Integration using maple

Question

I am trying to evaluate the following integral using maple but it returns the unsolved integral. Can anybody help me in using maple to solve this integral?

$$ \int {2\pi\lambda R e^{-\lambda \pi R^2}} D^{a\pi R^2} (a \pi R^2)^C dR $$

Answer 1

Regrouping and ignoring non-essential constants, the integral can be rewritten $$\int R^{2c+1}\alpha^{R^2}dR.$$ And by the change of variable $t=R^2$: $$\int t^c\alpha^tdt.$$ Maple should recognize the exponential integral function.

Answer 2

The antiderivative is $$\pi ^{C+1} \lambda \left(-a^C\right) R^{2 C+2} E_{-C}\left(\pi R^2 (\lambda -a \log (D))\right)$$ where $E_n(x)$ is a (generalized) exponential integral. It is important that the argument to $E_{-C}$ be positive (otherwise certain steps in the calculation are potentially invalid). With no bounds on the parameters in your integral, I can't determine whether this happened. See here for an exponential integral implementation in Maple, where it would be $Ei(-C,\dots)$.

Answer 3

Maple finds it, when putting D (which is reserved for the differentiation operator) to be a local variable and the other parameters to be real: $$local D; int(2*Pi*R*exp(-lambda*Pi*R^2)*D^{a*Pi*R^2}*(a*Pi*R^2)^C, R) $$ $$\, assuming \, real$$ outputs that, where the $\Gamma$ function is described here.