Identify the function f(t) that has the following Laplace transform,

$$\stackrel{~}{f}(s)={\int}_{0}^{\mathrm{\infty}}f(t){e}^{-st}dt={\left(\frac{1+\alpha s}{1+\alpha (s-{s}_{0})}\right)}^{p}$$

where $\alpha ,{s}_{0},p$ are positive parameters. Any suggestions about how to approach this problem?

$$\stackrel{~}{f}(s)={\int}_{0}^{\mathrm{\infty}}f(t){e}^{-st}dt={\left(\frac{1+\alpha s}{1+\alpha (s-{s}_{0})}\right)}^{p}$$

where $\alpha ,{s}_{0},p$ are positive parameters. Any suggestions about how to approach this problem?