I have the following integral
$$ Z = \frac{1}{2\pi i} \int dx \, \frac{1}{(x-a_1)(x-a_2)(x-a_3)}\times \frac{1}{(x+\epsilon - a_1)(x + \epsilon - a_2)(x+ \epsilon - a_3)} $$
and this integral has poles at $x=a_1,a_2,a_3$ from the first factor and so on for the second one $x=a_i - \epsilon$ where $\epsilon$ is just a number. I want to evaluate this integral around $a_1,a_2,a_3$ by taking small contours around these points or shifting them into the complex plane (i.e. by a small deformation like $a_i \to a_i + 0i $). This is motivated from instanton partition functions in physics. So are there any tips or guidelines on how to "nicely" do this?