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