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A study of $B_c^+\to J/\psi D_s^+$ and $B_c^+\to J/\psi D_s^{*+}$ decaysusing 139 fb$^{-1}$ of integrated luminosity collected with the ATLAS detectorfrom $\sqrt{s} = 13$ TeV $pp$ collisions at the LHC is presented. The ratios ofthe branching fractions of the two decays to the branching fraction of the$B_c^+\to J/\psi \pi^+$ decay are measured: $\mathcal B(B_c^+\to J/\psiD_s^+)/\mathcal B(B_c^+\to J/\psi \pi^+) = 2.76\pm 0.47$ and $\mathcalB(B_c^+\to J/\psi D_s^{*+})/\mathcal B(B_c^+\to J/\psi \pi^+) = 5.33\pm 0.96$.The ratio of the branching fractions of the two decays is found to be $\mathcalB(B_c^+\to J/\psi D_s^{*+})/\mathcal B(B_c^+\to J/\psi D_s^+) = 1.93\pm0.26$.For the $B_c^+\to J/\psi D_s^{*+}$ decay, the transverse polarization fraction,$\Gamma_{\pm\pm}/\Gamma$, is measured to be $0.70\pm0.11$. The reporteduncertainties include both the statistical and systematic components added inquadrature. The precision of the measurements exceeds that in all previousstudies of these decays. These results supersede those obtained in the earlierATLAS study of the same decays with $\sqrt{s} = 7$ and 8 TeV $pp$ collisiondata. A comparison with available theoretical predictions for the measuredquantities is presented.

Study of $B_c^+\to J/\psi D_s^+$ and $B_c^+\to J/\psi D_s^{*+}$ decays in $pp$ collisions at $\sqrt{s} = 13$ TeV with the ATLAS detector