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