  Using $2.93 \rm fb^{-1}$ of $e^+e^-$ collision data collected at a
center-of-mass energy of 3.773 GeV with the BESIII detector, the first
observation of the doubly Cabibbo-suppressed decay $D^+\to K^+\pi^+\pi^-\pi^0$
is reported. After removing decays that contain narrow intermediate resonances,
including $D^+\to K^+\eta$, $D^+\to K^+\omega$, and $D^+\to K^+\phi$, the
branching fraction of the decay $D^+\to K^+\pi^+\pi^-\pi^0$ is measured to be
$(1.13 \pm 0.08_{\rm stat} \pm 0.03_{\rm syst})\times 10^{-3}$. The ratio of
branching fractions of $D^+\to K^+\pi^+\pi^-\pi^0$ over $D^+\to
K^-\pi^+\pi^+\pi^0$ is found to be $(1.81\pm0.15)$\%, which corresponds to
$(6.28\pm0.52)\tan^4\theta_C$, where $\theta_C$ is the Cabibbo mixing angle.
This ratio is significantly larger than the corresponding ratios for other
doubly Cabibbo-suppressed decays. The asymmetry of the branching fractions of
charge-conjugated decays $D^\pm\to K^\pm\pi^\pm\pi^\mp\pi^0$ is also
determined, and no evidence of $CP$ violation is found. In addition, the first
evidence of the $D^+\to K^+\omega$ decay, with a statistical significance of
3.3$\sigma$, is presented and its decay branching fraction is determined to be
$({5.7^{+2.5}_{-2.1}}_{\rm stat}\pm0.2_{\rm syst})\times10^{-5}$.
