I consider the following function on $\mathbb{R}$ :
$$ f(z)=\int\limits_0^{+\infty} dx\frac{1+x}{\sqrt{x}}\frac{1}{x\, e^x +z}$$
I would like to know its singular expansion at $z=0$.
Doing a series expansion of the denominator does not help me much, so I don't really know where to start...
I don't know if this is accurate enough for your purposes, but just to make some progress, we can proceed as follows. First of all, $\int_1^{\infty}\ldots$ is of course holomorphic near $z=0$, so has a Taylor expansion about $z=0$. Next, we can simplify and consider $$ I(z)= \int_0^1\frac{1+x}{\sqrt{x}}\frac{dx}{x+z} ; $$ the difference between $I$ and the actual integral (over $(0,1)$) stays bounded as $z\to 0+$.
Now $I$ is easy to analyze. We can substitute $x=t^2$ to obtain that $$ I(z) = 2\int_0^1 \frac{t^2+1}{t^2+z}\, dt = 2 + 2(1-z)\int_0^1 \frac{dt}{t^2+z}=\frac{\pi}{\sqrt{z}}+O(1) , $$ and, as discussed, this is also the asymptotic behavior of $f(z)$ as $z\to 0+$.