How find this $3\sqrt{x^2+y^2}+5\sqrt{(x-1)^2+(y-1)^2}+\sqrt{5}(\sqrt{(x-1)^2+y^2}+\sqrt{x^2+(y-1)^2})$ find this follow minimum
$$3\sqrt{x^2+y^2}+5\sqrt{(x-1)^2+(y-1)^2}+\sqrt{5}\left(\sqrt{(x-1)^2+y^2}+\sqrt{x^2+(y-1)^2}\right)$$
I guess  This minimum is $6\sqrt{2}$
But I can't prove,Thank you 

Too long for a comment, I wonder if the problem can be solved geometrically.
Let $A,B, C, D$ be the points of coordinates $(0,0), (1,1), (1,0)$ and $(0,1)$ and let $P$ be the point in plane of coordinates $(x,y)$.
The problem asks you to find the minimum of 
$$3PA+5PB+\sqrt{5}(PC+PD) (*)$$
The issue now is that we cannot split the problem in two halves: $\sqrt{5}(PC+PD) \geq \sqrt{5} CD$ with equality if and only if $P \in (C,D)$, but $3PA+5PB \geq 3AB$ with equality if and only if $P=B$. 
But one might be able to find the min of $(*)$ geometrically...Or maybe physically (put four objects of weights $3,5,\sqrt{5}, \sqrt{5}$ at $A,B,C,D$ then $P$ must have some physical meaning).
