Dissatisfaction with the strict and intellectual rules of the french royal academy gave rise to which art movement?

Impressionism

The French Royal Academy was founded in 1648 as the Royal Academy of Painting and Sculpture.  After being shut down during the French Revolution, when reestablished, it was renamed the Academy of Painting and Sculpture.  (The "royals" weren't on the throne anymore.)  In 1816 two other academies (music and architecture) were merged into it and it became the Academy of Fine Arts.
A problem with the Academy was that it developed essentially a monopoly over  the visual arts, and tended to corral artists into adopting an "Academy" style that adhered to specific rules and methods.  Not until the Impressionist movement of the late 19th century did a group of non-Academic style artists manage to challenge the rigid aesthetic governance of the Academy.