Laplace transform and output of affine systems in Simulink

I have a system $h$ which takes as input $u(t)$ and outputs $y(t)$. It is known that the relation between $y(t)$ and $u(t)$ is affine (i.e. $y(t) = h(u(t)) = a u(t) + b$). Apply Laplace’s transform to $h(t) = a t + b$ gives me the expression:

$$
H(s) = dfrac{a}{s^2} + dfrac{b}{s} = bdfrac{(s + frac{a}{b})}{s^2}
$$

If I understood correctly, the Laplace inverse of this expression is always $at + b$. Hence I assume that the two expressions $h(t)$ and $H(s)$ are related, if not the same (but only different in representation).

If I introduce a unit step into the input, in the time domain representation I expect a change of value from $b$ to $a+b$. As I understand, this should also be the same expected behavior with e.g. a continuous-time block. In Simulink, I tried to implement both representations (time domain using multiplier and addition blocks, Laplace domain using the transfer function block). I had no problem with the time domain representation, but the Laplace domain one did not give what I expected (it became a sort of ramp).

What could I have done wrong?