6. Given the sequence $\left\{a_{n}\right\}$ with the general term formula
$$
a_{n}=\frac{1}{(n+1) \sqrt{n}+n \sqrt{n+1}}\left(n \in \mathbf{N}_{+}\right) \text {, }
$$

and its partial sum $S_{n}$. Then, in the sequence $S_{1}, S_{2}, \cdots, S_{2008}$, the number of rational terms is ( ).
(A) 43
(B) 44
(C) 45
(D) 46
6.A.
$$
\begin{array}{l}
\text { Given } a_{k}=\frac{1}{\sqrt{k(k+1)}(\sqrt{k+1}+\sqrt{k})} \\
=\frac{\sqrt{k+1}-\sqrt{k}}{\sqrt{k(k+1)}}=\frac{1}{\sqrt{k}}-\frac{1}{\sqrt{k+1}}(k=1,2, \cdots),
\end{array}
$$

then $S_{n}=\sum_{k=1}^{n} a_{k}=\sum_{k=1}^{n}\left(\frac{1}{\sqrt{k}}-\frac{1}{\sqrt{k+1}}\right)$
$$
=1-\frac{1}{\sqrt{n+1}} \text {. }
$$

Also, $\sqrt{2009}<45$, so $n+1=2^{2}, 3^{2}, \cdots, 44^{2}$.
Therefore, among $S_{1}, S_{2}, \cdots, S_{2008}$, there are 43 rational terms.