We show that there exists c>0 such that any subset of {1,…,N} of density at least (loglogN) ^−c contains a nontrivial progression of the form x, x+y, x+y ². This is the first quantitatively effective version of the Bergelson–Leibman polynomial Szemerédi theorem for a progression involving polynomials of differing degrees. Our key innovation is an inverse theorem characterising sets for which the number of configurations x, x+y, x+y ² deviates substantially from the expected value. In proving this, we develop the first effective instance of a concatenation theorem of Tao and Ziegler, with polynomial bounds.