Fortunately, Wolfram Alpha gives the roots as functions of p. These roots contain a cube root of a square root. 

I explored a range of the p values in the regime where the square root within the cube root is real. 

For p within about 6 of the origin, there are two maxima of the area. In all these cases,

there is one real root and a pair of complex conjugate roots: 

x1=A(p), x2=B(p)+ib(p), and x3=B(p)-ib(p). 

Therefore, the triangle is isosceles with  

Area = (1/2) base x height = (1/2) |b-(-b)| |A(p) - B(p)|

Area = |A(p)-B(p)| |b| 

Let's define c(p) as:

 

c(p) = (2 p^3 -15 p^2 + term(p) + 15 p - 29)^(1/3)

where term(p) = (3) 3^(1/2) (-p^4 + 2 p^3 + 29 p^2 -30 p +31)^1/2

For term to be real, (1/2)-5 sqrt(5)/2 < p < (1/2) + 5 sqrt(5)/2

We have:

A(p) =         c(p)/(3 2^(1/3)) - 2^(1/3) q(p)/(3         c(p)) + (p-1)/3

B(p) =        -c(p)/(6 2^(1/3)) +         q(p)/(3 2^(2/3) c(p)) + (p-1)/3

b(p) = 3^(1/2) c(p)/(6 2^(1/3)) + 3^(1/2) q(p)/(3 2^(2/3) c(p))

where q(p) =  -p^2 +5 p - 1

So

|A(p)-B(p)| = |c(p)/(2 2^(1/3)) - ( 2^(1/3)/3 + 1/(3 2^(2/3)) ) q(p)/c(p)|

(See plot in link above) 

The triangle's area is maximized at p=4.5504 and area = 2.761

I am currently exploring outside this p-range as well. The exploration involves thinking about which of the three roots of the cube root and which of the two roots of the square root to mix together when making complex roots x1, x2, and x3 while the square root is complex.