$I=\int_{0}^{\pi/2}\frac{\sin^{3/2}x}{\sin^{3/2}x+\cos^{3/2}x}\,dx$

$I+\int_{0}^{\pi/2}\frac{\cos^{3/2}x}{\sin^{3/2}x+\cos^{3/2}x}\,dx=\frac{\pi}{2}$
⇒ $2I=\frac{\pi}{2} \Rightarrow I=\frac{\pi}{4}$

Required integral:
$J=\int_{0}^{2I}\frac{x\sin x\cos x}{\sin^{4}x+\cos^{4}x}\,dx=\int_{0}^{\pi/2}\frac{x\sin x\cos x}{\sin^{4}x+\cos^{4}x}\,dx$

Using symmetry:
$2J=\frac{\pi}{2}\int_{0}^{\pi/2}\frac{\sin x\cos x}{\sin^{4}x+\cos^{4}x}\,dx$

Now,
$\int_{0}^{\pi/2}\frac{\sin x\cos x}{\sin^{4}x+\cos^{4}x}\,dx=\frac{\pi}{4}$

So,
$2J=\frac{\pi}{2}\cdot\frac{\pi}{4}=\frac{\pi^2}{8}$
⇒ $J=\frac{\pi^2}{16}$

$\boxed{\frac{\pi^2}{16}}$