\[ \int \sin ^{\frac {5}{2}}(a+b x) \, dx \]
Optimal antiderivative \[ -\frac {6 \sqrt {\frac {1}{2}+\frac {\sin \left (b x +a \right )}{2}}\, \EllipticE \left (\cos \left (\frac {a}{2}+\frac {\pi }{4}+\frac {b x}{2}\right ), \sqrt {2}\right )}{5 \sin \left (\frac {a}{2}+\frac {\pi }{4}+\frac {b x}{2}\right ) b}-\frac {2 \cos \left (b x +a \right ) \left (\sin ^{\frac {3}{2}}\left (b x +a \right )\right )}{5 b} \]
command
integrate(sin(b*x+a)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right )^{\frac {3}{2}} - 3 i \, \sqrt {2} \sqrt {-i} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + 3 i \, \sqrt {2} \sqrt {i} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right )}{5 \, b} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-{\left (\cos \left (b x + a\right )^{2} - 1\right )} \sqrt {\sin \left (b x + a\right )}, x\right ) \]