probability - Evaluating $int_{-infty}^infty e^{tx^2} frac{1}{sqrt{2pi} sigma} exp [ frac{-x^2}{2sigma^2} ] dx$

I have so far $$M(t) = E(e^{tX^2}) = \int_{-\infty}^\infty e^{tx^2} \frac{1}{\sqrt{2\pi} \sigma} \exp \left[ \frac{-x^2}{2\sigma^2} \right ] \ dx = \int_{-\infty}^\infty \frac{1}{\sqrt{2\pi}\sigma} \exp \left [ \frac{-x^2(1-t)}{2\sigma^2} \right ] \ dx$$

I've tried numerous substitutions and playing around but with no luck. There is a hint to use the fact that the integral of any density of a normal dist. is $1$.

Any help please

edit: answer should be $$M(t) = (1-2\sigma^2 t)^{-1/2}$$