**Hint** : It is solvable by setting up a system of four equations in four unknowns.
We are given:
$$\int_{-1}^{1} af(-1) + bf(1) +cf'(-1)+df'(1)$$
I will do the first and you can do the rest:
$$f(x) = x^0 = 1 \implies \displaystyle \int_{-1}^1 ~ 1~ dx = 2 = a(1) + b(1) + c(0) + d(0) $$
Now repeat this for $f(x) = x^1, x^2, x^3$ and set up and solve the $4x4$ system.
You should arrive at:
$$a = 1, b = 1, c = \dfrac 13, d = -\dfrac 13$$