\[ \int \frac {(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (c -d \right ) \cos \left (f x +e \right ) \left (a^{3}+a^{3} \sin \left (f x +e \right )\right )}{d \left (c +d \right ) f \sqrt {c +d \sin \left (f x +e \right )}}-\frac {4 a^{3} \left (2 c -d \right ) \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}}{3 d^{2} \left (c +d \right ) f}+\frac {4 a^{3} \left (4 c^{2}-5 c d -3 d^{2}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (f x +e \right )}{2}}\, \EllipticE \left (\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ), \sqrt {2}\, \sqrt {\frac {d}{c +d}}\right ) \sqrt {c +d \sin \left (f x +e \right )}}{3 \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) d^{3} \left (c +d \right ) f \sqrt {\frac {c +d \sin \left (f x +e \right )}{c +d}}}-\frac {4 a^{3} \left (4 c -5 d \right ) \left (c -d \right ) \sqrt {\frac {1}{2}+\frac {\sin \left (f x +e \right )}{2}}\, \EllipticF \left (\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ), \sqrt {2}\, \sqrt {\frac {d}{c +d}}\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c +d}}}{3 \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) d^{3} f \sqrt {c +d \sin \left (f x +e \right )}} \]
command
integrate((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left ({\left (\sqrt {2} {\left (8 \, a^{3} c^{3} d - 10 \, a^{3} c^{2} d^{2} - 9 \, a^{3} c d^{3} + 15 \, a^{3} d^{4}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (8 \, a^{3} c^{4} - 10 \, a^{3} c^{3} d - 9 \, a^{3} c^{2} d^{2} + 15 \, a^{3} c d^{3}\right )}\right )} \sqrt {i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right ) + {\left (\sqrt {2} {\left (8 \, a^{3} c^{3} d - 10 \, a^{3} c^{2} d^{2} - 9 \, a^{3} c d^{3} + 15 \, a^{3} d^{4}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (8 \, a^{3} c^{4} - 10 \, a^{3} c^{3} d - 9 \, a^{3} c^{2} d^{2} + 15 \, a^{3} c d^{3}\right )}\right )} \sqrt {-i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right ) - 3 \, {\left (\sqrt {2} {\left (-4 i \, a^{3} c^{2} d^{2} + 5 i \, a^{3} c d^{3} + 3 i \, a^{3} d^{4}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (-4 i \, a^{3} c^{3} d + 5 i \, a^{3} c^{2} d^{2} + 3 i \, a^{3} c d^{3}\right )}\right )} \sqrt {i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right )\right ) - 3 \, {\left (\sqrt {2} {\left (4 i \, a^{3} c^{2} d^{2} - 5 i \, a^{3} c d^{3} - 3 i \, a^{3} d^{4}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (4 i \, a^{3} c^{3} d - 5 i \, a^{3} c^{2} d^{2} - 3 i \, a^{3} c d^{3}\right )}\right )} \sqrt {-i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right )\right ) - 3 \, {\left ({\left (a^{3} c d^{3} + a^{3} d^{4}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left (4 \, a^{3} c^{2} d^{2} - 5 \, a^{3} c d^{3} + 3 \, a^{3} d^{4}\right )} \cos \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}\right )}}{9 \, {\left ({\left (c d^{5} + d^{6}\right )} f \sin \left (f x + e\right ) + {\left (c^{2} d^{4} + c d^{5}\right )} f\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (3 \, a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3} + {\left (a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}}{d^{2} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2}}, x\right ) \]