While working on some research, I have encountered an infinite series and its improper integral analogue:

\begin{align}\sum_{m=1}^{\infty}\frac1{\sqrt{m(m+1)(m+2)+\sqrt{m^3(m+2)^3}}}&=\frac12+\frac1{\sqrt{2}}, \\ \int_0^{\infty}\frac{dx}{\sqrt{x(x+1)(x+2)+\sqrt{x^3(x+2)^3}}}&=2.\end{align} The evaluations were guessed using numerical evidence.

Can you provide proofs, or any reference (if available)?