\[ \int \frac {-3 b+a x^2}{\sqrt [4]{3 b-2 a x^2} \left (3 b-2 a x^2+3 x^4\right )} \, dx \]
Optimal antiderivative \[ -\frac {\arctan \left (\frac {\frac {3^{\frac {1}{4}} x^{2} \sqrt {2}}{2}-\frac {\sqrt {-2 a \,x^{2}+3 b}\, \sqrt {2}\, 3^{\frac {3}{4}}}{6}}{x \left (-2 a \,x^{2}+3 b \right )^{\frac {1}{4}}}\right ) \sqrt {2}\, 3^{\frac {3}{4}}}{12}-\frac {\arctanh \left (\frac {\sqrt {2}\, 3^{\frac {1}{4}} x \left (-2 a \,x^{2}+3 b \right )^{\frac {1}{4}}}{\sqrt {3}\, x^{2}+\sqrt {-2 a \,x^{2}+3 b}}\right ) \sqrt {2}\, 3^{\frac {3}{4}}}{12} \]
command
Integrate[(-3*b + a*x^2)/((3*b - 2*a*x^2)^(1/4)*(3*b - 2*a*x^2 + 3*x^4)),x]
Mathematica 13.1 output
\[ -\frac {\text {ArcTan}\left (\frac {3 x^2-\sqrt {9 b-6 a x^2}}{\sqrt {2} 3^{3/4} x \sqrt [4]{3 b-2 a x^2}}\right )+\tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{9 b-6 a x^2}}{\sqrt {3} x^2+\sqrt {3 b-2 a x^2}}\right )}{2 \sqrt {2} \sqrt [4]{3}} \]
Mathematica 12.3 output
\[ \int \frac {-3 b+a x^2}{\sqrt [4]{3 b-2 a x^2} \left (3 b-2 a x^2+3 x^4\right )} \, dx \]________________________________________________________________________________________