Your next step is NOT partial fractions. Don't listen to that.
Here is information fresh out of my Dynamic Systems and controls class.
Let's see: Your transfer function is Y(s)=e^-5s/((s+1)(s-8))
You have an unstable system with poles of s=-1 and s=8.
The trick to solving problems with a time delay (e^-theta*s), is that you ignore the time delay until the end. Your theta is 5. So let's ignore the time delay term. What is the inverse Laplace of 1/[(s+1)(s-8)]? First you have to realize it is not in standard time constant form. In standard time constant form, the transfer function looks like: 1/[(s+1)*(-1/8*s+1)]. Your first time constant is +1. Your second time constant is -1/8. The inverse Laplace of an equation of this form is: 1/(b1-b2)*(exp(-b2t)-exp(-b1t)).
The result is 1/(1- -8)*(exp(--8*t)-exp(-1*t)) or 1/9*(exp(8t)-exp(-t)).
Now for the time delay. Since your theta is 5, anywhere you see a t in the final equation y(t), put in (t-5). Your resulting equation becomes y(t)= 1/9*(exp(8*(t-5))-exp(-(t-5))).
EDIT: Took out standard time constant form because it gets really confusing, I just did that and would have to explain much more.
Feel free to pm me with questions. I will also try to check back on this thread.