By Cauchy's inequality, we have
$$\begin{aligned} & 2\sqrt{x+1} + \sqrt{2x-3} + \sqrt{15-3x} \\ &= \sqrt{x+1} + \sqrt{x+1} + \sqrt{2x-3} + \sqrt{15-3x} \\ &\le \sqrt{[(x+1) + (x+1) + (2x-3) + (15-3x)](1^2 + 1^2 + 1^2 + 1^2)} \\ &= 2\sqrt{x+14} \le 2\sqrt{19}, \end{aligned}$$
and the equality holds if and only if $\sqrt{x+1} = \sqrt{2x-3} = \sqrt{15-3x}$ and $x = 5$. But this is impossible. So $2\sqrt{x+1} + \sqrt{2x-3} + \sqrt{15-3x} < 2\sqrt{19}$.