Why would you choose SABR over a quadratic local vol model ? 

To have better fit and closed formulas?

A bit of a background, I built a quadratic model based, combined with an efficient numerical scheme. The results are comparable to sabr(when sticking to the quadratic form), and better with some extensions of the local vol form(but that is not the subject). The time computation is acceptable (8-10 times slower than SABR formula), hence my question.

The results for what -- what's the application?

Sorry, I meant European option pricing only

No, I meant, what is the underlying? For example, both models would be a very poor choice for equities or perhaps other traded underlyings -- but for different reasons. 

Interest rate , SABR is "the" model for that class asset. 

OK, fair enough. 

If you had said equities, I would have complained about the lack of mean-reversion in the SABR vol and the loss of martingality in the quadratic local vol.

Why don't you try this? https://papers.ssrn.com/sol3/Papers.cfm ... id=2793125

@Alan Exactly.
@Gamal, this is a Cheyette model with local vol function, that is too huge for European option pricing, and very slow. This model can be good for pricing more complex product.
 My point is that I always hear the same stories in the rate world. SABR has stochastic volatility , hence better than local volatilty models that assume that stochasticity only comes from the underlying itself. Who cares when we price Europeans, where only the final distribution matters? Well, the other explanation is then that the dynamics of the smile is bad with a local volatility model (non parametric). I agree that the whole market dynamics info is not contained in one smile, so if we use the non parametric form, we have no control of the ATM backbone in particular.
 All SABR parameters are calibrated to the smile, expect beta. Therefore, beta is the only exogenous parameter that controls the SABR dynamics , the rest is implied by the smile (hoping that it goes with us).
So then , why don't we parametrize the local volatility model and one of its parameters will control the backbone ?
I could see that a quadratic local volatility form can do the job, but  I am pretty sure that I am not the only one who has thought about it, hence my question

Read carefully the paper. The calibration is straightforward.

Now, we are talking about model XYZ vs SABR (not my question ).  How do I price European options in this paper ?  

After calibrating the model using the particle algorithm, we have computed rolling-tenor swaption smile using a (quasi) Monte Carlo pricer with N = 2^15 paths and a timestep ∆t = 1/250 (see Figures 1, 2). Swaption fair values are quoted in (normal) implied volatility and the strikes are quoted in standard deviation with respect to the ATM volatility σATM: K = s t,t,+θ 0 + σATM √ t stdev with stdev ∈ [−2, 2]. The computation time is around 8 seconds for maturities up to 10 year" .

That is very slow, and in terms of European option modelling, why would one introduce a local volatility model on an HJM framework , while you can do the same thing at the forward rate level ?  Plus, there is no way to control the ATM backbone that makes it a bad candidate.
This paper might be good for for Bermudan pricing to compare with Markov functionals and Cheyette( vol sto), but it is an overkill for European option pricing. 

There's also a tricky approximation there. You practically have a closed formula.

not convinced.

Wrong. It's working pretty well.