Here's what I did:
$$Q_s(Q_t(x)) = Q_s(\sum Q^j(x) \beta(t)^j + \sum \beta Q^j(x) \beta(t)^{j-1} t) =$$
$$= \sum Q_s(Q^j(x)) Q_s(\beta(t))^j + \sum Q_s(\beta Q^j(x)) Q_s(\beta(t))^{j-1} Q_s(t) = \sum Q_s(Q_j(x)) \beta(s)^{j(1-p)} \beta(t)^j (\beta(t)+\beta(s))^{j(p-1)} + \sum Q_s(\beta Q^j(x)) \beta(s)^{(j-1)(1-p)} \beta(t)^{j-1} (\beta(t)+\beta(s))^{(j-1)(p-1)} Q_s(t) $$