  The decay of the D meson into multibody final states is a complex process
that provides valuable insights into the fundamental interactions within the
Standard Model of particle physics. This study focuses on the decay cascade
$D^{+}\to K^{*}_{J} \ell^{+}\nu \to K^{\pm}\pi^{\mp} \ell^{+}\nu$ where the
$K^*_J$ resonance encompasses the $K^*(892),K^{*}(1410),K^{*}_0(1430)$ states.
We employ the helicity amplitude technique to derive the angular distributions
for the decay chain, enabling the extraction of one-dimensional and
two-dimensional distributions. Utilizing form factors for the $D\to K^*$
transition derived from the quark model, we calculate the differential and
integrated partial decay widths, explicitly considering the electron and muon
masses.
  Decay branching fractions are calculated, the ratios of the branching
fractions are found to be $\frac{\mathcal{B}r(D^{+}\to K^{*}(892)(\to K
^{-}\pi^{+}) \, \mu^{+}\nu_{\mu})}{\mathcal{B}r(D^{+}\to K^{*}(892)(\to K
^{-}\pi^{+}) \, e^{+}\nu_{e})} = 0.975$, $\frac{\mathcal{B}r(D^{+}\to
K^{*}(1410)(\to K ^{-}\pi^{+}) \, \mu^{+}\nu_{\mu})}{\mathcal{B}r(D^{+}\to
K^{*}(1410)(\to K ^{-}\pi^{+}) \, e^{+}\nu_{e})} = 0.714$ and
$\frac{\mathcal{B}r(D^{+}\to K^{*}_{0}(1430)(\to K ^{+}\pi^{-}) \,
\mu^{+}\nu_{\mu})}{\mathcal{B}r(D^{+}\to K^{*}_{0}(1430)(\to K ^{+}\pi^{-}) \,
e^{+}\nu_{e})} = 0.774$. Results in this work will serve a calibration for the
study of $c \to s $ in $D$ meson decays in future and provide useful
information towards the understanding of the properties of the $K^{*}$ meson,
as well as $K \pi$ system.
