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