While trying to do partial fraction decomposition on $$\frac{x^4 +1}{x(x^2+1)^2}$$ I first equated it to $$\frac{A}{x}+\frac{Bx+1}{x^2+1}+\frac{C}{(x^2+1)^2}$$
On solving this, by adding the fractions, you get $$\frac{x^2(A+B)+A+x}{x^3+x}+\frac{C}{x^4+2x^2+1}$$ so obviously $A$ has to be equal to $1$ for the numerator's constant to be 1, and similarly, going forward, $B=-1$ and $C$ is -2, but when you write it out, you get an extra $x^5+2x^2+x$ in the numerator.
As far as I know, the method works everywhere. Can someone tell me where I've messed up here?
If you don't use the standard form, then you will be getting nowhere in most cases… like right now :(
The most general case is the following one [screenshot of the section "Over the reals---General Result" in this link]: