You could be forgiven for being mystified by the fact that David O. Russell and Jennifer Lawrence decided to make the story of Joy Mangano, the inventor of the Miracle Mop, the subject of their third collaboration.

Cleaning products and the world of QVC and HSN might make for compelling drama on “Shark Tank,” but they don’t exactly come across as an ideal subject for a prestigious studio picture.

Yet here we are and somehow, against steep odds, Russell and Lawrence have reconfigured this story as a tribute to strong and independent women.

The filmmaker incorporates his characteristically frenzied style — aggressive camera movements, hyperactive overlapping dialogue — in service of a fine and even rousing piece of entertainment with the veneer, at least, of actual substance.

Lawrence plays Long Island’s Mangano, at the center of one of Russell’s characteristically dysfunctional cinematic families (i.e. “The Fighter”), where she’s the breadwinner, the voice of reason and a severely stretched-thin mom of two.

Russell spends a good amount of time fleshing out this dynamic, where Joy’s mother, Terry (Virginia Madsen), remains bedbound and her divorced father, Rudy (Robert De Niro), moves into the basement, which he shares with Joy’s deadbeat ex-husband, Tony (Édgar Ramírez), a struggling singer.

Adjusting to the atmosphere requires a mental re-calibration; Russell has always seemed so in love with barely organized chaos that occasionally he’s used it as a crutch to disguise weaknesses elsewhere.

Here, he knows the actual invention of the Miracle Mop is far less interesting than the circumstances that inspired it, that swirling mix of desperation and innovation that led Mangano toward it and compelled her to will it into existence.

It’s what the Mop represents that matters — a small triumph over the crushing reality that keeping a family together can be painful, all-consuming work. The filmmaker vividly drives home the point, even if his approach makes one as desperate as Joy for some time away from these people. Lawrence gives a characteristically strong-willed performance that inspires empathy for Joy and her plight, even though it sometimes feels as if something deeper is missing.

The movie’s quieter moments provide the necessary depth so that the ultimate litmus test is fulfilled: When Joy presents her mop on QVC for the first time, you feel every bit of her emotions.