Non integer derivative of $1/p(x)$ I need to find the $k$'th derivative of $1/p(x)$, where $p(x)$ is a polynomial and $k\in\mathbb{R}$
It dosen't have to be an explicit formula, an algorithm which finds a formula for some $k$ is fine.

Why don't you just use the formula?
For generic $f$ and for 0<$\alpha$<1 you have:
$$D^\alpha(f(x))=\frac{1}{\Gamma(1-k)}\frac{\partial}{\partial x}\int_0^x\frac{f(t)}{(x-t)^\alpha}\,\mathrm{d}t$$
So after deriving [k] (the integer part) times you have the remainder $\alpha\in[0,1]$:
$$D^\alpha(D^{[k]}\left(\frac{1}{p(x)}\right))=\frac{1}{\Gamma(1-k)}\frac{\partial}{\partial x}\int_0^x\frac{1}{(x-t)^k}D^{[k]}\left(\frac{1}{p(x)}\right)\,\mathrm{d}t$$
