Fully gapped superconducting state in interstitial-carbon-doped <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi mathvariant="normal">Zr</mml:mi><mml:mn>5</mml:mn></mml:msub><mml:msub><mml:mi mathvariant="normal">Pt</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:math>

We report a comprehensive study of the Zr$_5$Pt$_3$C$_x$ superconductors, with interstitial carbon comprised between 0 and 0.3. At a macroscopic level, their superconductivity, with $T_c$ ranging from 4.5 to 6.3 K, was investigated via electrical-resistivity-, magnetic-susceptibility-, and specific-heat measurements. The upper critical fields $\mu_0H_\mathrm{c2}$ $\sim$ 7 T were determined mostly from measurements of the electrical resistivity in applied magnetic fields. The microscopic electronic properties were investigated by means of muon-spin rotation and relaxation ($\mu$SR) and nuclear magnetic resonance (NMR) techniques. In the normal state, NMR relaxation data indicate an almost ideal metallic behavior, confirmed by band-structure calculations, which suggest a relatively high electronic density of states at the Fermi level, dominated by the Zr 4$d$ orbitals. The low-temperature superfluid density, obtained via transverse-field $\mu$SR, suggests a fully-gapped superconducting state in Zr$_5$Pt$_3$ and Zr$_5$Pt$_3$C$_{0.3}$, with a zero-temperature gap $\Delta_0$ = 1.20 and 0.60 meV and a magnetic penetration depth $\lambda_0$ = 333 and 493 nm, respectively. The exponential dependence of the NMR relaxation rates below $T_c$ further supports a nodeless superconductivity. The absence of spontaneous magnetic fields below the onset of superconductivity, as determined from zero-field $\mu$SR measurements, confirms a preserved time-reversal symmetry in the superconducting state of Zr$_5$Pt$_3$C$_x$. In contrast to a previous study, our $\mu$SR and NMR results suggest a conventional superconductivity in the Zr$_5$Pt$_3$C$_x$ family, independent of the C content.