\[ \int \frac {2 b+a x^2}{\sqrt [4]{b+a x^2} \left (b n+a n x^2+2 x^4\right )} \, dx \]
Optimal antiderivative \[ \frac {\arctan \left (\frac {\frac {x^{2} 2^{\frac {3}{4}} 3155^{\frac {3}{4}}}{6310}-\frac {3155^{\frac {1}{4}} \sqrt {a \,x^{2}+b}\, 2^{\frac {1}{4}}}{2}}{x \left (a \,x^{2}+b \right )^{\frac {1}{4}}}\right ) 2^{\frac {1}{4}} 3155^{\frac {1}{4}}}{6310}+\frac {\arctanh \left (\frac {\frac {x^{2} 2^{\frac {3}{4}} 3155^{\frac {3}{4}}}{6310}+\frac {3155^{\frac {1}{4}} \sqrt {a \,x^{2}+b}\, 2^{\frac {1}{4}}}{2}}{x \left (a \,x^{2}+b \right )^{\frac {1}{4}}}\right ) 2^{\frac {1}{4}} 3155^{\frac {1}{4}}}{6310} \]
command
Integrate[(2*b + a*x^2)/((b + a*x^2)^(1/4)*(b*n + a*n*x^2 + 2*x^4)),x]
Mathematica 13.1 output
\[ \frac {\text {ArcTan}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{n} \sqrt [4]{b+a x^2}}-\frac {\sqrt [4]{n} \sqrt [4]{b+a x^2}}{2^{3/4} x}\right )+\tanh ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{n} \sqrt [4]{b+a x^2}}+\frac {\sqrt [4]{n} \sqrt [4]{b+a x^2}}{2^{3/4} x}\right )}{2^{3/4} n^{3/4}} \]
Mathematica 12.3 output
\[ \int \frac {2 b+a x^2}{\sqrt [4]{b+a x^2} \left (b n+a n x^2+2 x^4\right )} \, dx \]________________________________________________________________________________________