Putnam 1993 A5

Show that

$$\begin{gathered}\int_{-100}^{-10} \left( \frac{x^2 - x}{x^3 - 3x + 1} \right)^2 dx + \\ \int_{\frac{1}{101}}^{\frac{1}{11}} \left( \frac{x^2 - x}{x^3 - 3x + 1} \right)^2 dx + \\ \int_{\frac{101}{100}}^{\frac{11}{10}} \left( \frac{x^2 - x}{x^3 - 3x + 1} \right)^2 dx\end{gathered}$$

is a rational number.