How solve a double integral analytically without using numeric methods

I need to solve a double integral analytically without using numeric methods, which means no quad function and the like.

The integral is from -2 to 2 (outside) and 0 to 4 (inside) of

x^2 -3y^2 +xy^3 dx dy

Answer 1

[Walter Roberson]

If you have the symbolic toolbox

syms x y
int(int(x^2 - 3*y^2 + x*y^3, x, 0, 4), y, -2, 2)