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I'm reading this paper where to determine the coefficient $C^{\phi\phi O}(x_{12},\partial_2)$ of the OPE (p.10) $$\phi^\alpha (x_1)\phi^\beta (x_2)=C_\phi \frac{\delta^{\alpha\beta}}{x_{12}^{2\eta}}+C^{\phi\phi O}(x_{12},\partial_2)O(x_2)\delta^{\alpha\beta}$$ they require consistency of the OPE with the 3-point correlation function, i.e. comparing the fixed form of the 3-point correlation function to what happens when we apply the OPE to two channels in the 3-point function and then use the form of the 2-point function.

Anyway, to solve the form of the coefficient that satisfies eq. 2.29 they use the integral representation $$\frac{1}{(ab)^\rho}=\frac{1}{B(\rho,\rho)}\int_0^1 dt \frac{[t(1-t)]^{\rho-1}}{[at+(1-t)b]^{2\rho}}$$ which comes from the book Table of Integrals, Series and Products by I.S. Gradshteyn and I.M. Ryzhik. However, looking at that book (I only have access to the 7th ed.) I only found the relation $$B(\mu,\nu)\frac{1}{(a+c)^\mu(b+c)^\nu}=\int_0^1 dx x^{\mu-1}(1-x)^{\nu-1}\frac{1}{[ax+(1-x)b+c]^{\mu+\nu}}$$ which reduces to the desired relation when $c=0$ however, in the book it says that the relation is valid for $c>0$, so how can they use that relation?

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Short answer : This is just Feynmann parametrization, so you could demonstrate the formula by recurrence.

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