Rationalize a fraction. Rationalize the denominator $\frac{1}{2^{\frac{1}{3}} + 3^{\frac{1}{3}} + 4^{\frac{1}{3}}}$. Is there a short solution for this task ? Thanks in advance.

I found the minimal polynomial for $x=\sqrt[3]{2}+\sqrt[3]{3}+\sqrt[3]{4}$
$x^9-18 x^7-27 x^6+108 x^5+162 x^4-459 x^3-972 x^2-81=0$
The I divided all terms by $81x$ and I got 
$$\frac{1}{x}=\frac{x^8}{81}-\frac{2 x^6}{9}-\frac{x^5}{3}+\frac{4 x^4}{3}+2 x^3-\frac{17 x^2}{3}-12 x$$
Then I substituted $x$ with $\sqrt[3]{2}+\sqrt[3]{3}+\sqrt[3]{4}$ in the RHS getting after some trivial simplification
$$\frac{1}{\sqrt[3]{2}+\sqrt[3]{3}+\sqrt[3]{4}}=\frac{1}{9} \left(6 \sqrt[3]{2}-3\sqrt[3]{4}-3\sqrt[3]{12}+\sqrt[3]{9}+2 \sqrt[3]{18} +3 \sqrt[3]{6}-2\sqrt[3]{36}\right)$$
Hope this helps
PS
Without a CAS it is unthinkable to simplify the fraction as I did
