A properly mounted, properly manned, and properly equipped western army probably could dispatch Isil's caliphate reasonably quickly, if it were prepared to inflict and sustain the casualties normal in traditional warfare.
But if France has that in mind, the evidence suggests she'll be on her own.
•The Americans - and the British at least among their allies - want Isil to go but think it will take a properly trained ground force to get rid of it and are not prepared to commit troops, for domestic political reasons. 44, additionally, believes that there is no point defeating Isil while the conditions that led to its rise are in place, such as localised sectarian conflict, though he has no real notion of how to get rid of those conditions.
•Some elements in the Iraqi government may genuinely want to see the country reunited at all costs but others realise that if the Sunnis in Iraq return to the national fold the Shia majority will have to share power with them. Currently, Shia supremacist parties hold sway. These Shia parties are linked to the Shia militias that are fighting Isil on the ground - but signally failing to attack them in Sunni Arab strongholds like Fallujah and Mosul.
•The Assad regime's strategy presumably sees it eventually taking on Isil, but over most of the civil war its main enemy has been non-Isil rebels, who pose a more direct threat to regime heartlands. Having Isil fight them simultaneously has not been a bad thing.
•The Iranians don’t want Isil to gain ground but having a Sunni bogeyman, for whose existence they can blame their Sunni foes in the Gulf, has a certain use. The existence of Isil also prevents Sunni Arab states presenting a united face against their own ambitions.
•The Gulf states want to keep Isil down but they also see Isil as a lesser enemy than Iran, and certainly aren’t going to waste forces attacking fellow Sunnis that they could be using to fight Shia proxies in Yemen.
•Russia has ruled out sending ground troops, and has mostly targeted non-Isil rebels with its bombing campaign, since it believes the priority is to preserve the Assad regime. As these rebels are also fighting Isil, this has had the unfortunate effect of actually helping Isil advance in some areas, even as it cedes ground to Russian-backed regime forces in others.
•Turkey has made clear its main enemy is the Kurdish PKK, rather than Isil.
•The Kurds have a track record of success against Isil in both Iraq and Syria - thanks in part to their alliance with the US and other western air forces. But they have little interest in pushing beyond Kurdish territory. In Iraq, the Peshmerga took Sinjar last week, not Mosul.