# Expand terms of series in denominator

I have equation:

$a+bx+cx^2+dx^3= \dfrac{1}{f+gx+hx^2+ix^3} \dfrac{\partial}{\partial r}(ax+bx^2+cx^3+dx^4) +\dfrac{1}{(f+gx+hx^2+ix^3)^2} \dfrac{\partial^2}{\partial r^2}(ax^2+bx^3+cx^4+dx^5)$

where I need to extract terms which are multipliers of $x^0$, $x^1$, $x^2$, $x^3$ (it need to be sum of particular terms, for example right side need to be in shape $Ax+Bx^2+Cx^3+D$). Constants are $a$, $b$, $c$, $d$, $g$, $f$, $i$, $h$.

The biggest problem for me are parts with denominator on the right side of equation $\dfrac{1}{f+gx+hx^2+ix^3}$ and $\dfrac{1}{(f+gx+hx^2+ix^3)^2}$,

I dont know how to extract these terms which are multipliers of $x^0$, $x^1$, $x^2$, $x^3$?

Until now functions in Mathematica Apart, Expand, ExpandAll, Collect simplify didnt give me results, am I looking in right area of math operations?

• If x is small you can use taylor series. Otherwise there is no way to express rational expression as a polynome. – Vsevolod A. Mar 6 '18 at 11:22
• I think you have to use Series command, because power is neagive that contain infinite term of x. – Gopal Verma Mar 6 '18 at 11:36

         Series[(f + g*x + h*x^2 + i*x^3)^-1, {x, 0, 5}]
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