Answer
$$ \displaystyle\int \dfrac{{x}^{2}}{{x}^{2}-x+1}\, \mathrm d x = {{\sqrt{3}\,\ln \left(x^2-x+1\right)-2\,\arctan \left({{2\,x-1 }\over{\sqrt{3}}}\right)+2\,\sqrt{3}\,x}\over{2\,\sqrt{3}}} $$
Explanation
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