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