Let the domain of the function \(\mathrm{f}(\mathrm{x})={\log }_{2}{\log }_{4}{\log }_{6}\left(3+4\mathrm{x}-{\mathrm{x}…

Let the domain of the function \(\mathrm{f}(\mathrm{x})={\log }_{2}{\log }_{4}{\log }_{6}\left(3+4\mathrm{x}-{\mathrm{x}}^{2}\right)\) be \((a,b)\). If \({\int }_{0}^{\mathrm{b}-\mathrm{a}}\left[{\mathrm{x}}^{2}\right]\mathrm{dx}\) \(=\mathrm{p}-\sqrt{\mathrm{q}}-\sqrt{\mathrm{r}},\mathrm{p},\mathrm{q},\mathrm{r}\in \mathrm{ℕ},\gcd (\mathrm{p},\mathrm{q},\mathrm{r})=1\), where \([\cdot ]\) is the greatest integer function, then \(\mathrm{p}+\mathrm{q}+\mathrm{r}\) is equal to

[JEE Main 2025, 3 Apr (Shift 1)]