Prove that $\sqrt{x^{3} + y^{3}+1} + \sqrt{z^{3} + y^{3}+1} + \sqrt{x^{3} + z^{3}+1} \ge 2 + \sqrt{2(x^{3} + y^{3}+z^{3})+1}$ Given $x, y,z \ge 0$. Prove that 
$$\sqrt{x^{3} + y^{3}+1} + \sqrt{z^{3} + y^{3}+1}  + \sqrt{x^{3} + z^{3}+1} \ge 2 + \sqrt{2(x^{3} + y^{3}+z^{3})+1} $$

Attempt
Notice that $$(x^{3} + y^{3}+1) + (z^{3} + y^{3}+1)  + (x^{3} + z^{3}+1) = 2 + 2(x^{3} + y^{3}+z^{3})+1  $$
So the sum of squares must be between two forms as below:
$$2 + \sqrt{2(x^{3} + y^{3}+z^{3})+1} \le \sqrt{x^{3} + y^{3}+1} + \sqrt{z^{3} + y^{3}+1}  + \sqrt{x^{3} + z^{3}+1}  \le 2 + 2(x^{3} + y^{3}+z^{3})+1$$
If I work backwards I will get, by letting the form in each root be $a,b,c$ respectively:
$$\sqrt{a} + \sqrt{b}  + \sqrt{c} \ge 2 + \sqrt{a+b+c-2} $$
$$ a + b  +c + 2(\sqrt{ab} + \sqrt{bc}  + \sqrt{ca}) \ge 4 + (a+b+c-2) + 4\sqrt{a+b+c-2}$$
$$  (\sqrt{ab} + \sqrt{bc}  + \sqrt{ca}) \ge 1 + 2\sqrt{a+b+c-2}$$
$$  ((\sqrt{ab} + \sqrt{bc}  + \sqrt{ca}) -1))^{2} \ge  4(a+b+c-2)$$
$$ (ab + bc  + ca + 2a^{2}\sqrt{bc} +2b^{2}\sqrt{ac} +2c^{2}\sqrt{ab}) - 2(\sqrt{ab} + \sqrt{bc}  + \sqrt{ca}) + 1 \ge  4(a+b+c-2)$$
How to continue? 

Apart from the attempt above, I have tried: 


*

*AM-GM for the sum of the 3 squares

*AM-GM for the sum of the 3 squares, then using Holder's inequality from the GM

*AM-GM for each square

The inequality $$\sum_{cyc}\sqrt{a+1}\geq2+\sqrt{a+b+c+1}$$ we can prove also by the following way:
$$\sum_{cyc}\sqrt{a+1}-2-\sqrt{a+b+c+1}=$$
$$=\sqrt{a+1}+\sqrt{b+1}-2-\left(\sqrt{a+b+c+1}-\sqrt{c+1}\right)=$$
$$=\sqrt{a+1}+\sqrt{b+1}-2-\frac{a+b}{\sqrt{a+b+c+1}+\sqrt{c+1}}\geq$$
$$\geq\sqrt{a+1}+\sqrt{b+1}-2-\frac{a+b}{\sqrt{a+b+1}+1}=$$
$$=\sqrt{a+1}+\sqrt{b+1}-2-(\sqrt{a+b+1}-1)=$$
$$=\sqrt{a+1}-1-\left(\sqrt{a+b+1}-\sqrt{b+1}\right)=$$
$$=\sqrt{a+1}-1-\frac{a}{\sqrt{a+b+1}+\sqrt{b+1}}\geq\sqrt{a+1}-1-\frac{a}{\sqrt{a+1}+1}=0.$$
