Given is an M/G/1 queue with $A_i$ i.i.d. $\sim$ exp and $B_i$ i.i.d. $\sim$ $F(t)$, $F(t)=1-\frac{1}{2}e^{-2t}-\frac{1}{2}e^{-\frac{1}{2}t}$ and $\rho=\frac{1}{2}$.
I am supposed to see the distribution of the waiting time in the Laplace-Stieltjes-Transform $$ \tilde{W}(s)=\frac{(1-\rho)s}{\lambda \tilde{B}(s)+s-\lambda} =\frac{(1-\rho)s}{\lambda \Big(\frac{1}{s+2}+\frac{1}{3s+2} \Big)+s-\lambda} $$ (you see I already worked out the LST of the service time. I also conclude from this form that the service time is hyperexponential distributed).
I was given the clue to rewrite this in the form $$a + b \cdot \frac{c}{c+s} + d \cdot \frac{e}{e+s}$$ and then derive the distribution. But this is where I am stuck.
Do you see how to do it?