We can rewrite the given equation to get:
$\dfrac{a}{b}+\dfrac{b}{c+1}+\dfrac{c}{a}=\dfrac{5}{2}$
$\dfrac{a}{b}+\dfrac{b}{c+1}+\dfrac{c+1-1}{a}=\dfrac{5}{2}$
$\dfrac{a}{b}+\dfrac{b}{c+1}+\dfrac{c+1}{a}-\dfrac{1}{a}=\dfrac{5}{2}$
$\dfrac{a}{b}+\dfrac{b}{c+1}+\dfrac{c+1}{a}=\dfrac{5}{2}+\dfrac{1}{a}$
Applying AM-GM inequality to the terms on the left, we have:
$\begin{align*}\dfrac{a}{b}+\dfrac{b}{c+1}+\dfrac{c}{a}&\ge 3\sqrt[3]{\dfrac{a}{b}\cdot \dfrac{b}{c+1}\cdot \dfrac{c+1}{a}}\\&\ge 3 \end{align*}$
This implies $\dfrac{1}{a}\ge \dfrac{1}{2}$, which means $a\le 2$.
We thus have to consider two cases, $a=2$ and $a=1$.
When $a=2$, equality must be attained, i.e. $\dfrac{2}{b}=\dfrac{b}{c+1}=\dfrac{c+1}{2}$.
It follows that $b(c+1)=4$ so we have two choices, $b=2$, $c=1$ or $b=1$, $c=3$.
However, note that the second pair of solution does not satisfy $\dfrac{2}{b}=\dfrac{b}{c+1}$, thus we get only one solution in this case, namely $(a,\,b,\,c)=(2,\,2,\,1)$.
If $a=1$, the equality becomes $\dfrac{1}{b}+\dfrac{b}{c+1}+c=\dfrac{5}{2}$, which implies $c<2$.
For $c=1$, we get $\dfrac{1}{b}+\dfrac{b}{2}=\dfrac{3}{2}$, i.e. $b^2-3b+2=0$. This gives two solutions, $b=1$ and $b=2$ that yield $(a,\,b,\,c)=(1,\,1,\,1),\,(1,\,2,\,1)$.
For $c=2$, we get $\dfrac{1}{b}+\dfrac{b}{3}=\dfrac{1}{2}$, i.e. $2b^2-3b+6=0$. This does not have positive integer solutions.
We conclude that all solutions are given by that yield $(a,\,b,\,c)=(1,\,1,\,1),\,(1,\,2,\,1),\,(2,\,2,\,1)$.
Opalg has overlooked the pair $(a,\,b,\,c)=(1,\,2,\,1)$ but I know that is purely an honest careless mistake.(Smile)