$\mathrm{I}=\int_1^3 \sqrt{3+\mathrm{x}+\mathrm{x}^2} \mathrm{dx}=\int_1^3 \sqrt{\left(\mathrm{x}+\frac{1}{2}\right)^2+\left(\frac{\sqrt{11}}{2}\right)^2} \mathrm{dx}$
$$
\begin{gathered}
=\left[\frac{\left(x+\frac{1}{2}\right)}{2} \sqrt{x^2+x+3}+\frac{11}{4 \times 2} \log \left(x+\frac{1}{2}\right)+\sqrt{x^2+x+3} \mid\right]_1^3 \\
=\left[\frac{7}{4} \sqrt{15}+\frac{11}{8} \log \left|\frac{7}{2}+\sqrt{15}\right|-\frac{3}{4} \sqrt{5}-\frac{11}{8} \log \left|\frac{3}{2}+\sqrt{5}\right|\right] \\
=\frac{1}{4}[7 \sqrt{15}-3 \sqrt{5}]+\frac{11}{8}\left[\log \left(\frac{7+2 \sqrt{15}}{3+2 \sqrt{5}}\right)\right] \\
=\frac{\sqrt{5}}{4}[7 \sqrt{3}-3]+\frac{11}{8} \log \left[\frac{7+2 \sqrt{15}}{3+2 \sqrt{5}}\right] \\
=5.100066+0.9344 \Rightarrow I-6.0344 \in(2 \sqrt{5}, 2 \sqrt{15})
\end{gathered}
$$