\[ \int \frac {1}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {d \left (3 c +5 d \right ) \cos \left (f x +e \right )}{3 a \left (c -d \right )^{2} \left (c +d \right ) f \left (c +d \sin \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {\cos \left (f x +e \right )}{\left (c -d \right ) f \left (a +a \sin \left (f x +e \right )\right ) \left (c +d \sin \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {d \left (3 c^{2}+20 c d +9 d^{2}\right ) \cos \left (f x +e \right )}{3 a \left (c -d \right )^{3} \left (c +d \right )^{2} f \sqrt {c +d \sin \left (f x +e \right )}}+\frac {\left (3 c^{2}+20 c d +9 d^{2}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (f x +e \right )}{2}}\, \EllipticE \left (\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ), \sqrt {2}\, \sqrt {\frac {d}{c +d}}\right ) \sqrt {c +d \sin \left (f x +e \right )}}{3 \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) a \left (c -d \right )^{3} \left (c +d \right )^{2} f \sqrt {\frac {c +d \sin \left (f x +e \right )}{c +d}}}-\frac {\left (3 c +5 d \right ) \sqrt {\frac {1}{2}+\frac {\sin \left (f x +e \right )}{2}}\, \EllipticF \left (\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ), \sqrt {2}\, \sqrt {\frac {d}{c +d}}\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c +d}}}{3 \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) a \left (c -d \right )^{2} \left (c +d \right ) f \sqrt {c +d \sin \left (f x +e \right )}} \]
command
integrate(1/(a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ {\rm integral}\left (\frac {\sqrt {d \sin \left (f x + e\right ) + c}}{a d^{3} \cos \left (f x + e\right )^{4} + a c^{3} + 3 \, a c^{2} d + 3 \, a c d^{2} + a d^{3} - {\left (3 \, a c^{2} d + 3 \, a c d^{2} + 2 \, a d^{3}\right )} \cos \left (f x + e\right )^{2} + {\left (a c^{3} + 3 \, a c^{2} d + 3 \, a c d^{2} + a d^{3} - {\left (3 \, a c d^{2} + a d^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}, x\right ) \]