Non-Elementary Integration · Superchallenge problem

Using $\displaystyle\int_0^{\pi/2}\sin^{2p-1}x\cos^{2q-1}x\,dx=\dfrac{\Gamma(p)\Gamma(q)}{2\Gamma(p+q)}$, find $\dfrac{\displaystyle\int_0^{\pi/2}\sin^9x\cos x\,dx}{\displaystyle\int_0^{\pi/2}\sin^5x\cos^5x\,dx}$ without evaluating either integral numerically.