Measurement of the branching fractions of $D_s^+\rightarrow \eta'X$ and $D_s^+\rightarrow \eta'\rho^+$ in $e^+e^-\to D^+_sD^-_s$

We study?D+s?decays to final states involving the?η′?with a 482pb?1?data sample collected at?s√?= 4.009GeV with the \mbox{BESIII} detector at the BEPCII collider. We measure the branching fractions?B(D+s→η′X)?= (8.8±1.8±0.5)%?and?B(D+s→η′ρ+)?= (5.8±1.4±0.4)%?where the first uncertainty is statistical and the second is systematic. In addition, we estimate an upper limit on the non-resonant branching ratio?B(D+s→η′π+π0)<5.1%?at the 90%?confidence level. Our results are consistent with CLEO's recent measurements and help to resolve the disagreement between the theoretical prediction and CLEO's previous measurement of?B(D+s→η′ρ+).