I managed to solve an initial value problem in the Laplace domain in terms of a special function
$ F(s) = c_2 \frac{1}{{{\left( {{s}^{1 +\beta}}-1\right) }^{\frac{1}{\beta+1}}}}+ c_1 \frac{s}{{{\left( {{s}^{1 +\beta}}-1\right) }^{\frac{1}{\beta+1}}}} \ {_{2}{F}_{1}}\left( \frac{1}{\beta+1},\frac{\beta}{\beta+1}; \frac{1}{\beta+1}+1;{{s}^{\beta+1}}\right) $ where $ 0 \leq \beta \leq 1$ and $ _{2}{F}_{1}$ if the hypergeometric function.
However, I am unable to find the ILT or give an approximation in the time domain in the general case.
Is there a way to invert the equation or at least to give an approximation for short times?