Proof for for convergence of $\int_{-∞}^0\frac{\sqrt{\left|x\right|}}{x^2+x+1}dx$ I want to show that the following integral converges. However I am troubled finding an estimate for this term which is sufficient for the proof. Could I use L'Hospital on this one?
$$\int_{-∞}^0\frac{\sqrt{\left|x\right|}}{x^2+x+1}dx$$

$$\int_{-\infty}^{0}\frac{\sqrt{|x|}\,dx}{x^2+x+1} = \int_{0}^{+\infty}\frac{\sqrt{x}\,dx}{x^2-x+1} \stackrel{x\mapsto z^2}{=} \int_{0}^{+\infty}\frac{2z^2}{z^4-z^2+1}\,dz$$
equals (by parity)
$$\int_{-\infty}^{+\infty}\frac{dz}{z^2+\frac{1}{z^2}-1}=\int_{\mathbb{R}}\frac{dz}{\left(z-\frac{1}{z}\right)^2+1}\stackrel{\text{GMT}}{=}\int_{\mathbb{R}}\frac{dz}{z^2+1}=\color{red}{\pi}<+\infty. $$
$\text{GMT}$ stands for Glasser's Master Theorem.
