Elongation of a hanging wire

Consider a non uniform wire of length $L$ hung vertically from the ceiling whose density varies $\rho(x) = \rho_{o} ( 1 - \frac{x}{L} )$ where $\rho_{o}$ is a constant and distance $x$ is measured from top.Young 's modulus of wire is known to be $Y$ .Assuming that the gravitational field strength is uniform in the region the elongation produced in the wire can be expressed as

$\Delta x = \alpha \rho_{o} g L^2 / \beta Y$ where $\alpha$ and $\beta$ are co-prime integers constants in their simplest form.What is $\alpha + \beta$ ?

Original. Inspiration from Irodov problem $1.298$ .