Using macsyma 2.4 I was able to invert the transform of
1/(s^3+a*s^2+b*s+c). It did ask about the relative values
of a,b and c, which is to be expected since there are a few
possible cases, that is that the roots are equal or all real
or 2 are imaginary. However there is a formula for each of
these cases. Using maxima 5.6, if you declare a,b and c constant
using solve, you can determine the roots. Using partfrac,
you can convert that expression to linear factors, and then
invert termwise. You will have to make sure that the roots are
not equal however as those are 2 special cases. This is just
a example of where the tools aren't up to the task and they
need some help.
Dan Stanger