Conditional inequality

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Let $a$, $b$, $c$ are positive real numbers and $abc=1$. Prove that
$\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b} \geq \frac{a}{b^2+c^2}+\frac{b}{c^2+a^2}+\frac{c}{a^2+b^2}$.

What I would like to know is it а well known problem?