Question Number 157873 by HongKing last updated on 29/Oct/21
$$\mathrm{if}:\:\boldsymbol{\alpha}\:=\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\:\frac{\sqrt[{\mathrm{3}}]{\mathrm{x}}}{\mathrm{1}\:+\:\sqrt[{\mathrm{3}}]{\mathrm{x}}}\:\:\mathrm{e}^{-\boldsymbol{\pi\mathrm{y}}\left(\mathrm{1}+\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\frac{\mathrm{1}}{\boldsymbol{\mathrm{x}}^{\mathrm{2}} }\right)} \:\mathrm{dydx} \\ $$$$\mathrm{find}:\:\sqrt{\mathrm{19683}\boldsymbol{\alpha}^{\mathrm{6}} \:-\:\mathrm{94041}\boldsymbol{\alpha}^{\mathrm{4}} \:+\:\mathrm{105786}\boldsymbol{\alpha}^{\mathrm{2}} } \\ $$$$ \\ $$