\[ \int \frac {b+2 a x^4}{\left (-b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[(b + 2*a*x^4)/((-b + a*x^4)*(b*x^2 + a*x^4)^(1/4)),x]
Mathematica 13.1 output
\[ \frac {\sqrt {x} \sqrt [4]{b+a x^2} \left (8 \left (\text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )+\tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )\right )+3 \sqrt [4]{a} \text {RootSum}\left [a^2-a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right )+\log \left (\sqrt [4]{b+a x^2}-\sqrt {x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{4 \sqrt [4]{a} \sqrt [4]{x^2 \left (b+a x^2\right )}} \]
Mathematica 12.3 output
\[ \int \frac {b+2 a x^4}{\left (-b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx \]________________________________________________________________________________________