# Integral of the modified Bessel function of the first kind zero order involving power and exponential functions

Is there a closed form expression for an integral of the modified Bessel function of the first kind zero order including the following?

$\int_0^\infty x^a e^{-bx^2} I_0(cx)\ x \,dx$

where a is positive integer, $b$ and $c$ are positive real.

Please also clarify if integrals including such expressions can be of closed form, e.g., for specific $a$,$b$, or $c$, or if they can only be numerically evaluated for any value included. If this can be only numerically evaluated, is there any closed formed formula for approximating this expression?

$$\int_{0}^{\infty}x^{\alpha-1}e^{-px^2}I_{\nu}(cx)dx=2^{-\nu-1}c^{\nu}p^{-\frac{\alpha+\nu}{2}}\frac{\Gamma(\frac{\alpha+\nu}{2})}{\Gamma(\nu+1)}\, _1F_1\left(\frac{\alpha+\nu}{2},\nu+1,\frac{c^2}{4p}\right).$$ Here $\mathrm{Re}\,p>0$, $\mathrm{Re}(\alpha+\nu)>0$, $-\pi<\arg c<\pi$, and $_1F_1$ denotes confluent hypergeometric function. For $\nu=0$ this gives $$\int_{0}^{\infty}x^{\alpha-1}e^{-px^2}I_{0}(cx)dx=\frac12p^{-\frac{\alpha}{2}}\Gamma(\alpha/2)\; _1F_1\left(\frac{\alpha}{2},1,\frac{c^2}{4p}\right).$$ Further, for integer $\alpha$ (evens are better than odds) $_1F_1$ simplifies. For example, $$_1F_1(1/2,1,z)=e^{z/2}I_0(z/2),\qquad _1F_1(1,1,z)=e^z.$$ Similar answers for greater integer values of $\alpha$ can be obtained using recursion formulas for $_1F_1$ w.r.t. parameters.