The 100100th derivative of (x2+1)/(x3−x)(x^2 + 1)/(x^3 – x)

I am reading a collection of problems by the Russian mathematician Vladimir Arnol’d, titled A Mathematical Trivium. I am taking a stab at this one:

Calculate the $100$th derivative of the function

The derivative is non-trivial (in the sense that I computed it for a few rounds, and it only became more assertive). My first thought was to let

and apply the Leibnitz rule for products,

Since $f$ is vanishing after the third differentiation, we get

This would be great if we could compute the last few derivatives of $g$. Indeed, we can boil this down: notice that

further, $h, i,$ and $j$ have friendly behavior under repeated differentation, e.g. $h^{(n)}(x) = \frac{(-1)^n n!}{(x-1)^{n + 1}}$.

So overall, it is possible to use Leibnitz again to beat a lengthy derivative out of this function, (namely,

with the details filled in).

However, this is really pretty far from computing the derivative.

So, my question: does anyone know how to either improve the above argument, or generate a new one, which can resolve the problem?