(5) Given the sequence $\left\{a_{n}\right\}$ with the general term formula $a_{n}=\frac{1}{(n+1) \sqrt{n}+n \sqrt{n+1}}$ $\left(n \in \mathbf{N}^{*}\right)$, and its first $n$ terms sum as $S_{n}$, then in the sequence $S_{1}, S_{2}, \cdots, S_{2009}$, the number of rational terms is $\qquad$ terms.
(5) 43 Hint: From
$$
\begin{aligned}
a_{k} & =\frac{1}{\sqrt{k(k+1)}(\sqrt{k+1}+\sqrt{k})} \\
& =\frac{\sqrt{(k+1)}-\sqrt{k}}{\sqrt{k(k+1)}} \\
& =\frac{1}{\sqrt{k}}-\frac{1}{\sqrt{k+1}}
\end{aligned}
$$

we get
$$
S_{n}=\sum_{k=1}^{n} a_{k}=\sum_{k=1}^{n}\left(\frac{1}{\sqrt{k}}-\frac{1}{\sqrt{k+1}}\right)=1-\frac{1}{\sqrt{n+1}},
$$

Since $44<\sqrt{2010}<45$, we have $n+1=2^{2}, 3^{2}, 4^{2}, \cdots, 44^{2}$. Therefore, among $S_{1}, S_{2}, \cdots, S_{2009}$, there are only 43 rational terms.