The diagram shows the curve \(y=\frac{1}{\sqrt{x^{2}+x+1}}\) for \(x \geqslant 0\), together with a set of \(n\) rectangles of unit width. By considering the sum of the areas of these rectangles, show that
\[\sum_{r=1}^{n} \frac{1}{\sqrt{r^{2}+r+1}}<\ln \left(\frac{1}{3}+\frac{2}{3} n+\frac{2}{3} \sqrt{n^{2}+n+1}\right) .\]