  We study the three-body scattering hypervolume $D$ of atoms whose scattering
length $a$ is on the order of or smaller than the typical range
$r_{\mathrm{vdW}}$ of the van der Waals attraction. We find that the real part
of $D$ behaves universally in this weakly interacting regime
($|a|/r_{\mathrm{vdW}}\lesssim 1$) in the absence of trimer resonances. This
universality originates from hard-spherelike collisions that dominate elastic
three-body scattering. We use this result to make quantitative predictions for
the thermodynamics and elementary excitations of an atomic Bose-Einstein
condensate in the vicinity of a quantum tricritical point, including quantum
droplets stabilized by effective three-body interactions.
