\[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{b^2+a x^2} \, dx \]
Optimal antiderivative \[ \frac {2 \arctan \left (\frac {\sqrt {a}\, x}{\sqrt {b}\, \sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}\right )}{\sqrt {a}\, \sqrt {b}} \]
command
Integrate[Sqrt[b + Sqrt[b^2 + a*x^2]]/(b^2 + a*x^2),x]
Mathematica 13.1 output
\[ \frac {2 \text {ArcTan}\left (\frac {\sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a} \sqrt {b}} \]
Mathematica 12.3 output
\[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{b^2+a x^2} \, dx \]________________________________________________________________________________________