problem with partial fraction decomposition I want to do partial fraction decomposition on the following rational function:
$$\frac{1}{x^2(1+x^2)^3}$$
So I proceed as follows:
$$\begin{align}
\frac{1}{x^2(1+x^2)^3} &= \frac{A}{x} + \frac{B}{x^2} + \frac{Cx + D}{1 + x^2} + \frac{Ex + F}{(1 + x^2)^2} + \frac{Gx + H}{(1 + x^2)^3} \\
1 = &Ax(1+x^2)^3 + B(1+x^2)^3 + (Cx + D)x^2(1+x^2)^2 \\
& + (Ex + F)x^2(1+x^2) + (Gx + H)(x^2) \\
1 = &(A+C)x^7 + (B+D)x^6 + (3A+2C+E)x^5 + (3B+2D)x^4 \\
& + (3A+C+E+G)x^3 + (3B+D+H)x^2 +(A)x + (B)\\
\end{align}$$
But from this, $B=1$, $B=-D$, and $3B+2D=0$, which is an inconsistent system. What am I doing wrong?

Your final expression is almost correct; you're just forgetting the $F$. The coefficient of $x^4$ should be:
$$
3B + 2D + F
$$
and the coefficient of $x^2$ should be:
$$
3B + D + F + H
$$
