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I agree, it must be a problem with normal. The following works as expected

sage: f(x,y)= x*y^3
sage: f
(x, y) |--> x*y^3
sage: g = f.integral(y,0,7)
sage: g
(x, y) |--> 2401/4*x
sage: plot(g,-5,5)

Where as this returns the error you've seen:

sage: f(x,y,z)= x*y^3+z
sage: g = f.integral(y,0,7)
sage: plot(g,-5,5)
ValueError: free variable: z

If you try to simply integrate the normal function, do you get a number, or another function of x?