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