\[ \int \left (a \sin ^3(x)\right )^{5/2} \, dx \]
Optimal antiderivative \[ -\frac {26 a^{2} \cot \left (x \right ) \sqrt {a \left (\sin ^{3}\left (x \right )\right )}}{77}-\frac {26 a^{2} \sqrt {\frac {1}{2}+\frac {\sin \left (x \right )}{2}}\, \EllipticF \left (\cos \left (\frac {\pi }{4}+\frac {x}{2}\right ), \sqrt {2}\right ) \sqrt {a \left (\sin ^{3}\left (x \right )\right )}}{77 \sin \left (\frac {\pi }{4}+\frac {x}{2}\right ) \sin \left (x \right )^{\frac {3}{2}}}-\frac {78 a^{2} \cos \left (x \right ) \sin \left (x \right ) \sqrt {a \left (\sin ^{3}\left (x \right )\right )}}{385}-\frac {26 a^{2} \cos \left (x \right ) \left (\sin ^{3}\left (x \right )\right ) \sqrt {a \left (\sin ^{3}\left (x \right )\right )}}{165}-\frac {2 a^{2} \cos \left (x \right ) \left (\sin ^{5}\left (x \right )\right ) \sqrt {a \left (\sin ^{3}\left (x \right )\right )}}{15} \]
command
integrate((a*sin(x)^3)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {195 \, \sqrt {2} \sqrt {-i \, a} a^{2} \sin \left (x\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) + 195 \, \sqrt {2} \sqrt {i \, a} a^{2} \sin \left (x\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right ) + 2 \, {\left (77 \, a^{2} \cos \left (x\right )^{7} - 322 \, a^{2} \cos \left (x\right )^{5} + 530 \, a^{2} \cos \left (x\right )^{3} - 480 \, a^{2} \cos \left (x\right )\right )} \sqrt {-{\left (a \cos \left (x\right )^{2} - a\right )} \sin \left (x\right )}}{1155 \, \sin \left (x\right )} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-{\left (a^{2} \cos \left (x\right )^{6} - 3 \, a^{2} \cos \left (x\right )^{4} + 3 \, a^{2} \cos \left (x\right )^{2} - a^{2}\right )} \sqrt {-{\left (a \cos \left (x\right )^{2} - a\right )} \sin \left (x\right )}, x\right ) \]