Nonrational del Pezzo fibrations

Let $X$ be a general divisor in $\mathrm{\mid }3M+nL\mathrm{\mid }$ on the rational scroll $\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{j}({\oplus }_{i=1}^4{\mathcal{O}}_{{\mathbb{P}}^1}(d_i))$, where $d_i$ and $n$ are integers, $M$ is the tautological line bundle, $L$ is a fibre of the natural projection to ${\mathbb{P}}^1$, and $d_1⩾\mathrm{.}\mathrm{.}\mathrm{.}⩾d_4=0$. We prove that $X$ is rational $⟺$ $d_1=0$ and $n=1$.