\[ \int \frac {1}{x^2 \sqrt [3]{-1-x+5 x^2+2 x^3-10 x^4+2 x^5+7 x^6-5 x^7+x^8}} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[1/(x^2*(-1 - x + 5*x^2 + 2*x^3 - 10*x^4 + 2*x^5 + 7*x^6 - 5*x^7 + x^8)^(1/3)),x]
Mathematica 13.1 output
\[ -\frac {\left (-1-x+x^2\right ) \left (-6+6 x+2 \sqrt {3} (-1+x)^{2/3} x \text {ArcTan}\left (\frac {1-2 \sqrt [3]{-1+x}}{\sqrt {3}}\right )-2 (-1+x)^{2/3} x \log \left (1+\sqrt [3]{-1+x}\right )+(-1+x)^{2/3} x \log \left (1-\sqrt [3]{-1+x}+(-1+x)^{2/3}\right )+6 (-1+x)^{2/3} x \text {RootSum}\left [-1+\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log \left (\sqrt [3]{-1+x}-\text {$\#$1}\right )+\log \left (\sqrt [3]{-1+x}-\text {$\#$1}\right ) \text {$\#$1}^3}{\text {$\#$1}^2+2 \text {$\#$1}^5}\&\right ]\right )}{6 x \sqrt [3]{(-1+x)^2 \left (-1-x+x^2\right )^3}} \]
Mathematica 12.3 output
\[ \int \frac {1}{x^2 \sqrt [3]{-1-x+5 x^2+2 x^3-10 x^4+2 x^5+7 x^6-5 x^7+x^8}} \, dx \]________________________________________________________________________________________