We make the change of variables $x = -w$, $dx = -dw$, and change the limits of integration to obtain
$\begin{eqnarray*} \int_{-\infty}^0 \frac{x^2}{1+x^4} \log|x| \; dx &=& -\int_0^\infty \frac{w^2}{1+w^4} \log|-w| \; dw \\ &=& -\int_0^\infty \frac{w^2}{1+w^4} \log w \; dw \end{eqnarray*}$
But this is clearly wrong, since $\frac{x^2}{1+x^4} \log|x|$ is an even function. Where is the mistake?