\begin{align*} M(t) & = \int_0^{\infty} e^{tx} \frac{x^{r/2-1} e^{-x/2} }{\Gamma(r/2) 2^{r/2} } dx\\ & = \int_0^{\infty} \frac{x^{r/2-1} e^{-x(1-2 t)/2} }{\Gamma(r/2) 2^{r/2}} dx\\ & = \int_0^{\infty} \frac{\left ( \frac{u}{1-2t} \right )^{r/2-1} e^{-u/2} }{\Gamma(r/2) 2^{r/2} } \frac{1}{1-2t} du\\ & = \left ( \frac{1}{1-2t} \right )^{r/2} \int_0^{\infty} \frac{u^{r/2-1} e^{-u/2} }{\Gamma(r/2) 2^{r/2}} du\\ & = \left ( \frac{1}{1-2t} \right )^{r/2} \end{align*}