TITLE: Closed form for $\int x^ne^{-x^m} \ dx\ ?$
QUESTION [10 upvotes]: While entertaining myself by answering a question, the following problem arose.
For what natural numbers $n,m$ does the following undefined integral have a closed form
$$\int x^ne^{-x^m} \ dx\ ?$$
Closed form means that the antiderivative consists only of powers of $x^{...}$ and $x$ in $e^{-x^{...}}$.
I created the following matrix showing for different pairs of $n$ and $m$ the nature of the antiderivative.
$$\begin{matrix}
& m&1&2&3&4&5&6&7\\
n\\
1&&\checkmark&\checkmark&\Gamma&\text{erf}&\Gamma&\Gamma&\Gamma\\
2&&\checkmark&\text{erf}&\checkmark&\Gamma&\Gamma&\text{erf}&\Gamma&\\
3&&\checkmark&\checkmark&\Gamma&\checkmark&\Gamma&\Gamma&\Gamma&\\
4&&\checkmark&\text{erf}&\Gamma&\Gamma&\checkmark&\Gamma&\Gamma\\
5&&\checkmark&\checkmark&\checkmark&\text{erf}&\Gamma&\checkmark&\Gamma\\
6&&\checkmark&\text{erf}&\Gamma&\Gamma&\Gamma&\Gamma&\checkmark\\
7&&\checkmark&\checkmark&\Gamma&\checkmark&\Gamma&\Gamma&\Gamma\\
\end{matrix}$$
$$$$
The $\checkmark$ sign stands for a closed form, "erf" signals that the antiderivative contains the  erf function , and $\Gamma$ signals that the antiderivative contains the upper incomplete $\Gamma$ function.
I have no clue. Does anybody?

REPLY [5 votes]: Let's start with a simple substitution $x=u^{1/m}$.  This gives us
$$I=\frac1m\int u^{(n+1)/m-1}e^{-u}\ du=\frac1m\gamma\left(\frac{n+1}m,x^m\right)+c$$
This trivially has closed forms for $\frac{n+1}m\in\mathbb N$ due to integration by parts.  Indeed, checking your table, it corresponds with every checkmark perfectly.
And just for the record, when $k\in\mathbb N$,
$$\int x^ke^{-x}\ dx=-e^{-x}\sum_{n=0}^k(k-n)!x^n+c$$