Differentiation under the integral sign gives a pretty fast way. Let:

$$ I(\alpha) = \int_{0}^{+\infty}x^{\alpha}e^{-x^2}\,dx = \frac{1}{2}\int_{0}^{+\infty}x^{\frac{\alpha-1}{2}}e^{-x}\,dx = \frac{1}{2}\,\Gamma\left(\frac{\alpha+1}{2}\right).\tag{1}$$
Our integral is just $I'(0)$: since $\Gamma' = \psi\cdot\Gamma$,
$$ \int_{0}^{+\infty}e^{-x^2}\log(x)\,dx = \frac{1}{4}\Gamma\left(\frac{1}{2}\right)\psi\left(\frac{1}{2}\right)=\color{red}{-\frac{\sqrt{\pi}}{4}\left(\gamma+2\log 2\right)}.\tag{2} $$
$\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$ is well-known and $\psi\left(\frac{1}{2}\right)$ can be computed from $\psi(1)=-\gamma$ and the duplication formula for the $\psi$ function:
$$ \psi(z)+\psi\left(z+\frac{1}{2}\right) = -2\log 2+2\,\psi(2z).\tag{3} $$