\[ \int \frac {\sqrt {b+a^2 x^4}}{\sqrt {a x^2+\sqrt {b+a^2 x^4}}} \, dx \]
Optimal antiderivative \[ \frac {b x}{8 \left (a \,x^{2}+\sqrt {a^{2} x^{4}+b}\right )^{\frac {3}{2}}}+\frac {x \sqrt {a \,x^{2}+\sqrt {a^{2} x^{4}+b}}}{4}+\frac {5 \sqrt {b}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {a}\, x \sqrt {a \,x^{2}+\sqrt {a^{2} x^{4}+b}}}{\sqrt {b}}\right ) \sqrt {2}}{16 \sqrt {a}} \]
command
Integrate[Sqrt[b + a^2*x^4]/Sqrt[a*x^2 + Sqrt[b + a^2*x^4]],x]
Mathematica 13.1 output
\[ \frac {1}{16} \left (\frac {2 x \left (b+2 \left (a x^2+\sqrt {b+a^2 x^4}\right )^2\right )}{\left (a x^2+\sqrt {b+a^2 x^4}\right )^{3/2}}+\frac {5 \sqrt {2} \sqrt {b} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b}}\right )}{\sqrt {a}}\right ) \]
Mathematica 12.3 output
\[ \int \frac {\sqrt {b+a^2 x^4}}{\sqrt {a x^2+\sqrt {b+a^2 x^4}}} \, dx \]________________________________________________________________________________________