$$\displaystyle\frac{d}{d x } \left( \sqrt[ 3 ]{ x+1 } - \frac{ 1 }{ \sqrt{ { x }^{ 2 } -x+1 } } -(3 { x }^{ 2 } +1) \right)$$
$-6x+\frac{1}{3\left(x+1\right)^{\frac{2}{3}}}+\frac{x}{\left(x^{2}-x+1\right)^{\frac{3}{2}}}-\frac{1}{2\left(x^{2}-x+1\right)^{\frac{3}{2}}}$
$-6-\frac{2}{9\left(x+1\right)^{\frac{5}{3}}}-\frac{2x^{2}}{\left(x^{2}-x+1\right)^{\frac{5}{2}}}+\frac{2x}{\left(x^{2}-x+1\right)^{\frac{5}{2}}}+\frac{1}{4\left(x^{2}-x+1\right)^{\frac{5}{2}}}$
