\[ \int \frac {\sqrt {a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 a \cos \left (f x +e \right )}{3 \left (c +d \right ) f \left (c +d \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {a +a \sin \left (f x +e \right )}}-\frac {4 a \cos \left (f x +e \right )}{3 \left (c +d \right )^{2} f \sqrt {a +a \sin \left (f x +e \right )}\, \sqrt {c +d \sin \left (f x +e \right )}} \]
command
integrate((a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {4 \, \sqrt {2} {\left ({\left (\frac {3 \, {\left (c^{6} d^{4} - 2 \, c^{5} d^{5} - c^{4} d^{6} + 4 \, c^{3} d^{7} - c^{2} d^{8} - 2 \, c d^{9} + d^{10}\right )} \tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, f x + \frac {1}{4} \, e\right )^{2}}{c^{7} d^{4} - c^{6} d^{5} - 3 \, c^{5} d^{6} + 3 \, c^{4} d^{7} + 3 \, c^{3} d^{8} - 3 \, c^{2} d^{9} - c d^{10} + d^{11}} + \frac {2 \, {\left (3 \, c^{6} d^{4} - 14 \, c^{5} d^{5} + 21 \, c^{4} d^{6} - 4 \, c^{3} d^{7} - 19 \, c^{2} d^{8} + 18 \, c d^{9} - 5 \, d^{10}\right )}}{c^{7} d^{4} - c^{6} d^{5} - 3 \, c^{5} d^{6} + 3 \, c^{4} d^{7} + 3 \, c^{3} d^{8} - 3 \, c^{2} d^{9} - c d^{10} + d^{11}}\right )} \tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, f x + \frac {1}{4} \, e\right )^{2} + \frac {3 \, {\left (c^{6} d^{4} - 2 \, c^{5} d^{5} - c^{4} d^{6} + 4 \, c^{3} d^{7} - c^{2} d^{8} - 2 \, c d^{9} + d^{10}\right )}}{c^{7} d^{4} - c^{6} d^{5} - 3 \, c^{5} d^{6} + 3 \, c^{4} d^{7} + 3 \, c^{3} d^{8} - 3 \, c^{2} d^{9} - c d^{10} + d^{11}}\right )} \sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, f x + \frac {1}{4} \, e\right )}{3 \, {\left (c \tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, f x + \frac {1}{4} \, e\right )^{4} + d \tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, f x + \frac {1}{4} \, e\right )^{4} + 2 \, c \tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, f x + \frac {1}{4} \, e\right )^{2} - 6 \, d \tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, f x + \frac {1}{4} \, e\right )^{2} + c + d\right )}^{\frac {3}{2}} f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________