Need help with $\int_0^\infty x^{-\frac{3}{2}}\ \text{Li}_{\sqrt{2}}(-x)\ dx$

I need help with solving this integral: $$\int_0^\infty x^{-\frac{3}{2}}\ \text{Li}_{\sqrt{2}}(-x)\ dx,$$ where $\text{Li}_{s}(z)$ is the polylogarithm.

Solution 1:

$$\int_0^\infty x^{-\frac{3}{2}}\ \text{Li}_{\sqrt{2}}(-x)\ \mathrm dx=-2^{\sqrt{2}}\pi.$$

Use formula (3) from here to get an integral representation of the polylogarithm: $$\text{Li}_s(-x)=-\frac{1}{\Gamma(s)}\int_0^\infty\frac{k^{s-1}}{\frac{e^k}{x}-1}\mathrm dk.$$ Then, changing the order of integration, $$\int_0^\infty x^{-p}\ \text{Li}_s(-x)\mathrm dx=-\frac{1}{\Gamma(s)}\int_0^\infty\int_0^\infty\frac{x^{-p}\,\ k^{s-1}}{\frac{e^k}{x}-1}\mathrm dx\ \mathrm dk=\\\frac{\pi}{\Gamma(s)\sin \pi p}\int_0^\infty e^{k(1-p)}\ k^{s-1}\mathrm dk=\frac{\pi}{(p-1)^s \sin \pi p}$$