The rate of vertical ascent, $\frac{dh}{dt}$ in metres per minute, of a weather balloon is modelled by the expression:

$$\frac{dh}{dt}=(3t-2)(4t+3)(3t+2)$$

where $t$ is the time in minutes. Find the general expression for the height $h$ of the balloon at time $t$, writing your answer in simplest form.