A special integral related to Beta function

Question

Consider the integral $$ I(a,b) = \int_0^\infty \big(z^a-(z-1)_+^a\big)z^{-b}dz $$ for some $a\in \mathbb R$, $b\in (a,a+1)$ (here $z_+ = \max(z,0)$). When $a>0$, we can write $$ I(a,b) = a\int_0^\infty \int_0^1 (z-s)_+^{a-1}ds\, z^{-b}dz = a\int_0^1 \int_s^\infty (z-s)^{a-1}z^{-b} dz\, ds\\ = a \int_0^1 s^{a-b} \int_1^\infty (u-1)^{a-1} u^{-b}du \, ds= a\int_0^1 s^{a-b}\, \mathrm{B}(a,b-a)ds \\ = \frac{a}{a-b+1}\, \mathrm{B}(a,b-a)= \frac{b}{a-b+1}\mathrm{B}(a+1,b-a). $$ Since both $I(a,b)$ and $\mathrm{B}(a+1,b-a)$ are well defined for $a>-1$ and $b\in (a,a+1)$, it seems that the formula $$ I(a,b) = \frac{b}{a-b+1}\mathrm{B}(a+1,b-a) $$ must be valid in this case too. Is there any simple way to prove this or any book which contains it? (Mathematica confirms this, but I don't want to cite it.)