Using Cardano's method to find an algebraic equation whose root is $\sqrt{2} +\sqrt[3]{3}$ $\sqrt{2} +\sqrt[3]{3}$ solution of an algebraic equation $ a_{0}x^{n}+a_{1}x^{n-1}+a_{2}x^{n-2}+\cdots+a_{0}=0 $
How to find this equation?
I tried Cardano's method, noting that 
$$\sqrt{2} +\sqrt[3]{3} = \sqrt[3]{\sqrt{8}} +\sqrt[3]{3}$$ 
It means
$$\begin{align}
-\frac{q}{2}+\sqrt{ \frac{q^{2}}{4}+\frac{p^{3}}{27}}&=3 \\[4pt]
-\frac{q}{2}-\sqrt{ \frac{q^{2}}{4}+\frac{p^{3}}{27}}&=\sqrt{8} 
\end{align}$$
but this system doesn't have natural solutions.
Maybe it has rational $p$ and $q$?  

Let $x=\sqrt 2 + \sqrt[3]{3}$. Then,
$$
x-\sqrt 2 = \sqrt[3]{3}
$$
Cubing both sides and rearranging we have,
$$
x^3+6x-3=\sqrt 2 (3x^2+2)
$$
Squaring both sides and rearranging we have,
$$
x^6-6x^4-6x^3+12x^2-36x+1=0
$$
