I am struggling to get the inverse Laplace transform of the ratio of two incomplete gamma function as follow. $\frac{{{\Gamma ^m}\left[ {\alpha, \left( {\frac{1}{\beta } + s} \right)\eta } \right]}}{{{\Gamma ^m}\left( {\alpha, \frac{\eta }{\beta }} \right)}}$
where $m$ is a positive integer,$\alpha$,$\beta$ and $\eta $ are positive real number. $s$ is the complex number frequency parameter of Laplace transform. The inverse Laplace transform is as follow $\frac{1}{{2\pi j}}\int_{r - j\infty }^{r + j\infty } {\frac{{{\Gamma ^m}\left( {\alpha ,\left( {\frac{1}{\beta } + s} \right)\eta } \right)}}{{{\Gamma ^m}\left( {\alpha ,\frac{\eta }{\beta }} \right)}}{e^{sx}}ds} $
It seems to be hard work. Do someone have a way to get the result. An approximation is also a good result.