5. Let $n$ be a positive integer, and the function $f_{n}(x)=\frac{n+x+\frac{1}{n+x}}{n+1}, x \in(0,1)$ has a range $I_{n}, I=\bigcup_{n=1}^{\infty} I_{n}$, then $I=$ $\qquad$ .
Since $f_{n}(x)$ is monotonically increasing on $(0,1)$, then $I_{n}=\left(\frac{n^{2}+1}{n(n+1)}, \frac{(n+1)^{2}+1}{(n+1)^{2}}\right)$,
For $\frac{n^{2}+1}{n(n+1)}=1-\frac{n-1}{n(n+1)} \xlongequal{t=n-1} 1-\frac{t}{(t+1)(t+2)}=1-\frac{1}{t+\frac{2}{t}+3}$
$\geqslant 1-\frac{1}{6}=\frac{5}{6}, \frac{(n+1)^{2}+1}{(n+1)^{2}}=1+\frac{1}{(n+1)^{2}} \leqslant 1+\frac{1}{4}=\frac{5}{4} \Rightarrow I=\bigcup_{n=1}^{\infty} I_{n}=\left(\frac{5}{6}, \frac{5}{4}\right)$.