\[ \int \frac {\sqrt [3]{x+x^3} \left (b+a x^6\right )}{d+c x^6} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[((x + x^3)^(1/3)*(b + a*x^6))/(d + c*x^6),x]
Mathematica 13.1 output
\[ \frac {\sqrt [3]{x+x^3} \left (a d \left (6 x^{4/3} \sqrt [3]{1+x^2}-2 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{1+x^2}}\right )-2 \log \left (c \left (-x^{2/3}+\sqrt [3]{1+x^2}\right )\right )+\log \left (x^{4/3}+x^{2/3} \sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right )\right )+(-2 b c+2 a d) \text {RootSum}\left [c-d+3 d \text {$\#$1}^3-3 d \text {$\#$1}^6+d \text {$\#$1}^9\&,\frac {-2 \log \left (\sqrt [3]{x}\right ) \text {$\#$1}+\log \left (\sqrt [3]{1+x^2}-x^{2/3} \text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^3}\&\right ]\right )}{12 c d \sqrt [3]{x} \sqrt [3]{1+x^2}} \]
Mathematica 12.3 output
\[ \int \frac {\sqrt [3]{x+x^3} \left (b+a x^6\right )}{d+c x^6} \, dx \]________________________________________________________________________________________