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

$\Large{\mathfrak{T}=\ln {\dfrac{\color{#EC7300}{\varphi^{\varphi}}}{8}}}$

$\huge{ \color{#EC7300}{\varphi}=?}$

Given

$\large\color{forestgreen}{\phi}=\dfrac{\color{#3D99F6}{\psi}}{4}+1$

$\large\color{#3D99F6}{\psi}={\sqrt{8\sqrt{216\sqrt{8\sqrt{216\cdots}}}}}$