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