\[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)} \, dx \]
Optimal antiderivative \[ -\frac {2 a c \left (g \cos \left (f x +e \right )\right )^{\frac {5}{2}} \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}{21 f g \sqrt {c -c \sin \left (f x +e \right )}}+\frac {2 c \left (g \cos \left (f x +e \right )\right )^{\frac {5}{2}} \left (a +a \sin \left (f x +e \right )\right )^{\frac {5}{2}}}{9 f g \sqrt {c -c \sin \left (f x +e \right )}}-\frac {22 a^{3} c \left (g \cos \left (f x +e \right )\right )^{\frac {5}{2}}}{45 f g \sqrt {a +a \sin \left (f x +e \right )}\, \sqrt {c -c \sin \left (f x +e \right )}}+\frac {22 a^{3} c g \sqrt {\frac {\cos \left (f x +e \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {f x}{2}+\frac {e}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (f x +e \right )\right ) \sqrt {g \cos \left (f x +e \right )}}{15 \cos \left (\frac {f x}{2}+\frac {e}{2}\right ) f \sqrt {a +a \sin \left (f x +e \right )}\, \sqrt {c -c \sin \left (f x +e \right )}}-\frac {22 a^{2} c \left (g \cos \left (f x +e \right )\right )^{\frac {5}{2}} \sqrt {a +a \sin \left (f x +e \right )}}{105 f g \sqrt {c -c \sin \left (f x +e \right )}} \]
command
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(5/2)*(c-c*sin(f*x+e))^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {-231 i \, \sqrt {2} \sqrt {a c g} a^{2} g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 231 i \, \sqrt {2} \sqrt {a c g} a^{2} g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) - 2 \, {\left (90 \, a^{2} g \cos \left (f x + e\right )^{2} + 7 \, {\left (5 \, a^{2} g \cos \left (f x + e\right )^{2} - 11 \, a^{2} g\right )} \sin \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{315 \, f} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-{\left (a^{2} g \cos \left (f x + e\right )^{3} - 2 \, a^{2} g \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, a^{2} g \cos \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}, x\right ) \]