  The study of the moments of the partially transposed density matrix provides
a new and effective way of detecting bipartite entanglement in a many-body
mixed state. This is valuable for cold-atom and ion-trap experiments, as well
as in the general context of quantum simulation of many-body systems. In this
work we study the time evolution after a quantum quench of the moments of the
partial transpose, and several related quantities, such as the R\'enyi
negativities. By combining Conformal Field Theory (CFT) results with
integrability, we show that, in the space-time scaling limit of long times and
large subsystems, a quasiparticle description allows for a complete
understanding of the R\'enyi negativities. We test our analytical predictions
against exact numerical results for free-fermion and free-boson lattice models,
even though our framework applies to generic interacting integrable systems.
