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