Inequality - GM, AM, HM and SM means I've got stuck at this problem :

Prove that for any $a > 0$ and any $b > 0$ the following inequality is true:
  $$
{3} {\left(\frac{a^3}{b^3} + \frac{b^3}{a^3}\right)} \geq \frac{a}{b} +\frac{b}{a} + 4$$

The first thing that I've thought was the AM-GM inequality (the extended version - heard that is also known as The power mean inequality):
$$
HM \leq GM \leq AM \leq SM
$$
where $HM$, $GM$, $AM$, and $SM$ refer to the harmonic, geometric, arithmetic, and square mean, respectively. CBS(Cauchy - Buniakowsky -  Schwartz) also come to my mind, but I think it isn't helpful in this case.
I would be greatful for some hints.
Thanks!

Let $ x = \frac{a}b+\frac{b}a \ge 2$ (by AM-GM).  Then
$$x^3=\frac{a^3}{b^3}+\frac{b^3}{a^3} + 3x$$
So we want to show, for $x \ge 2$, 
$$3(x^3-3x) \ge x+4 \iff (x-2)(3x^2+6x+2) \ge 0$$
which is obvious.
