Stock-weighted call payoff in Black-Scholes

Work in the Black-Scholes model with rate $r$ and volatility $\sigma$. A contract pays at time $T$ the amount $S_T\max(S_T-K,0)$, that is, the stock price multiplied by a call payoff. Find its price in closed form.

$S_0^2e^{(r+\sigma^2)T}N(d_1+\sigma\sqrt T)-KS_0N(d_1)$, with $d_1$ the usual Black-Scholes $d_1$. Split the payoff into $S_T^2\mathbf 1_{S_T>K}-KS_T\mathbf 1_{S_T>K}$. The second piece is $K$ asset-or-nothing calls worth $KS_0N(d_1)$; the first uses $\mathbb E[S_T^2]=S_0^2e^{(2r+\sigma^2)T}$ and a change of measure that shifts the log-drift by $2\sigma^2$.