Let $(W_t)_{t\in [0,T]}$ a standard Brownian Motion and let $c\in\mathbb{R}$. How to compute
$$\mathbb{E}\left[\int_0^TW^2_sdW_s\mid W_T=c\right]?$$
We thought about rewriting the stochastic integral via Itô's Lemma to find that
$$\int_0^TW^2_sdW_s = \frac{1}{3} W^3_T - \int_0^TW_sds$$
and further
$$\int_0^TW_sds = TW_T-\int_0^TsdW_s.$$
This didn't lead to anything better. Does one a have to construct a markov kernel instead?