  Using $2.93~\mathrm{fb}^{-1}$ of $e^+e^-$ collision data collected with the
BESIII detector at the center-of-mass energy of 3.773 GeV, we investigate the
semileptonic decays $D^+\to \pi^+\pi^- \ell^+\nu_\ell$ ($\ell=e$ and $\mu$).
The $D^+\to f_0(500)\mu^+\nu_\mu$ decay is observed for the first time. By
analyzing simultaneously the differential decay rates of $D^+\to f_0(500)
\mu^+\nu_\mu$ and $D^+\to f_0(500) e^+\nu_e$ in different $\ell^+\nu_\ell$
four-momentum transfer intervals, the product of the relevant hadronic form
factor $f^{f_0}_{+}(0)$ and the magnitude of the $c\to d$
Cabibbo-Kobayashi-Maskawa matrix element $|V_{cd}|$ is determined to be
$f_{+}^{f_0} (0)|V_{cd}|=0.0787\pm0.0060_{\rm stat}\pm0.0033_{\rm syst}$ for
the first time. With the input of $|V_{cd}|$ from the global fit in the
standard model, we determine $f_{+}^{f_0} (0)=0.350\pm0.027_{\rm
stat}\pm0.015_{\rm syst}$. The absolute branching fractions of $D^+\to
f_0(500)_{(\pi^+\pi^-)}\mu^+\nu_\mu$ and $D^+\to \rho^0_{(\pi^+\pi^-)}
\mu^+\nu_\mu$ are determined as $(0.72\pm0.13_{\rm stat}\pm0.10_{\rm
syst})\times10^{-3}$ and $(1.64\pm0.13_{\rm stat}\pm0.11_{\rm syst})\times
10^{-3}$. Combining these results with those of previous BESIII measurements on
their semielectronic counterparts from the same data sample, we test lepton
flavor universality by measuring the branching fraction ratios ${\mathcal
B}_{D^+\to \rho^0 \mu^+\nu_\mu}/{\mathcal B}_{D^+\to \rho^0
e^+\nu_e}=0.88\pm0.10$ and ${\mathcal B}_{D^+\to f_0(500)
\mu^+\nu_\mu}/{\mathcal B}_{D^+\to f_0(500) e^+\nu_e}=1.14\pm0.28$, which are
compatible with the standard model expectation.
