\[ \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx \]
Optimal antiderivative \[ \frac {10 a \,c^{2} \left (g \cos \left (f x +e \right )\right )^{\frac {5}{2}} \left (c -c \sin \left (f x +e \right )\right )^{\frac {3}{2}}}{77 f g \sqrt {a +a \sin \left (f x +e \right )}}+\frac {2 a c \left (g \cos \left (f x +e \right )\right )^{\frac {5}{2}} \left (c -c \sin \left (f x +e \right )\right )^{\frac {5}{2}}}{33 f g \sqrt {a +a \sin \left (f x +e \right )}}-\frac {2 a \left (g \cos \left (f x +e \right )\right )^{\frac {5}{2}} \left (c -c \sin \left (f x +e \right )\right )^{\frac {7}{2}}}{11 f g \sqrt {a +a \sin \left (f x +e \right )}}+\frac {2 a \,c^{4} \left (g \cos \left (f x +e \right )\right )^{\frac {5}{2}}}{3 f g \sqrt {a +a \sin \left (f x +e \right )}\, \sqrt {c -c \sin \left (f x +e \right )}}+\frac {2 a \,c^{4} g \sqrt {\frac {\cos \left (f x +e \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {f x}{2}+\frac {e}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (f x +e \right )\right ) \sqrt {g \cos \left (f x +e \right )}}{\cos \left (\frac {f x}{2}+\frac {e}{2}\right ) f \sqrt {a +a \sin \left (f x +e \right )}\, \sqrt {c -c \sin \left (f x +e \right )}}+\frac {2 a \,c^{3} \left (g \cos \left (f x +e \right )\right )^{\frac {5}{2}} \sqrt {c -c \sin \left (f x +e \right )}}{7 f g \sqrt {a +a \sin \left (f x +e \right )}} \]
command
integrate((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(7/2)*(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
\[ \frac {-231 i \, \sqrt {2} \sqrt {a c g} c^{3} g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 231 i \, \sqrt {2} \sqrt {a c g} c^{3} g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) - 2 \, {\left (21 \, c^{3} g \cos \left (f x + e\right )^{4} - 132 \, c^{3} g \cos \left (f x + e\right )^{2} + 77 \, {\left (c^{3} g \cos \left (f x + e\right )^{2} - c^{3} g\right )} \sin \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{231 \, f} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-{\left (3 \, c^{3} g \cos \left (f x + e\right )^{3} - 4 \, c^{3} g \cos \left (f x + e\right ) - {\left (c^{3} g \cos \left (f x + e\right )^{3} - 4 \, c^{3} g \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}, x\right ) \]