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