\[{\mathfrak T=\int_{2}^{\color{forestgreen}{\phi}}\left(\dfrac{x^2+x}{\displaystyle\sum_{r=1}^{x}\left(\int_{0}^1\left( 3r^2y^2+4ry+r^2\right)\,dy\right)}\right)\,dx}\]

\[\Large{\mathfrak{T}=\ln {\dfrac{\color{orange}{\varphi^{\varphi}}}{8}}}\]

\[\huge{ \color{orange}{\varphi}=?}\]

Given

\[\large\color{forestgreen}{\phi}=\dfrac{\color{blue}{\psi}}{4}+1\]

\[\large\color{blue}{\psi}={\sqrt{8\sqrt{216\sqrt{8\sqrt{216\cdots}}}}}\]

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