  Resonant elastic X-ray scattering is a powerful technique for measuring
multipolar order parameters. In this paper, we theoretically and experimentally
study the possibility of using this technique to detect the proposed multipolar
order parameters in URu$_2$Si$_2$ at the U-$L_{3}$ edge with the electric
quadrupolar transition. Based on an atomic model, we calculate the azimuthal
dependence of the quadrupolar transition at the U-$L_{3}$ edge. The results
illustrate the potential of this technique for distinguishing different
multipolar order parameters. We then perform experiments on ultra-clean single
crystals of URu$_2$Si$_2$ at the U-$L_{3}$ edge to search for the predicted
signal, but do not detect any indications of multipolar moments within the
experimental uncertainty. We theoretically estimate the orders of magnitude of
the cross-section and the expected count rate of the quadrupolar transition and
compare them to the dipolar transitions at the U-$M_4$ and U-$L_3$ edges,
clarifying the difficulty in detecting higher order multipolar order parameters
in URu$_2$Si$_2$ in the current experimental setup.
