Find the greatest value of this expression Let $x$, $y$, $z$ be nonnegative real numbers with $x + y + z = 3$.  Find the greatest value of the expression 
\begin{equation}
P = \sqrt{(x+1)(y^2 + 2)(z^3 + 3)} + \sqrt{(y+1)(z^2 + 2)(x^3 + 3)} + \sqrt{(z+1)(x^2 + 2)(y^3 + 3)}.
\end{equation}
I tried. We have
$$\sqrt{(x+1)(y^2 + 2)(z^3 + 3)}=\dfrac{1}{\sqrt{24}}\sqrt{(x+1)(y^2 + 2)(z^3 + 3)\cdot 24}  \leqslant \dfrac{1}{\sqrt{24}}\cdot \dfrac{(x+1)(y^2 + 2)(z^3 + 3)+24}{2}, $$
$$\sqrt{(y+1)(z^2 + 2)(x^3 + 3)}=\dfrac{1}{\sqrt{24}}\sqrt{(y+1)(z^2 + 2)(x^3 + 3)\cdot 24}  \leqslant \dfrac{1}{\sqrt{24}}\cdot \dfrac{(y+1)(z^2 + 2)(x^3 + 3)+24}{2}, $$
$$\sqrt{(z+1)(x^2 + 2)(y^3 + 3)}=\dfrac{1}{\sqrt{24}}\sqrt{(z+1)(x^2 + 2)(y^3 + 3)\cdot 24}  \leqslant \dfrac{1}{\sqrt{24}}\cdot \dfrac{(z+1)(x^2 + 2)(y^3 + 3)+24}{2}. $$
Therefore
$$P \leqslant \dfrac{1}{\sqrt{24}}\dfrac{ (x+1)(y^2 + 2)(z^3 + 3) + (y+1)(z^2 + 2)(x^3 + 3) +(z+1)(x^2 + 2)(y^3 + 3) + 72 }{2}$$
Expand the above expression, we get
\begin{align*}
P\leqslant \dfrac{1}{\sqrt{24}}(90 + 6(x+y+z) + 2(x^3 + y^3 + z^3) + 2(x^3 y + y^3 z + z^3 x)\\
3(x^2 + y^2 + z^2) + 3(xy^2 + yz^2 + zx^2) + xyz (xy^2 + yz^2 + zx^2). 
\end{align*}
I can not solve from here. 


Hint: Above is plot using Mathematica. Looks like maximum is for $(3,0,0),(0,3,0),(0,0,3)$.
