Answer
$\frac{A}{x} + \frac{B}{2x-5} + \frac{C}{(2x-5)^2} + \frac{D}{(2x-5)^3} + \frac{Ex+F}{x^2+2x+5} + \frac{Gx+H}{(x^2+2x+5)^2}$
Work Step by Step
For each factor $(ax+b)^n$ of the denominator we add $n$ fractions $\frac{A_k}{(ax+b)^k}$, where $k=1,2,3,\ldots n$.
For each factor $(ax^2+bx+c)^n$, where $ax^2+bc+c$ is irreducible, we add $n$ fractions $\frac{p_k x+q_k}{(ax^2+bx+c)^k}$, where $k=1,2,3,\ldots n$.
$\frac{x^3+x+1}{x(2x-5)^3(x^2+2x+5)^2}=\frac{A}{x} + \frac{B}{2x-5} + \frac{C}{(2x-5)^2} + \frac{D}{(2x-5)^3} + \frac{Ex+F}{x^2+2x+5} + \frac{Gx+H}{(x^2+2x+5)^2}$
