\[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}{(a+a \sin (e+f x))^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {4 c \left (g \cos \left (f x +e \right )\right )^{\frac {5}{2}}}{f g \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {c -c \sin \left (f x +e \right )}}-\frac {6 c g \sqrt {\frac {\cos \left (f x +e \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {f x}{2}+\frac {e}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (f x +e \right )\right ) \sqrt {g \cos \left (f x +e \right )}}{\cos \left (\frac {f x}{2}+\frac {e}{2}\right ) a f \sqrt {a +a \sin \left (f x +e \right )}\, \sqrt {c -c \sin \left (f x +e \right )}} \]
command
integrate((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {4 \, \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} g + 3 \, \sqrt {a c g} {\left (-i \, \sqrt {2} g \sin \left (f x + e\right ) - i \, \sqrt {2} g\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 3 \, \sqrt {a c g} {\left (i \, \sqrt {2} g \sin \left (f x + e\right ) + i \, \sqrt {2} g\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}{a^{2} f \sin \left (f x + e\right ) + a^{2} f} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} g \cos \left (f x + e\right )}{a^{2} \cos \left (f x + e\right )^{2} - 2 \, a^{2} \sin \left (f x + e\right ) - 2 \, a^{2}}, x\right ) \]