Find maximum area of rectangle

Calculus Level 4

A rectangle's bottom is at \(y=0\)

while its top corners are on the curve \(y=x{ (x-1) }^{ 2 }\) between \(x=0\) and \(x=1\). The maximum area of this rectangle can be expressed as

\[ \dfrac { a\sqrt { a } -b }{ c\sqrt { d } }\]

where \( a\) and \( d\) are prime numbers. What is the sum \(a+b+c+d?\)

(Don't count \( a\) twice)


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