\begin{align*} M(t) & = \int_0^{\infty} e^{tx} \frac{x^{r-1} \cdot e^{-\frac{x}{\mu}}}{\Gamma(r) \cdot \mu^r} dx\\ M(t) & = \int_0^{\infty} \frac{x^{r-1} \cdot e^{-x \left ( \frac{1}{\mu} - t \right )}}{\Gamma(r) \mu^r \left ( \frac{1}{\mu} - t \right )} dx\\ M(t) & = { \frac{1}{\left ( 1-t \mu \right )^r} } \int_0^{\infty} \frac{x^{r-1} \cdot e^{-\frac{x}{ \left ( \frac{\mu}{1-t \mu} \right )}}}{\Gamma(r) \cdot { \left ( \frac{\mu}{1-t \mu} \right )}^r} dx\\ & = \frac{1}{\left(-\mu t + 1 \right )^{r}}. \end{align*}