Area of a Region

Calculus Level pending

The region bounded by the curve \({\bf 2 x^2 + 12 xy - 3 y^2 - 42 = 0 }\) and the parallel lines \({\bf -3\sqrt{13}\:x - 2\sqrt{13}\:y + 39 = 0 }\) and \({\bf -3\sqrt{13}\:x - 2\sqrt{13}\:y - 39 = 0 }\) can be expressed as \({\bf b\sqrt{a * b} (a\sqrt{\frac{b}{c}} + \sqrt{c}\:ln(\frac{a + \sqrt{b}}{\sqrt{c}})) }\), \(\) where \({\bf a\:,b\:,and\: c }\) are coprime positive integers.

\(\)

Find: \({\bf a + b + c. }\)

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