\[ \int \frac {-b+a x^6}{\sqrt [3]{-x+x^3} \left (-d+c x^6\right )} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[(-b + a*x^6)/((-x + x^3)^(1/3)*(-d + c*x^6)),x]
Mathematica 13.1 output
\[ -\frac {\sqrt [3]{-1+\frac {1}{x^2}} x \left (3 a d \left (-2 \sqrt {3} \text {ArcTan}\left (\frac {1-2 \sqrt [3]{-1+\frac {1}{x^2}}}{\sqrt {3}}\right )-2 \log \left (c \left (1+\sqrt [3]{-1+\frac {1}{x^2}}\right )\right )+\log \left (1-\sqrt [3]{-1+\frac {1}{x^2}}+\left (-1+\frac {1}{x^2}\right )^{2/3}\right )\right )+2 (b c-a d) \text {RootSum}\left [c-d-3 d \text {$\#$1}^3-3 d \text {$\#$1}^6-d \text {$\#$1}^9\&,\frac {\log \left (\sqrt [3]{-1+\frac {1}{x^2}}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{12 c d \sqrt [3]{x \left (-1+x^2\right )}} \]
Mathematica 12.3 output
\[ \int \frac {-b+a x^6}{\sqrt [3]{-x+x^3} \left (-d+c x^6\right )} \, dx \]________________________________________________________________________________________