# Is there a name for this family of integral?

This one: $$\int_{0}^{\bar{x}}e^{-x^{a}}x^{b}(1-x)^{c}dx,a,b,c\ge0$$. When $$a=1,c=0,\bar{x}=\infty$$ it is the gamma function.

• The integral does not even converge for $a>0$. – Bullet51 Jun 13 at 1:41
• no name, no closed form evaluation. – Carlo Beenakker Jun 13 at 5:47

Maple evaluates $$\int e^{-x^a}x^b\;dx$$ in terms of the Whittaker M function, which may also be written as $${}_1F_1$$ hypergeometrics. $$\int \!{{\rm e}^{-{x}^{a}}}{x}^{b}\,{\rm d}x= \\ {\frac {{{\rm e}^{-{ x}^{a}/2}}}{ \left( b+1 \right) \left( a+b+1 \right) \left( 2\,a+b+1 \right) } \left( \left( \left( a+b+1 \right) {x}^{b/2+1/2-3/2\,a}+{ x}^{b/2+1/2-a/2}a \right) a{{\rm M}_{-{\frac {a-b-1}{2a}},\,{ \frac {2\,a+b+1}{2a}}}\left({x}^{a}\right)}+{x}^{b/2+1/2-3/2\,a}{ {\rm M}_{{\frac {a+b+1}{2a}},\,{\frac {2\,a+b+1}{2a}}}\left({x }^{a}\right)} \left( a+b+1 \right) ^{2} \right) }$$