$$ \int\exp\left(n \over x^{1 + a}\right)\, x^{m}\,\mathrm{d}x $$ This integral is aim at calculating an expression like the expression between stress and strain, but there is little effects on using integration by parts, so I have no idea which way to choose.
If impossible, can you use Taylor's formula to calculate an approximate expression way ?
Thanks for your help !
$$I=\int x^m \exp \left(x^{\frac{n}{\alpha+1}}\right)\,\mathrm d x$$
Substitute $u=x^{m+1}$ to get
$$I=\frac{1}{m+1}\int\exp\left({u^{\frac{n}{(\alpha+1)(m+1)}}}\right)\,\mathrm d u$$
This integral is expressible in terms of upper incomplete gamma function using the identity $$\int \exp{x^a}\,\mathrm d x=-\frac{\mathop\Gamma \left( \frac 1a, -x^a\right)}{a(-1)^{\frac 1a}}+C$$
We thus get \begin{align*}I&=\frac{1}{m+1}\cdot\left(-\frac{\left(\alpha+1\right)\left(m+1\right)\mathop{\Gamma}\left(\frac{\left(\alpha+1\right)\left(m+1\right)}{n},-u^\frac{n}{\left(\alpha+1\right)\left(m+1\right)}\right)}{n\left(-1\right)^\frac{\left(\alpha+1\right)\left(m+1\right)}{n}}\right)\\ &= -\dfrac{\left(\alpha+1\right)\operatorname{\Gamma}\left(\frac{\left(\alpha+1\right)\left(m+1\right)}{n},-x^\frac{n}{\alpha+1}\right)}{n\left(-1\right)^\frac{\left(\alpha+1\right)\left(m+1\right)}{n}}+C \end{align*}
which is equivalent to Dr. Sonnhard Graubner's answer in comments.