This Letter reports the first extraction of individual antineutrino spectra from $^{235}$U and $^{239}$Pu fission and an improved measurement of the prompt energy spectrum of reactor antineutrinos at Daya Bay. The analysis uses $3.5times 10^6$ inverse beta-decay candidates in four near antineutrino detectors in 1958 days. The individual antineutrino spectra of the two dominant isotopes, $^{235}$U and $^{239}$Pu, are extracted using the evolution of the prompt spectrum as a function of the isotope fission fractions. In the energy window of 4--6~MeV, a 7% (9%) excess of events is observed for the $^{235}$U ($^{239}$Pu) spectrum compared with the normalized Huber-Mueller model prediction. The significance of discrepancy is $4.0sigma$ for $^{235}$U spectral shape compared with the Huber-Mueller model prediction. The shape of the measured inverse beta-decay prompt energy spectrum disagrees with the prediction of the Huber-Mueller model at $5.3sigma$. In the energy range of 4--6~MeV, a maximal local discrepancy of $6.3sigma$ is observed.
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