Let $$F(s)=\frac{\Gamma(1+b)\Gamma(1+s)}{s(\Gamma(1+b+s)-\Gamma(1+b)\Gamma(1+s))}$$
What is the inverse Laplace transform of this function?
$b$ are positive real numbers. Even if this could be solved for $b$ an integer, it will be helpful.
Or if it cannot be done directly, can we do it approximately?
Many thanks!