Why can’t $frac{dy}{dt} = ay+b$ be integrated as is?

In my ODEs course, I have seen that the general solution to the equation $frac{dy}{dt} = ay+b$, if $a neq 0$ is:
$$
frac{dy/dt}{ay+b} = 1 \
vdots \
y = Ce^{at}-frac{b}{a}
$$

However, I fail to see why a solution could not be derived as follows:
$$
frac{dy}{dt} = ay+b \
int frac{dy}{dt} dt = int{ay+b} dt \
y = ayt + bt + C \
y (1-at) = bt + C \
y = frac{bt + C}{1 – at}
$$

I suppose this must be erroneous but I don’t see where my mistake is. What is the problem with this solution? Thank you!