\(\int \frac{2{x}^{2}+5x+9}{\sqrt{{x}^{2}+x+1}}dx=x\sqrt{{x}^{2}+x+1}+\alpha \sqrt{{x}^{2}+x+1}+\beta \ln \left(x+\frac{…

\(\int \frac{2{x}^{2}+5x+9}{\sqrt{{x}^{2}+x+1}}dx=x\sqrt{{x}^{2}+x+1}+\alpha \sqrt{{x}^{2}+x+1}+\beta \ln \left(x+\frac{1}{2}+\sqrt{{x}^{2}+x+1}\right)+c\)

then \(\alpha +4\beta\) is equal to

[Jee Main 2025]