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)?



sum, integral of certain functions
For the integral, notice that the expression under the square root is
 $$ x(x+1)(x+2)+x(x+2)\sqrt{x(x+2)} = \frac12\,x(x+2)(\sqrt x+\sqrt{x+2})^2. $$
Consequently,
\begin{align\*} 
 \frac1{\sqrt{x(x+1)(x+2)+x(x+2)\sqrt{x(x+2)}}} 
 &= \frac{\sqrt 2}{(\sqrt x+\sqrt{x+2}) \sqrt{x(x+2)}} \\
 &= \frac1{\sqrt 2}\,\frac{\sqrt{x+2}-\sqrt{x}}{\sqrt{x(x+2)}} \\
 &= \frac1{\sqrt 2} \left( \frac1{\sqrt x}-\frac1{\sqrt{x+2}}\right);
\end{align\*}
thus, the indefinite integral is
 $$ \sqrt{2}\, (\sqrt x-\sqrt{x+2})+C $$
and the result follows easily.


As Antony Quas noticed, this also works for the sum showing that the partial sum over $m\in[1,M]$ is
 $$ \frac1{\sqrt 2} \sum\_{m=1}^M \frac1{\sqrt m} - \frac1{\sqrt 2} \sum\_{m=3}^{M+2} \frac1{\sqrt m} = \frac1{\sqrt 2} \left( 1+\frac1{\sqrt 2}\right) + o(1). $$


