\[ \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {12 c^{2} \left (g \cos \left (f x +e \right )\right )^{\frac {5}{2}} \left (c -c \sin \left (f x +e \right )\right )^{\frac {3}{2}}}{a f g \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {4 c \left (g \cos \left (f x +e \right )\right )^{\frac {5}{2}} \left (c -c \sin \left (f x +e \right )\right )^{\frac {5}{2}}}{5 f g \left (a +a \sin \left (f x +e \right )\right )^{\frac {5}{2}}}+\frac {154 c^{4} \left (g \cos \left (f x +e \right )\right )^{\frac {5}{2}}}{5 a^{2} f g \sqrt {a +a \sin \left (f x +e \right )}\, \sqrt {c -c \sin \left (f x +e \right )}}+\frac {462 c^{4} g \sqrt {\frac {\cos \left (f x +e \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {f x}{2}+\frac {e}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (f x +e \right )\right ) \sqrt {g \cos \left (f x +e \right )}}{5 \cos \left (\frac {f x}{2}+\frac {e}{2}\right ) a^{2} f \sqrt {a +a \sin \left (f x +e \right )}\, \sqrt {c -c \sin \left (f x +e \right )}}+\frac {66 c^{3} \left (g \cos \left (f x +e \right )\right )^{\frac {5}{2}} \sqrt {c -c \sin \left (f x +e \right )}}{5 a^{2} f g \sqrt {a +a \sin \left (f x +e \right )}} \]
command
integrate((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(7/2)/(a+a*sin(f*x+e))^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (8 \, c^{3} g \cos \left (f x + e\right )^{2} - 146 \, c^{3} g - {\left (c^{3} g \cos \left (f x + e\right )^{2} + 162 \, c^{3} g\right )} \sin \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} + 231 \, {\left (-i \, \sqrt {2} c^{3} g \cos \left (f x + e\right )^{2} + 2 i \, \sqrt {2} c^{3} g \sin \left (f x + e\right ) + 2 i \, \sqrt {2} c^{3} g\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 231 \, {\left (i \, \sqrt {2} c^{3} g \cos \left (f x + e\right )^{2} - 2 i \, \sqrt {2} c^{3} g \sin \left (f x + e\right ) - 2 i \, \sqrt {2} c^{3} g\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}{5 \, {\left (a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \sin \left (f x + e\right ) - 2 \, a^{3} f\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (3 \, c^{3} g \cos \left (f x + e\right )^{3} - 4 \, c^{3} g \cos \left (f x + e\right ) - {\left (c^{3} g \cos \left (f x + e\right )^{3} - 4 \, c^{3} g \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{3 \, a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3} + {\left (a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3}\right )} \sin \left (f x + e\right )}, x\right ) \]