Using e^+e^- data taken at the center-of-mass energy of 3. 773 GeV with the BESIII detector, corresponding to an integrated luminosity of 20. 3 fb^-1, we report the first measurement of the decay dynamics of the semileptonic decays D^+ (0) K^-π^+π^0 (-) e^+νₑ. The amplitude analysis gives the hadronic form factors of the semileptonic D transitions into the axial-vector meson K₁ (1270) to be rA= (-11. 21. 0 ₒₓ₀ₓ0. 9 ₒₘₒₓ) 10^-2 and rV = (-4. 3 1. 0 ₒₓ₀ₓ2. 5 ₒₘₒₓ) 10^-2. This is the first in the semileptonic decays of heavy mesons into axial-vector mesons. The angular analysis yields an up-down asymmetry A^ₔ₃ = 0. 010. 11, which is consistent with the Standard Model prediction. In addition, the branching fractions of D^+ K₁ (1270) ⁰ e^+νₑ and D⁰ K₁ (1270) ^- e^+νₑ are determined with improved precision to be (2. 270. 11 ₒₓ₀ₓ0. 07 ₒₘₒₓ0. 07 ₈₍ₔₓ) 10^-3 and (1. 020. 06 ₒₓ₀ₓ0. 06 ₒₘₒₓ0. 03 ₈₍ₔₓ) 10^-3, respectively. No significant signals of D^+ K₁ (1400) ⁰ e^+νₑ and D⁰ K₁ (1400) ^- e^+νₑ are observed and their branching fraction upper limits are set as 1. 410^-4 and 0. 710^-4 at 90\% confidence level, respectively.
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