} \sqrt {a +a \sin \left (f x +e \right )}} \]
command
integrate((a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{8 \, c^{\frac {5}{2}} f \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________