\[ \int \frac {\sqrt [4]{b x^2+a x^4} \left (-b-a x^4+x^8\right )}{-b+a x^4} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[((b*x^2 + a*x^4)^(1/4)*(-b - a*x^4 + x^8))/(-b + a*x^4),x]
Mathematica 13.1 output
\[ \frac {x^{3/2} \left (b+a x^2\right )^{3/4} \left (2 a^{3/4} x^{3/2} \sqrt [4]{b+a x^2} \left (-96 a^3-7 b^2+32 a^2 x^4+4 a b \left (24+x^2\right )\right )-3 b \left (-32 a^3+32 a b+7 b^2\right ) \text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )+3 b \left (-32 a^3+32 a b+7 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )-96 a^{7/4} \left (2 a^2-b\right ) b \text {RootSum}\left [a^2-a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right ) \text {$\#$1}+\log \left (\sqrt [4]{b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}}{-a+\text {$\#$1}^4}\&\right ]\right )}{384 a^{15/4} \left (x^2 \left (b+a x^2\right )\right )^{3/4}} \]
Mathematica 12.3 output
\[ \int \frac {\sqrt [4]{b x^2+a x^4} \left (-b-a x^4+x^8\right )}{-b+a x^4} \, dx \]________________________________________________________________________________________