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