Integral with exponential function

Question

I'm trying to determine the order of an integral involving the derivative of the heat kernel on the real line. Based on some numerics, it appears as though the following identity holds: $$ \int_{\mathbb{R}} |x|^{3/2} e^{-x^2/t}dx = C t^{5/4} $$ for some constant involving a gamma function. Is this true? If so, do you have any suggestions about tackling this integral?

According to the Gamma function wiki, it looks like the identity concerning the "stretched exponential function" might apply, though that might only work for integer exponents of $x$. However, assuming that identity holds for rational exponents, I get a different result $(t^{5/2})$, so I'm tempted to think that identity won't work in this context.

Answer 1

If $t\in\mathbb{R}^+$, through the substitution $x=z\sqrt{t}$, followed by $z=\sqrt{u}$, we have: $$I(t)=\int_{0}^{+\infty}x^{3/2} e^{-x^2/t}\,dx = t^{\frac{5}{4}}\int_{0}^{+\infty}z^{3/2}e^{-z^2}\,dz = \frac{1}{2}t^{\frac{5}{4}}\int_{0}^{+\infty}u^{1/4}e^{-u}\,du$$ so:

$$ I(t) = t^{\frac{5}{4}}\cdot\frac{\Gamma\left(\frac{5}{4}\right)}{2}=\color{red}{t^{\frac{5}{4}}\cdot\frac{\Gamma\left(\frac{1}{4}\right)}{8}}.$$