For $H>L$ , $p,q,\alpha,\beta>0$, and B(.,.) the beta functon, trying to solve this integral:
$$\mathbb{E}(X)_0^H=\frac{\alpha H }{\beta B(p,q)}\int_0^H \frac{x \left(\frac{-H \log \left(\frac{H-x}{H}\right)}{\beta }\right)^{\alpha p-1} \left(\left(\frac{-H \log \left(\frac{H-x}{H}\right)}{\beta }\right)^{\alpha }+1\right)^{-p-q}}{H-x} \, \mathrm{d}x$$
${\bf Motivation: }$ This is the partial expectation of the random variable $X \in[0,H]$, a transformation of a r.v. following the generalized Beta distribution of second kind (also known as the Beta prime distribution).
${\bf Note: }$ I simplified the question and changed the support from $X \in [L,H]$ to $X \in [0,H]$.