Is there a closed form for $\int x^n e^{cx}\,\mathrm dx$?
It means you differentiate with respect to c, n times
Use Wolfram Online Integrator, for example. The general answer is given in terms of the Incomplete Gamma Function.
It is the (n-fold because of the exponent) derivative of $\frac{e^{cx}}{c}$ with respect to $c$, considering $x$ to be fixed. So for $n=1$ it is $\frac{c^2e^{cx}-e^{cx}}{c^2}$