An idea to compute the following integral $ \int_0^1 s^a (1-s)^b e^{-c/s} \, ds$

Question

Someone has an idea to calculate the following integral $$I_{a,b,c} = \int_{0}^{1} s^{a} (1-s)^b e^{-\frac{c}{s}} \, ds; \quad a<0, \mbox{and}\, b,c>0.$$ Thank you in advance

Answer 1

$\int_0^1s^a(1-s)^be^{-\frac{c}{s}}~ds$

$=\int_\infty^1\left(\dfrac{1}{t}\right)^a\left(1-\dfrac{1}{t}\right)^be^{-ct}~d\left(\dfrac{1}{t}\right)$

$=\int_1^\infty t^{-a-b-2}(t-1)^be^{-ct}~dt$

$=\int_0^\infty(t+1)^{-a-b-2}t^be^{-c(t+1)}~d(t+1)$

$=e^{-c}\int_0^\infty t^b(t+1)^{-a-b-2}e^{-ct}~dt$

$=e^{-c}~\Gamma(b+1)U(b+1,-a,c)$ (according to https://en.wikipedia.org/wiki/Confluent_hypergeometric_function#Integral_representations)