inverse_laplace of a fraction whose denominator has real roots (or complex)

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Okay. Now, we have two parts :

1. How to get an exact partial fraction decomposition of L
2. How to get ins inverse Laplace transform.

Let's be lazy and first solve the second one. We'll do it symbolically.

sage: var("a,b,c,d,e,s,t")
(a, b, c, d, e, s, t)
sage: Ls=(s-a)*(s-b)/((s-c)*(s-d)*(s-e))
sage: inverse_laplace(Ls,s,t)
(a*b - (a + b)*c + c^2)*e^(c*t)/(c^2 - c*d - (c - d)*e) - (a*b - (a + b)*d + d^2)*e^(d*t)/(c*d - d^2 - (c - d)*e) + (a*b - (a + b)*e + e^2)*e^(e*t)/(c*d - (c + d)*e + e^2)

which is $\frac{{\left(a b - {\left(a + b\right)} c + c^{2}\right)} e^{\left(c t\right)}}{c^{2} - c d - {\left(c - d\right)} e} - \frac{{\left(a b - {\left(a + b\right)} d + d^{2}\right)} e^{\left(d t\right)}}{c d - d^{2} - {\left(c - d\right)} e} + \frac{{\left(a b - {\left(a + b\right)} e + e^{2}\right)} e^{\left(e t\right)}}{c d - {\left(c + d\right)} e + e^{2}}$.

Our problem is now reduced to find exact (if possible closed_form) expressions of $a,~b,~c,~d$ and $e$, respective roots of the numerator and denominator of L.

The second part of my answer to your previous question shows how to obtain the radical expression of an element of F if such an expression can be computed by Sage.

This can be applied either to the result of L.partial_fraction_decomposition() or to the factorization of the numerator or denominator of L, with identification to the monomials into which its numerator and denominator can be factorized into.

But I'm not sure that such explicit representation of the resulting function is worthy...

Back to your original problem : I'll have a look at this (survival analysis is an important case in biostatistics). What do you want to show ? I'd also like to question the (implicit) assumption of interest in the (total) survival function : in competitive risks models, there are more interesting things to do...