How to estimate time to fall with air drag and variable density?

I would like to estimate the time that it takes for an object to fall if there is a quadratic drag force and a variable density.

In a vacuum, the falling time $t_mathrm{vacuum}$ has a simple closed-form expression, which is simply $$t_mathrm{vacuum}=sqrt{frac{2h}{g}},$$ where $h$ is the height through which the object falls from rest.

In reality, however, we are not in a vacuum, and there is a drag force, which is a function of the velocity of the object. For most falling objects, it is proportional to the square of the velocity, in the form $F_d=frac{1}{2} rho C_D A v^2$.

Also, we know that the density $rho$ of the atmosphere is not constant, it depends on altitude. Here the Laplace’s formula for an isothermic atmosphere will be used, and it is written as $rho(z)=rho_0 e^{-kz}$, where $k$ is an experimentally determined constant and $rho_0$ is the density of air at sea level.

Knowing all of this, from Newton’s second law we get the differential equation that we need to solve to find the vertical position $z$ (height) as a function of time $t$:
$$mddot{z}(t)=-mg-frac{1}{2}mathrm{sgn} (dot{z}(t))C_DArho_0e^{-kz(t)}dot{z}(t)^2$$

This differential equation is not easy to solve, and it requires a numerical solution.

Now, here’s my question: How can one estimate the time that the object will take to fall using the equation above? – i.e. the time $t_f$ such that $z(t_f)=0$. Or I am modelling it wrong (I ask this because of the minus sign in front of the $mg$).