So here's the question:
If $f$ is a quadratic function such that $f(0) = 1$ and $\int \frac{f(x)}{x^2(x+1)^3}\,dx$ is a rational function, find the value of $fâ(0)$.
What I've done so far is try to solve the integral using partial fractions i.e.
$\frac{f(x)}{x^2(x+1)^3} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{(x+1)} + \frac{D}{(x+1)^2} + \frac{E}{(x+1)^3}$ Multiply out the denominator from the LHS to get:
$f(x) = Ax(x+1)^3 + B(x+1)^3 + Cx^2(x+1)^2 + Dx^2(x+1) + Ex^2$ when $x = 0$ I get that $B=1$.
At this point I'm stuck. I tried solving for the other variables but it gets insanely complicated. Wondering if anyone has a better strategy to solving the problem.
Thank you.