If $\int \frac{2x^{2} + 5x + 9}{\sqrt{x^{2} + x + 1}} \, \mathrm{d}x = x\sqrt{x^{2} + x + 1} + \alpha\sqrt{x^{2} + x + 1} + \beta \log_{e}\left| x + \frac{1}{2} + \sqrt{x^{2} + x + 1} \right| + \mathrm{C}$, where $C$ is the constant of integration, then $\alpha + 2\beta$ is equal to $\_\_\_\_$.