Interesting integral

$\begin{aligned} I & = \int_0^\infty \frac {x^{-\frac 13}}{x+1} dx & \small \color{#3D99F6}{\text{Convert to beta function}} \\ & = B \left(\frac 23, \frac 13 \right) & \small \color{#3D99F6}{B(m+1, n+1) = \int_0^\infty \frac {u^m}{(1+u)^{m+n+2}} du} \\ & = \frac {\Gamma \left(\frac 23\right)\Gamma \left(\frac 13\right)}{\Gamma \left(1\right)} & \small \color{#3D99F6}{\Gamma (\cdot) \text{ is gamma function}} \\ & = \frac {\frac \pi{\sin \frac \pi 3}}{0!} \\ & = \frac {2\pi}{\sqrt 3} \end{aligned}$

$\implies a+b = 2+3 = \boxed{5}$

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References

• Beta function
• Gamma function