We have $$I(n)=\int z^n e^{az}dz=\int z^n \left (\frac{1}{a}e^{az}\right )'dz=\frac{1}{a}z^ne^{az}-\frac{1}{a}\int nz^{n-1}e^{az}dz \\ \Rightarrow I(n)=\frac{1}{a}z^ne^{az}-\frac{1}{a}nI(n-1) \ \ \text{ with } \ \ I(1)=\int ze^{az}dz=\frac{1}{a}ze^{az}-\frac{1}{a^2}e^{az}$$ 

When we solve this recursive relation, we get an element of the ring $\mathbb{C}[z, e^{\lambda z}\mid \lambda \in \mathbb{C}]$, so an expression that contains exponentials, polynomials and their products. 

Is this correct?