(More difficult question) Let $f$ be the power series expansion of a rational function $P/Q \in \mathbf{Q}(x)$ all of whose poles are rational numbers. Suppose that the antiderivative $\int_0^x f(t)\,dt$ is globally bounded. Show that $\int_0^x f(t)\,dt$ is then the power series expansion of a rational function in $\mathbf{Q}(x)$.