\[ \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{\sqrt {a+a \sin (e+f x)}} \, dx \]
Optimal antiderivative \[ \frac {2 c \left (g \cos \left (f x +e \right )\right )^{\frac {5}{2}} \left (c -c \sin \left (f x +e \right )\right )^{\frac {3}{2}}}{7 f g \sqrt {a +a \sin \left (f x +e \right )}}+\frac {22 c^{3} \left (g \cos \left (f x +e \right )\right )^{\frac {5}{2}}}{15 f g \sqrt {a +a \sin \left (f x +e \right )}\, \sqrt {c -c \sin \left (f x +e \right )}}+\frac {22 c^{3} g \sqrt {\frac {\cos \left (f x +e \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {f x}{2}+\frac {e}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (f x +e \right )\right ) \sqrt {g \cos \left (f x +e \right )}}{5 \cos \left (\frac {f x}{2}+\frac {e}{2}\right ) f \sqrt {a +a \sin \left (f x +e \right )}\, \sqrt {c -c \sin \left (f x +e \right )}}+\frac {22 c^{2} \left (g \cos \left (f x +e \right )\right )^{\frac {5}{2}} \sqrt {c -c \sin \left (f x +e \right )}}{35 f g \sqrt {a +a \sin \left (f x +e \right )}} \]
command
integrate((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e))^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {-231 i \, \sqrt {2} \sqrt {a c g} c^{2} g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 231 i \, \sqrt {2} \sqrt {a c g} c^{2} g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) - 2 \, {\left (15 \, c^{2} g \cos \left (f x + e\right )^{2} + 63 \, c^{2} g \sin \left (f x + e\right ) - 140 \, c^{2} g\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{105 \, a f} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (c^{2} g \cos \left (f x + e\right )^{3} + 2 \, c^{2} g \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, c^{2} g \cos \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{\sqrt {a \sin \left (f x + e\right ) + a}}, x\right ) \]