$$F(S)=\frac{se^{-3s}}{s^2+22s+125}$$
My first reflex was to attempt a decomposition into partial fractions, but I am simply left with a fraction that is just as "complicated", instead of the usual decomposition into multiple simpler fractions separated by addition/subtraction/etc.
P.S. To save you some time, it is sufficient to show me how to get a decomposed fraction from this fraction, I can do the actual inverse laplace transforms from there.
Thank you!
Hint:
Solve $s^2+22s+125=0$. This yields $$s_1= -11 - 2i,\quad s_2=-11+2i$$
Now $$f(t)= \operatorname{Res}_{s=s_1}(F(s)\cdot e^{st})+\operatorname{Res}_{s=s_2}(F(s)\cdot e^{st}).$$
Calculate the residues by factoring the denominator with $s_1$ and $s_2$ and then calculate the limit, without any indeterminate form.