\[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{9/2}} \, dx \]
Optimal antiderivative \[ \frac {4 a \left (g \cos \left (f x +e \right )\right )^{\frac {5}{2}}}{13 f g \left (c -c \sin \left (f x +e \right )\right )^{\frac {9}{2}} \sqrt {a +a \sin \left (f x +e \right )}}-\frac {2 a \left (g \cos \left (f x +e \right )\right )^{\frac {5}{2}}}{39 c f g \left (c -c \sin \left (f x +e \right )\right )^{\frac {7}{2}} \sqrt {a +a \sin \left (f x +e \right )}}-\frac {2 a \left (g \cos \left (f x +e \right )\right )^{\frac {5}{2}}}{65 c^{2} f g \left (c -c \sin \left (f x +e \right )\right )^{\frac {5}{2}} \sqrt {a +a \sin \left (f x +e \right )}}-\frac {2 a \left (g \cos \left (f x +e \right )\right )^{\frac {5}{2}}}{65 c^{3} f g \left (c -c \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {a +a \sin \left (f x +e \right )}}+\frac {2 a g \sqrt {\frac {\cos \left (f x +e \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {f x}{2}+\frac {e}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (f x +e \right )\right ) \sqrt {g \cos \left (f x +e \right )}}{65 \cos \left (\frac {f x}{2}+\frac {e}{2}\right ) c^{4} f \sqrt {a +a \sin \left (f x +e \right )}\, \sqrt {c -c \sin \left (f x +e \right )}} \]
command
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(9/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (12 \, g \cos \left (f x + e\right )^{2} - {\left (3 \, g \cos \left (f x + e\right )^{2} - 23 \, g\right )} \sin \left (f x + e\right ) + 7 \, g\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} - 3 \, {\left (i \, \sqrt {2} g \cos \left (f x + e\right )^{4} - 8 i \, \sqrt {2} g \cos \left (f x + e\right )^{2} + 4 \, {\left (i \, \sqrt {2} g \cos \left (f x + e\right )^{2} - 2 i \, \sqrt {2} g\right )} \sin \left (f x + e\right ) + 8 i \, \sqrt {2} g\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) - 3 \, {\left (-i \, \sqrt {2} g \cos \left (f x + e\right )^{4} + 8 i \, \sqrt {2} g \cos \left (f x + e\right )^{2} + 4 \, {\left (-i \, \sqrt {2} g \cos \left (f x + e\right )^{2} + 2 i \, \sqrt {2} g\right )} \sin \left (f x + e\right ) - 8 i \, \sqrt {2} g\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}{195 \, {\left (c^{5} f \cos \left (f x + e\right )^{4} - 8 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f + 4 \, {\left (c^{5} f \cos \left (f x + e\right )^{2} - 2 \, c^{5} f\right )} \sin \left (f x + e\right )\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} g \cos \left (f x + e\right )}{5 \, c^{5} \cos \left (f x + e\right )^{4} - 20 \, c^{5} \cos \left (f x + e\right )^{2} + 16 \, c^{5} - {\left (c^{5} \cos \left (f x + e\right )^{4} - 12 \, c^{5} \cos \left (f x + e\right )^{2} + 16 \, c^{5}\right )} \sin \left (f x + e\right )}, x\right ) \]