How do you show that [math]\displaystyle\sum_{m\geq 3}\left[4\binom{1/2}{m}+\binom{1/2}{m-1}\right]\frac{1}{\Gamma\left(m-\tfrac{3}{2}\right)}\int\limits_{0}^{\infty}\frac{x^{m-3/2}}{e^x-1}\,dx=\int\limits_{0}^{\infty}\frac{3 e^{-x} \left(2-2 e^x+x+e^x x\right)}{2 \sqrt{\pi } x^{5/2}(e^x-1)}\,dx[/math]?
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