$$
\begin{align}
\int_0^\infty\frac{x^{3/2}}{(1+x)^4}\,\mathrm{d}x
&=\int_0^\infty\frac{2u^4}{(1+u^2)^4}\,\mathrm{d}u\\
&=\int_{-\infty}^\infty\frac{u^4}{(1+u^2)^4}\,\mathrm{d}u\\
&=\int_\gamma\frac{z^4}{(1+z^2)^4}\,\mathrm{d}z\\
&=\int_\gamma\frac{z^4}{(z+i)^4(z-i)^4}\,\mathrm{d}z\\
&=\int_\gamma\frac{(w+i)^4}{(w+2i)^4w^4}\,\mathrm{d}w\\
&=\frac1{16}\int_\gamma\frac{(1-iw)^4}{(1-iw/2)^4w^4}\,\mathrm{d}w\\
&=\frac1{16}\int_\gamma\left(\begin{array}{c}1-4iw-6w^2+4iw^3+O(w^4)\\\times\\1+2iw-\frac52w^2-\frac52iw^3+O(w^4)\end{array}\right)\frac{\mathrm{d}w}{w^4}\\