5. The sum of the coefficients of the integer power terms of $x$ in the expansion of $(\sqrt{x}+2)^{2 n+1}$ is ( ).
(A) $\frac{1}{2}\left(3^{2 n+1}+1\right)$
(B) $\frac{1}{2}\left(2^{2 n+1}+1\right)$
(C) $\frac{1}{3}\left(3^{2 n+1}+1\right)$
(D) $\frac{1}{3}\left(2^{2 n+1}+1\right)$
5. A.

Let $A=(\sqrt{x}+2)^{2 n+1}, B=(-\sqrt{x}+2)^{2 n+1}$.
By the binomial theorem, the sum of the integer power terms of $x$ in $A$ and $B$ is the same, denoted as $f(x)$, and the sum of the non-integer power terms are opposite numbers, which cancel each other out when added. Therefore, we have
$$
2 f(x)=(\sqrt{x}+2)^{2 n+1}+(-\sqrt{x}+2)^{2 n+1} \text {. }
$$

Let $x=1$, then the sum of the coefficients is
$$
f(1)=\frac{1}{2}\left(3^{2 n+1}+1\right) \text {. }
$$