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