  We study Boolean stable laws, $\mathbf{b}_{\alpha,\rho}$, with stability
index $\alpha$ and asymmetry parameter $\rho$. We show that the classical scale
mixture of $\mathbf{b}_{\alpha,\rho}$ coincides with a free mixture and also a
monotone mixture of $\mathbf{b}_{\alpha,\rho}$. For this purpose we define the
multiplicative monotone convolution of probability measures, one is supported
on the positive real line and the other is arbitrary.
  We prove that any scale mixture of $\mathbf{b}_{\alpha,\rho}$ is both
classically and freely infinitely divisible for $\alpha\leq1/2$ and also for
some $\alpha>1/2$. Furthermore, we show the multiplicative infinite
divisibility of $\mathbf{b}_{\alpha,1}$ with respect classical, free and
monotone convolutions.
  Scale mixtures of Boolean stable laws include some generalized beta
distributions of second kind, which turn out to be both classically and freely
infinitely divisible. One of them appears as a limit distribution in
multiplicative free laws of large numbers studied by Tucci, Haagerup and
M\"oller.
  We use a representation of $\mathbf{b}_{\alpha,1}$ as the free multiplicative
convolution of a free Bessel law and a free stable law to prove a conjecture of
Hinz and M{\l}otkowski regarding the existence of the free Bessel laws as
probability measures. The proof depends on the fact that
$\mathbf{b}_{\alpha,1}$ has free divisibility indicator 0 for $1/2<\alpha$.
