RO-NMO 2015 Problem 124

Find all non-constant polynoms $f\in\mathbb{Q} [X]$ that don't have any real roots in the interval $[0,1]$ and for which there exists a function $\xi :[0,1]\longrightarrow\mathbb{Q} [X]\times\mathbb{Q} [X], \xi (x):=\left( g_x,h_x \right)$ such that $h_x(x)\neq 0$ and $\int_0^x \frac{dt}{f(t)} =\frac{g_x(x)}{h_x(x)} ,$ for all $x\in [0,1] .$