Integrable optics is an innovation in particle accelerator design that enables strong nonlinear focusing without generating parametric resonances. I present an integrable rapid-cycling synchrotron designed as a high-intensity replacement of the Fermilab Booster. I use the Synergia space-charge tracking code to investigate the stability of a beam in this lattice with a space-charge tune-shift up to 0.4 and a RMS momentum spread up to 0.25%. The strong nonlinearity of the lattice suppresses the beam halo generated by a mismatched KV or WB beam. A new higher periodicity lattice design shows a marked improvement over a previous lattice design. Experimental tests of high-intensity beams in integrable lattices will take place over the next several years at the Fermilab Integrable Optics Test Accelerator (IOTA) and the University of Maryland Electron Ring (UMER).