What is a tractable calculus of Integral of $\exp(a x^n +b)$

Question

What is the anti-derivative in $x$ of $$f(x)=\exp(a x^n +b)$$ with $(a,b)\in \mathbb{R}^2$ and $n>0$ for $x\in \mathbb{R}$ ?

Answer 1

$$I=\int\exp(ax^n+b)\,dx$$ take the $b$ outside the integral as it is just a constant, and let $u=a^{1/n}x\Rightarrow dx=a^{-1/n}u$ and you are left with: $$I=a^{-1/n}e^b\int\exp(u^n)\,du$$ now with another simple substitution you should see it looks a lot like the Exponential integral