Find the minimum of $(x(1+y)+y(1+z)+z(1+x))/\sqrt{xyz}$ over positive integers $x,y,z$ Let $x,y,z$ be positive integers.The least value 
$$\frac{x(1+y)+y(1+z)+z(1+x)}{(xyz)^{\frac 12}}$$
I tried sum using arithmetic-geometric means inequality (seems promising as the denominator is similar to geometric mean of $x,y,z$). However, so far this didn't help my cause.
Notice

$$\frac{x(1+y)+y(1+z)+z(1+x)}{(xyz)^{\frac 12}} = \frac{x + xy + y + yz + z + zx}{(xyz)^{1/2}} \geq \frac{ 6 (x^3y^3z^3)^{1/6}}{(xyz)^{1/2}} = 6$$

By the AM-GM inequality
Take partial derivatives and then solve for them = $0$.