  Recently we have reported (Knoop et al. [arXiv:1404.4826]) on an experimental
determination of metastable triplet $^4$He+$^{87}$Rb scattering length by
performing thermalization measurements for an ultracold mixture in a quadrupole
magnetic trap. Here we present our experimental apparatus and elaborate on
these thermalization measurements. In particular we give a theoretical
description of interspecies thermalization rate for a quadrupole magnetic trap,
i. e. in the presence of Majorana heating, and a general procedure to extract
the scattering length from the elastic cross section at finite temperature
based on knowledge of the $C_6$ coefficient alone. In addition, from our
thermalization data we obtain an upper limit of the total interspecies two-body
loss rate coefficient of $1.5\times 10^{-12}$ cm$^3$s$^{-1}$.
