Points $D$ and $E$ are chosen on side $BC$ of triangle $ABC$ such that $E$ is between $B$ and $D$ and $BE=1$, $ED=DC=3$.  If $\angle BAD=\angle EAC=90^\circ$, the area of $ABC$ can be expressed as $\tfrac{p\sqrt q}r$, where $p$ and $r$ are relatively prime positive integers and $q$ is a positive integer not divisible by the square of any prime.  Compute $p+q+r$.

[asy]
import olympiad;
size(200);
defaultpen(linewidth(0.7)+fontsize(11pt));
pair D = origin, E = (3,0), C = (-3,0), B = (4,0);
path circ1 = arc(D,3,0,180), circ2 = arc(B/2,2,0,180);
pair A = intersectionpoint(circ1, circ2);
draw(E--A--C--B--A--D);
label("$A$",A,N);
label("$B$",B,SE);
label("$C$",C,SW);
label("$D$",D,S);
label("$E$",E,S);
[/asy]