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The Gaussian decay makes the integral absolutely convergent, so odd symmetry gives $\boxed{{\int }_{-\infty }^{\infty }f(x)dx=0}$. On the positive half-line substitute $u=3x^2/2$, so $x^{3}dx=2udu/9$. ThusThe last integral is $\Gamma (2)=1$, an instance of the odd Gaussian moments on a half-line.