Is it possible to factor out monomials in a rational function?

Question

after thinking about the problem for some hours I thought I come here to ask. My problem is that I want to do a coordinate transformation on the following equation

$y=\frac{a}{x^2}+\frac{b}{x}+c+dx+ex^2$

now let's subsitute $x=gx'+f$ so that we have

$y=\frac{a}{(gx'+f)^2}+\frac{b}{gx'+f}+c+d(gx'+f)+e(gx'+f)^2$

now for the fun part. Is it possible to somehow put this equation into the following form:

$y=\frac{\alpha}{x^2}+\frac{\beta}{x}+\gamma+\delta x'+\varepsilon x'^2$.

I'd say that this is not possible. Any other opinions?

Thanks for your help!

Answer 1

No, you can't; the form you want to put your equation into describes a function that only has singularities at $x = 0$, while the equation you have can have singularities elsewhere (say, if $g = f = a = 1$).