I'm struggling with this integral:

$$\int\_{-\infty}^{\infty}x^{2}e^{-ax^{2}}dx = \frac{\sqrt{\pi}}{2a^{3/2}}$$

Is there a way to do this integration without using integration by parts and then explicitly relying on the Gaussian integral, instead using the Gamma function $\Gamma(a)=\int\_{0}^{\infty}x^{a-1}e^{-x} dx$ ? I had tried the substitution $u=ax^2$ to give $\frac{2}{a}\int\_{0}^{\infty}ue^{-u}du$ but that doesn't seem to get me far. 

Thanks in advance for any help or hints !