Finding the value of $f'(0)$ 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, how can we find the value of $f'(0)$? I am totally clueless to this. Any tip on how to start? If you wish to give details, then many thanks to you.

Let $f(x) = ax^2 + bx + c.$ As $f(0) = 1,$ we have that $f(x) = ax^2 + bx + 1.$ Then $$I=\int \frac{f(x) dx}{x^2(x+1)^3} = \int \frac{a}{(x+1)^3}+\frac{b}{x(x+1)^3}+\frac{1}{x^2(x+1)^3}\,dx \\\stackrel{Partial Fractions}{=} -\frac 1x + \frac{b-a-1}{2 (1 + x)^2} + \frac{-2 + b}{1 + x} + (b-3)\log(x) - (b-3)\log(1 + x) + C.$$
As the function must be rational, we must have $b=3,$ which leads $f'(0) = \left.2ax+b\right|_{x=0} = \boxed{3}$
