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