Show that $\sum_{cyc}\frac{\sqrt{x}\left(\frac{1}{x}+y\right)}{\sqrt{1+x^2}}\ge 3\sqrt{2}$

Question

$x,y,z$ are positive reals, show that $$\sum_{cyc}\frac{\sqrt{x}\left(\frac{1}{x}+y\right)}{\sqrt{1+x^2}} \ge 3\sqrt{2} $$ Here is my approach $$\sum_{cyc}\frac{\sqrt{x}\left(\frac{1}{x}+y\right)}{\sqrt{1+x^2}} = \sum_{cyc}\frac{xy+1}{\sqrt{x^3+x}} $$ I tried to use Hölder with weights $p=2$ and $ q=1$, $$\left(\sum_{cyc}\frac{xy+1}{\sqrt{x^3+x}}\right)^2\left(\sum_{cyc}(x^3+x)(xy+1) \right)\ge \left(\sum_{cyc}xy+1\right)^3 $$ The next step is to prove that the quotient of the second term in the LHS and the term in the RHS is $\ge 18$ .But the terms are so messy I can’t even work with them, what should I do? Is the use of Holder the right thing to do?