7. In the expansion of $(2+\sqrt{x})^{2 n+1}$, the sum of the coefficients of the terms where the power of $x$ is an integer is
7. $\frac{3^{2 n+1}+1}{2}$.

Notice, $T_{r+1}=\mathrm{C}_{2 n+1}^{r} x^{\frac{2 n+1-r}{2}} 2^{r}$.
Since the power index of $x$ is an integer, $r$ must be odd.
$$
\begin{array}{l}
\text { Let } S=\mathrm{C}_{2 n+1}^{1} 2+\mathrm{C}_{2 n+1}^{3} 2^{3}+\cdots+\mathrm{C}_{2 n+1}^{2 n+1} 2^{2 n+1} . \\
\text { Also, }(1+2)^{2 n+1} \\
=\mathrm{C}_{2 n+1}^{0}+\mathrm{C}_{2 n+1}^{1} 2+\mathrm{C}_{2 n+1}^{2} 2^{2}+\cdots+\mathrm{C}_{2 n+1}^{2 n+1} 2^{2 n+1}, \\
(1-2)^{2 n+1} \\
=\mathrm{C}_{2 n+1}^{0}-\mathrm{C}_{2 n+1}^{1} 2+\mathrm{C}_{2 n+1}^{2} 2^{2}-\cdots-\mathrm{C}_{2 n+1}^{2 n+1} 2^{2 n+1},
\end{array}
$$

Subtracting the above two equations gives $S=\frac{3^{2 n+1}+1}{2}$.