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Mathematica evaluates the integral in terms of a hypergeometric function, $$ \int_{0}^{\infty} \int_{-1}^{1} \frac{(ku + w)f_{U}(u)f_{W}(w)}{ku+\frac{1}{ku}+2w} dwdu=$$ $$\frac{k^2 4^M u^{2 M+1}\Gamma(M+\tfrac{1}{2})}{\sqrt{\pi } \left(k^2 u^2-1\right)^3\left(u^2+1\right)^{2 M}} \left[\frac{1}{(M-1)!} \left(k^4 u^4+4 M k^2 u^2+4M+3\right)-\frac{4 M}{(M+1)!} \left(k^2 u^2+1\right) (\tfrac{3}{4}+m^2+2m)\, \, _2{F}_1\left(-\tfrac{1}{2},1;M+2;\frac{4 k^2 u^2}{\left(k^2 u^2+1\right)^2}\right)\right].$$