<s>[INST] Proving that $\frac{ab}{c^3}+\frac{bc}{a^3}+\frac{ca}{b^3}> \frac{1}{a}+\frac{1}{b}+\frac{1}{c}$ 
Prove that $\dfrac{ab}{c^3}+\dfrac{bc}{a^3}+\dfrac{ca}{b^3}> \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}$, where $a,b,c$  are different positive real numbers.

First, I tried to simplify the proof statement but I got an even more complicated: $$a^4b^4+b^4c^4+a^4c^4> a^2b^3c^3+b^2c^3a^3+a^3b^3c^2$$
Then I used Power mean inequality on $\dfrac{1}{a},\dfrac{1}{b},\dfrac {1}{c}$ but that wasn't useful here.
Finally, I tried to solve it using AM-HM inequality but couldn't.
What would be an efficient method to solve this problem? Please provide only a hint and not the entire solution since I wish to solve it myself.
 [/INST] \begin{eqnarray*}
a^4(b^2-c^2)^2+b^4(a^2-c^2)^2+c^4(b^2-a^2)^2+a^2b^2c^2((a-b)^2+(c-b)^2+(a-c)^2) \geq 0.
\end{eqnarray*}
Rearrange to 
\begin{eqnarray*}
a^4b^4+b^4c^4+c^4a^4 \geq a^2b^2c^2(ab+bc+ca).
\end{eqnarray*}
Now divide by $a^3b^3c^3$ and we have
\begin{eqnarray*}
\frac{ab}{c^3}+\frac{bc}{a^3}+\frac{ca}{ b^3}  \geq \frac{1}{a}+\frac{1}{b}+\frac{1}{c}.
\end{eqnarray*}
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