Consider the following Diophantine equation
$$z^3 = 3(x^3 +y^3+2xyz)$$
Is there any elementary proof for the non solubility in positive integers for this Diophantine equation, where $x, y$ and $z$ are pairwise coprime positive integers?
I have proved the impossibility of solution of this Diophantine equation in positive integers, but that took quite a few pages and I consider this too long. I hope for a much more elementary and shorter proof for this puzzle.
Hint: It should be noted that is also true for the non-availability of the similar form (replacing coefficient 3 by 1) as the following:
$$z^3=x^3+y^3+2xyz,$$ but this case is very simple to prove, where $x, y\;\&\; z$ are nonzero integers.