\[ \int \frac {b+a x^8}{\sqrt [4]{-b x^2+a x^4} \left (-b+a x^8\right )} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[(b + a*x^8)/((-(b*x^2) + a*x^4)^(1/4)*(-b + a*x^8)),x]
Mathematica 13.1 output
\[ \frac {\sqrt [4]{-a+\frac {b}{x^2}} x \left (2 \sqrt {2} \left (-\text {ArcTan}\left (\frac {-\sqrt {a}+\sqrt {-a+\frac {b}{x^2}}}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a+\frac {b}{x^2}}}\right )+\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a+\frac {b}{x^2}}}{\sqrt {a}+\sqrt {-a+\frac {b}{x^2}}}\right )\right )+\sqrt [4]{a} \text {RootSum}\left [a^4-a b^3+4 a^3 \text {$\#$1}^4+6 a^2 \text {$\#$1}^8+4 a \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {\log \left (\sqrt [4]{-a+\frac {b}{x^2}}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{4 \sqrt [4]{a} \sqrt [4]{-b x^2+a x^4}} \]
Mathematica 12.3 output
\[ \int \frac {b+a x^8}{\sqrt [4]{-b x^2+a x^4} \left (-b+a x^8\right )} \, dx \]________________________________________________________________________________________