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