\[ \int \frac {(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^{7/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (c -d \right ) \cos \left (f x +e \right ) \left (a^{3}+a^{3} \sin \left (f x +e \right )\right )}{5 d \left (c +d \right ) f \left (c +d \sin \left (f x +e \right )\right )^{\frac {5}{2}}}+\frac {8 a^{3} \left (c -d \right ) \left (c +3 d \right ) \cos \left (f x +e \right )}{15 d^{2} \left (c +d \right )^{2} f \left (c +d \sin \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {4 a^{3} \left (4 c^{2}+15 c d +27 d^{2}\right ) \cos \left (f x +e \right )}{15 d^{2} \left (c +d \right )^{3} f \sqrt {c +d \sin \left (f x +e \right )}}+\frac {4 a^{3} \left (4 c^{2}+15 c d +27 d^{2}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (f x +e \right )}{2}}\, \EllipticE \left (\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ), \sqrt {2}\, \sqrt {\frac {d}{c +d}}\right ) \sqrt {c +d \sin \left (f x +e \right )}}{15 \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) d^{3} \left (c +d \right )^{3} f \sqrt {\frac {c +d \sin \left (f x +e \right )}{c +d}}}-\frac {4 a^{3} \left (4 c^{2}+11 c d +15 d^{2}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (f x +e \right )}{2}}\, \EllipticF \left (\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ), \sqrt {2}\, \sqrt {\frac {d}{c +d}}\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c +d}}}{15 \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) d^{3} \left (c +d \right )^{2} f \sqrt {c +d \sin \left (f x +e \right )}} \]
command
integrate((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^(7/2),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 \[ {\rm integral}\left (-\frac {{\left (3 \, a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3} + {\left (a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}}{d^{4} \cos \left (f x + e\right )^{4} + c^{4} + 6 \, c^{2} d^{2} + d^{4} - 2 \, {\left (3 \, c^{2} d^{2} + d^{4}\right )} \cos \left (f x + e\right )^{2} - 4 \, {\left (c d^{3} \cos \left (f x + e\right )^{2} - c^{3} d - c d^{3}\right )} \sin \left (f x + e\right )}, x\right ) \]