\[\int _{ 0 }^{ \infty }{ \sin { ( { x }^{ 2 } ) } { \left( \frac { x+4 }{ x } \right) }^{ 2 }dx } =a\pi +\sqrt { \frac { b }{ c } \pi } \]

If the above is true and \(b\) and \(c\) are coprime positive integers, find \(a+b+c\).

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