<s>[INST] 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. 
 [/INST] your inequalitiy is equivalent to
$$a^4c+b^4a+c^4b-a^2b^2c-a^2bc^2-ab^2c^2\geq 0$$
this is equivalent to
$$(a^2-b^2)(a^2c-bc^2)+(b^2-c^2)(ab^2-bc^2)\geq 0$$ we assume that $$a\geq b\geq c$$ thus $$a^2\geq b^2$$ and $$a^2\geq bc$$ and $$b^2\geq c^2$$ and $$ab\geq c^2$$
in the case $$a\geq c\geq b$$ we have
$$(a^2-c^2)(a^2c-c^2b)+(a^2c-b^2a)(c^2-b^2)\geq 0$$
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