\[ \int \frac {(a+b \sin (e+f x)) \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right )}{\sin ^{\frac {3}{2}}(e+f x)} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (b B -a \left (A -C \right )\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (f x +e \right )}{2}}\, \EllipticE \left (\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ), \sqrt {2}\right )}{\sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) f}-\frac {2 \left (3 A b +3 B a +b C \right ) \sqrt {\frac {1}{2}+\frac {\sin \left (f x +e \right )}{2}}\, \EllipticF \left (\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ), \sqrt {2}\right )}{3 \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) f}-\frac {2 a A \cos \left (f x +e \right )}{f \sqrt {\sin \left (f x +e \right )}}-\frac {2 b C \cos \left (f x +e \right ) \left (\sqrt {\sin }\left (f x +e \right )\right )}{3 f} \]
command
integrate((a+b*sin(f*x+e))*(A+B*sin(f*x+e)+C*sin(f*x+e)^2)/sin(f*x+e)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {\sqrt {2} \sqrt {-i} {\left (3 \, B a + {\left (3 \, A + C\right )} b\right )} \sin \left (f x + e\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + \sqrt {2} \sqrt {i} {\left (3 \, B a + {\left (3 \, A + C\right )} b\right )} \sin \left (f x + e\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) - 3 \, \sqrt {2} \sqrt {-i} {\left (i \, {\left (A - C\right )} a - i \, B b\right )} \sin \left (f x + e\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) - 3 \, \sqrt {2} \sqrt {i} {\left (-i \, {\left (A - C\right )} a + i \, B b\right )} \sin \left (f x + e\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) - 2 \, {\left (C b \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 3 \, A a \cos \left (f x + e\right )\right )} \sqrt {\sin \left (f x + e\right )}}{3 \, f \sin \left (f x + e\right )} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left ({\left (C a + B b\right )} \cos \left (f x + e\right )^{2} - {\left (A + C\right )} a - B b + {\left (C b \cos \left (f x + e\right )^{2} - B a - {\left (A + C\right )} b\right )} \sin \left (f x + e\right )\right )} \sqrt {\sin \left (f x + e\right )}}{\cos \left (f x + e\right )^{2} - 1}, x\right ) \]