calculus and analysis - Integral does not converge (when it should)

Bug introduced in 10.0 and fixed in 10.0.2

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I was looking at integrals like:

Integrate[HermiteH[50, x]*Exp[-x^2], {x, 0, Infinity}]

which gave me a "does not converge on $(0,\infty)$" error. On the other hand something like

Integrate[(x^50)*Exp[-x^2]], {x, 0, Infinity}]
Integrate[HermiteH[4,x]*Exp[-x^2]], {x, 0, Infinity}]

works just fine. In general, for $n\geq 16$, $\int_0^\infty H_n(x)e^{-x^2}dx$ is reported as divergent.

Is there an easy way to resolve this bug?

I should mention that I'm using Mathematica 10.0.0.0.