Inequality Proof: $\frac{a(a^2-b^2)}{13a^2+5b^2} + \frac{b(b^2-c^2)}{13b^2+5c^2} + \frac{c(c^2-a^2)}{13c^2+5a^2} \geq 0$

Question

Question

Prove that for $a,b,c \geq 0$, we have $$\frac{a(a^2-b^2)}{13a^2+5b^2} + \frac{b(b^2-c^2)}{13b^2+5c^2} + \frac{c(c^2-a^2)}{13c^2+5a^2} \geq 0$$

Attempts

I've tried the proving using:

• Cauchy's inequality
• AM-GM inequality
• Trying to find a way for Jensen's inequality

but I don't get anywhere.

Note

I know this might be done using critical points and verifying that the minimum is positive, but the calculations get really messy. I'm just wondering if there's a clean way of proving it.