What is the general form of a transfer function
You may write the second equation on the first form by using $b_i = 0$ corresponding to those poles appearing at $s = 0$. Hence, the only difference is that in the second form you know that there are $m$ poles at $s=0$ while in the first form they may still be there, but one will have to check the values of $b_i$ to determine if they are there.

Note that a system with poles at $s=0$ is a marginally stable system and that the poles are typically introduced by integrators, which have a transfer function $1/s$.