The first integral follows from Lemma 7.18 applied to $\beta=(\alpha_{0},\dots,\alpha_{r-1})$ and $f(z)=\frac{z^{|\alpha|+r}}{(|\alpha|+r)!}$. The second one follows similarly, applying Lemma 7.18 to the function $f(z)=\frac{z^{|\alpha|+r}}{(|\alpha|+r)!}\left(\log(z)-\sum_{j=\alpha_{i}+1}^{|\alpha|+r}\frac{1}{j}\right)$, after some possible permutation (for $i=1,\dots,r-1$) or linear change of variables (for $i=0$). ∎