I have the following triple integral
$$\int_{y_1=0}^{\infty}\int_{y_2=0}^{\infty}\int_{x_2=0}^{zy_2}\exp\left(-\min\left[x_2,\,y_1(z-\frac{x_2}{y_2})\right]\right)\,dx_2dy_1dy_2$$
I want to divide the integrals into intervals to evaluate it. What I did was that I divided the inner most integral into two interval as following
$$\int_{x_2=0}^{\frac{z\,y_1y_2}{y_1+y_2}}\exp\left(-x_2\right)\,dx_2 + \int_{x_2=\frac{z\,y_1y_2}{y_1+y_2}}^{zy_2}\exp\left(-y_1(z-\frac{x_2}{y_2})\right)\,dx_2$$
Although this allows us to evaluate the integral over $x_2$ easily, the integration over $y_1$ and $y_2$ will be harder.
My actual problem is a little more complicated, and this question is used to capture the essence of the problem. Thus, I am not looking for a solution for the integrals as they are, but rather I need a method where I can divide all the integrals (not just the inner most one) into intervals that makes the evaluation of all integrals much simpler. Is this feasible?
EDIT: My actual problem is the following
$$\int_{y_1=0}^{\infty}\int_{y_2=0}^{\infty}\int_{x_2=0}^{zy_2}\exp\left(-\min\left[x_2,\,y_1(z-\frac{x_2}{y_2})\right]\right)e^{-y_1}\,e^{-y_2}\,dx_2dy_1dy_2$$