I noticed that the fraction is expressible as (a+b+c)/(a*b) + (a+b+c)/(a*c) + (a+b+c)/(b*c). However, the fractions may or may not be integers for the sum to be an integer.
I also noticed that for the fraction to be an integer, then (a+b)^2 must be a multiple of c, similarily (a+c)^2 is a multiple of b and (b+c)^2 is a multiple of a.
Also, if any two of a,b,c have a common factor, that number is also a factor of the third. Similarily, if any two of a,b,c are coprime, then they are also coprime to the third.