TITLE: Prove ${\large\int}_0^1\frac{\ln(1+8x)}{x^{2/3}\,(1-x)^{2/3}\,(1+8x)^{1/3}}dx=\frac{\ln3}{\pi\sqrt3}\Gamma^3\!\left(\tfrac13\right)$
QUESTION [18 upvotes]: Here is one more numerically discovered conjecture that I was not able to prove, and asking you for help:
$${\large\int}_0^1\frac{\ln(1+8x)}{x^{\small2/3}\,(1-x)^{\small2/3}\,(1+8x)^{\small1/3}}dx\stackrel{\color{#A0A0A0}{\small?}}=\frac{\ln3}{\pi\sqrt3}\Gamma^3\!\left(\tfrac13\right),\tag1$$
or, equivalently,
$$\frac{d}{da}{_2F_1}\!\left(a,\frac13;\,\frac23;\,-8\right)\Bigg|_{a=\frac13}\stackrel{\color{#A0A0A0}{\small?}}=-\frac{2\ln3}3.\tag2$$

Update: I can suggest a generalization of this conjecture that might be easier to prove:
$$_2F_1\!\left(-a,\frac16-\frac{a}2;\,\frac23;\,-8\right)=2\times3^{\frac{3a-1}2}\cos\left(\frac\pi6(3a+1)\right).\tag3$$
This is actually the identity 07.23.03.0658.01 from the Wolfram Functions Site generalized to non-integer values of $a$.

REPLY [6 votes]: I think I have found something for you.
See http://authors.library.caltech.edu/43489/ for the Integral Tables of the Bateman Project.
In Volume 1, p.310 (PDF p. 322) Formula (23), which I hope should be applicable for your case.
The last formula of david-h's computation is an example of a Mellin integral transformation.
$\int_0^\infty (1+x)^\nu (1+\alpha x)^\mu x^{s-1} dx = B(s,-\mu-\nu-s)
\ {_2F_1}(-\mu,s;-\mu-\nu;1-\alpha)$,
with B the Beta-function, $|\text{arg}\alpha| \le \pi$ and $-\mathcal{Re}(\mu+\nu) > \mathcal{Re}(s) > 0$.