$f(x)$ is a quadratic function such that $f(0) = 1$ and $\int {f(x) \over x^2(x + 1)^3} dx $ is a rational function. Find $f(x)$. 
$f(x)$ is a quadratic function such that $f(0) = 1$ and $$\int {f(x) \over x^2(x + 1)^3 } dx $$ is a rational function. Find $f(x)$.

Now first of all we don't even know what the degree of the numerator and denominator of the rational function is, so I suppose differentiating both sides is not an option or will become tedious task and since $f(x)$ is also obviously not given, so we can also not integrate then how to go about solving this question.

Let $$f(x)=Z(x+1)(2x+1)~~~~(1)$$ where $Z$ is a constant  then $$I=\int \frac{f(x)dx}{x^2(x+1)^3}=Z\int \frac{2x+1}{(x(x+1))^2} dx= \frac{-Z}{x^2+x}+C.$$
Choose $Z=1$ in (1) to have $f(0)=1$.
Explaination: Since $f(0)=1$, $x$ or $x^2$ cannot be ia factor of $f(x)$. Try to have $(x+1)$ as a factor of $f(x)$ so that your integrand reduces. You may let the other factor be $(Kx+L)$. Then $$I=\int\frac{Kx+L}{(x^2+x)^2}dx.$$
Now, $K=2Z, L=Z$ is the obvious choice for $I$ to be algebraic and rational.
Else,
A Hint
You may do it by assuming $f(x)=Ax^2+Bx+C$,
Then by partial fractions $$I=\int\frac{Ax^2+Bx+C}{x^2(x+1)^3}=\int \left(\frac{P}{x}+\frac{Q}{x^2}+\frac{R}{x+1}+\frac{S}{(x+1)^2}+\frac{T}{(x+1)^3}\right) dx$$
For RHS to be rational and Algebraic $P=0=R$, this is to kill $\log x$ and $\log (x+1)$ on the RHS. Then you can choose sich values of $A,B,C$. Also you are given that $C=1$.
