Given that the indefinite integral

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\[ \large \int \sin^{\frac {5}{2}} x \cos^3 x \space \mathrm{d}x \]

equals to

\[ \large 2 \sin^{ \frac {A}{2} } x \left [ \frac {1}{B} - \frac {1}{C} \sin^2 x \right ] \]

neglecting the arbitrary constant.

Find the value of \(A + B - C \).

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