Prove that if $a,b,$ and $c$ are positive real numbers, then $\frac{a^3}{b}+\frac{b^3}{c}+\frac{c^3}{a} \geq ab + bc + ca$. 
Prove that if $a,b,$ and $c$ are positive real numbers, then $\dfrac{a^3}{b}+\dfrac{b^3}{c}+\dfrac{c^3}{a} \geq ab + bc + ca$.

I tried AM-GM and it doesn't look like AM-GM or Cauchy-Schwarz work here. The $ab+bc+ca$ reminds of a cyclic expression, so that may help by factoring the inequality and getting a true statement. 

We have:
$$ \frac{a^4}{ab}+\frac{b^4}{bc}+\frac{c^4}{ca} ≥ ab + bc + ca $$
Also note that
$$ \frac{a^4}{ab}+\frac{b^4}{bc}+\frac{c^4}{ca} ≥ \frac{(a^2+b^2+c^2)^2}{ab+bc+ca}$$
(Because of the Cauchy Schwarz lemma or the Titu's Lemma)
So if we can prove that 
$$\frac{(a^2+b^2+c^2)^2}{ab+bc+ca} ≥ ab + bc + ca$$
Then our work is done; so
$$ (a^2+b^2+c^2)^2 ≥ (ab+bc+ca)^2$$
$$ ⇒ a^2+b^2+c^2 ≥ ab+bc+ca   \ \square$$
This can simply be proven using the rearrangement inequality.
