\begin{multline*} \scriptstyle
  P^{(2)} =
    \frac{T^2 \left(\omega ^2 \left(2 \dot{p}_1 (T-1)^2+T \left(3 \dot{\omega}^2 T-2 (\dot{\omega}+\ddot{\omega} T) \omega \right)\right)-p_2\right)-p_1^2 (T-1)^4-2 p_1 (T-1) T \omega  (2 \dot{\omega} (T-1) T-(1+T) \omega )}{2 T^2} \\ \scriptstyle
    + \frac{2 \omega  \left(p_1 \left(T^2-1\right) \omega -2 \dot{\omega} p_1 (T-1)^2 T+T \left(\dot{p}_1 (T-1)^2+\dot{\omega}^2 T^2\right) \omega -T^2 (\dot{\omega}+\ddot{\omega} T) \omega ^2\right)}{T} a \\ \scriptstyle
    + 2 T \omega ^2 \left(\dot{\omega}^2 T-\dot{\omega} \omega -\ddot{\omega} T \omega \right) a^2
    + 2 \omega  \left(2 \dot{\omega} p_1 (T-1)+\frac{\dot{\omega}^2 T^2 \omega }{T-1}-\frac{\left(\dot{p}_1 (T-1) T+p_1 (1+T)\right) \omega }{T}\right) x y \\ \scriptstyle
    + \frac{4 T \omega ^2 \left(\ddot{\omega} (T-1) T \omega -\dot{\omega}^2 (T-1) T-\dot{\omega} \omega \right)}{(T-1)^2} a x y
    + \frac{T \omega ^2 \left(2 \dot{\omega}^2 (T-1) T-\dot{\omega} (T-3) \omega -2 \ddot{\omega} (T-1) T \omega \right)}{(T-1)^3} x^2y^2.
\end{multline*}