a,b,c fraction integer

Given positive integers $a,b,c$ such that
$$\frac{a}{b} + \frac{b}{c} + \frac{c}{a}$$
is also an integer, show that
$$\frac{a^2b^2}{c},\frac{b^2c^2}{a}, \frac{c^2a^2}{b}$$
are all cubic numbers.