We know that due to expansion, the Quadrupole potential equals $$1/4\pi \epsilon r^3 . \int (r^\prime)^2(\frac32. \cos^2\theta^\prime-\frac12)\rho(r^\prime)d\tau$$
but what is the equation for point charges? for example, we have two charge $q$ and one charge $-2q$ with defined locations.
You're trying to understand how a point charge is represented in the charge density $\rho(r)$.
The concept you need is the Dirac delta function $\delta(r)$, which can describe the density of a finite amount of stuff packed into an infinitesimal point:
$$\delta(r) = \left\{ ^{\infty \text{ if } r=0 }_{0 \text{ if } r\ne0} \right\} $$
$$\int_{-\infty}^{+\infty} \delta(x) dx = 1$$
And it has the following really useful property:
$$\int_{-\infty}^{+\infty} f(x)\delta(x-a) dx = f(a)$$
And in your case the the charge density could be written
$$\rho(r') = q\delta(r'-r_1) + q\delta(r'-r_2) - 2q\delta(r'-r_3)$$
Where $r_i$ is the location of the $i$th charge. The useful property above makes it easy to calculate your integral.
