Let's work out the LCD. First we need a, b and c in the denominator, so that gives us abc as one factor.
I have the feeling that the middle term should be (1/b(b-a)(b-c)) by symmetry with the other terms.
Let's assume that's what it is and let's change the sign of the middle term to minus:
1/a(a-b)(a-c)-1/b(a-b)(b-c)+1/c(a-c)(b-c). Notice I changed the last term to keep the variables in alphabetical order. This makes it easier to see what we're doing.
Back to the LCD. We had abc as a factor. Now it's easy to see what the other factors are: (a-b)(b-c)(a-c).
So the LCD is abc(a-b)(b-c)(a-c).
So the numerator becomes: bc(b-c)-ac(a-c)+ab(a-b). See the pattern?
Now expand the numerator: b^2c-bc^2-a^2c+ac^2+a^2b-ab^2.
The final fraction is:
(b^2c-bc^2-a^2c+ac^2+a^2b-ab^2)/(abc(a-b)(b-c)(a-c)).
I suspect that more is possible, because of the symmetry.
Let's expand (a-b)(b-c)(a-c)=a^2b-a^2c-ab^2+abc-abc+ac^2+b^2c-bc^2. Recognise this? It's the same as the numerator, because abc cancels out. So we are left with 1/abc.
If we let a=1, b=2, c=3 and substitute these in the original expression see get 1/2-1/2+1/6=1/abc.