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