\[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{x^2} \, dx \]
Optimal antiderivative \[ -\frac {\sqrt {a \,x^{2}+\sqrt {a^{2} x^{4}+b}}}{x}+\frac {\sqrt {a}\, \ln \left (i a \,x^{2}+i \sqrt {a^{2} x^{4}+b}+i \sqrt {2}\, \sqrt {a}\, x \sqrt {a \,x^{2}+\sqrt {a^{2} x^{4}+b}}\right ) \sqrt {2}}{2} \]
command
Integrate[Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]/x^2,x]
Mathematica 13.1 output
\[ -\frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{x}+\frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {2} \sqrt {a} x}\right )}{\sqrt {2}} \]
Mathematica 12.3 output
\[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{x^2} \, dx \]________________________________________________________________________________________