I would like to calculate stochastic differential of:
$$X_t=\left(\int_0^t(s^3+B_s) \,dB_s \right)(2t+tB_t)=Y_tZ_t$$
I would like to use: $d(Y_tZ_t)=Z_t \, dY_t +Y_t \, dZ_t+dY_t \, dZ_t\tag{$*$}$
$$Y_t:=\int_0^t(s^3+B_s) \, dB_s$$
$$Z_t:=2t+tB_t$$
$$dY_t=(t^3+B_t)\,dB_t$$
$$dZ_t=(2+B_t)\,dt+t\,dB_t$$
I put everything in $(*)$ and get a proper result, right?