For the positive reals $a,b$ and $c$ so that $a+b+c=3$. Show that $$27+7\left(ab+bc+ca\right)\le 8\left(\sqrt{a+3b}+\sqrt{b+3c}+\sqrt{c+3a}\right)$$

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That inequality has been created by Imad Zak, i think it is interesting problem.

We can see that $$\text{L.H.S}=3(a+b+c)^2+7(ab+bc+ca)$$

$$=3(a^2+b^2+c^2)+13(ab+bc+ca)$$

$$=\sum\_{cyc} (a+3b)(b+3c)$$

Let $\sqrt{a+3b}=x;\sqrt{b+3c}=y$ and $z=\sqrt{c+3a}$ where $x,y,z>0$ and $x^2+y^2+z^2=12$

We prove $$8(x+y+z)\ge x^2y^2+y^2z^2+z^2x^2$$

I am stuck here. I tried to homogenize the last inequality but i only get wrong inequalities. I also used C-S or Holder but failed.