Example 8 Let $a, b, c$ be positive rational numbers, and $a+\frac{1}{b c}$, $b+\frac{1}{c a}$, $c+\frac{1}{a b}$ are all integers. Find the value of $a+b+c$.
$$
M=\left(a+\frac{1}{b c}\right)\left(b+\frac{1}{a c}\right)\left(c+\frac{1}{a b}\right)=\frac{(a b c+1)^{3}}{a^{2} b^{2} c^{2}}
$$

is a positive integer.
Let $a b c$ be the irreducible fraction $\frac{p}{q}\left(p, q \in \mathbf{Z}_{+}\right)$. Then $M=\frac{(p+q)^{3}}{p^{2} q} \Rightarrow(p+q)^{3}=M p^{2} q$.
From $p\left|(p+q)^{3}, q\right|(p+q)^{3}$, we can respectively deduce $p\left|q^{3}, q\right| p^{3}$.
Combining $(p, q)=1$ we get
$$
p=1, q=1 \text {. }
$$

From $a b c=1$, we get
$$
\begin{array}{l}
a+\frac{1}{b c}=2 a \in \mathbf{N}_{+}, b+\frac{1}{c a}=2 b \in \mathbf{N}_{+}, \\
c+\frac{1}{a b}=2 c \in \mathbf{N}_{+} .
\end{array}
$$

Assume without loss of generality that $a \leqslant b \leqslant c$. Then we can obtain three sets of solutions
$$
(a, b, c)=\left(\frac{1}{2}, \frac{1}{2}, 4\right),\left(\frac{1}{2}, 1,2\right),(1,1,1) \text {. }
$$

Thus, the values of $a+b+c$ are $5$, $3.5$, and $3$.