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We report the results of muon spin rotation and relaxation ($\rm \mu$SR)measurements on the recently discovered layered Cu-based superconductingmaterial La$_{2}($Cu$_{1-x}$Ni$_{x}$)$_{5}$As$_{3}$O$_{2}$ ($x =$ 0.40, 0.45).Transverse-field $\rm \mu$SR experiments on both samples show that thetemperature dependence of superfluid density is best described by a two-bandmodel. The absolute values of zero-temperature magnetic penetration depth$\lambda_{\rm ab}(0)$ were found to be 427(1.7) nm and 422(1.5) nm for $x =$0.40 and 0.45, respectively. Both compounds are located between theunconventional and the standard BCS superconductors in the Uemura plot. Noevidence of time-reversal symmetry (TRS) breaking in the superconducting stateis suggested by zero-field $\rm \mu$SR measurements.

Superconducting Properties of La$_2$(Cu$_{1-x}$Ni_x)$_5$As$_3$O$_2$: A $\rm \mu$SR Study