Firstly,
$$I(c)=\int\limits_0^1 \dfrac{\ln x\,\text dx}{\sqrt{x(1-x)(1-cx)\large\mathstrut}}
=4\int\limits_0^1 \dfrac{\ln\sqrt x\,\text d\sqrt x}{\sqrt{(1-x)(1-cx)\large\mathstrut}}
=4\int\limits_0^1 \dfrac{\ln y\,\text dy}{\sqrt{(1-y^2)(1-cy^2)\large\mathstrut}}.$$
If $\mathbf{c=0},$ then
$$I(0) = 4\int\limits_0^1\dfrac{\ln y}{1-y^2}\,\text dy =-\dfrac{\pi^2}2
\approx-4.9348.$$
If $\mathbf{c=1},$ then
$$I(1) = 4\int\limits_0^1\dfrac{\ln y}{\sqrt{1-y^2}}\,\text dy =-2\pi\ln2\approx-4.35517.$$
Assume $\mathbf{c\in(0,1).}$
Using elliptic antiderivative
$$\int \dfrac{\text dy}{\sqrt{(1-y^2)(1-c y^2)}} = \operatorname{F}\big(\arcsin y\ |\,c\big)\tag1$$
and integration by parts, one can get:
$$I(c)=4\int\limits_0^1 \ln y\,\text d\operatorname{F}\big(\arcsin(y)\,|\,c\big)
=4\ln y\operatorname{F}\big(\arcsin(y)\,|\,c\big)\bigg|_0^1
-4\int\limits_0^1 \operatorname{F}\big(\arcsin(y)\,|\,c\big)\,\dfrac{\text dy}{y},$$
$$\color{brown}{\mathbf{I(c)=-4\int\limits_0^1 \operatorname{F}\big(\arcsin(y)\,|\,c\big)\,\dfrac{\text dy}{y}.}}\tag1$$
Applying Maclaurin series in the form of
$$\operatorname{F}\big(\arcsin(y)\,|\,c\big)=
\dfrac{c^k \left(1/2\right)_{(k)}}{(1 + 2 k) k!}
\operatorname{_2F_1}\left(\dfrac12, -k, \dfrac12 - k, \dfrac1c\right)
y^{2k+1}, \qquad (|y|<1)$$
easily tu get
$$I(c)=-4\int_0^1\dfrac{c^k \left(1/2\right)_{(k)}}{(1 + 2 k)^2 k!}
\operatorname{_2F_1}\left(\dfrac12, -k, \dfrac12 - k, \dfrac1c\right)
y^{2k+1}, \qquad (|y|<1)\tag3.$$
Sadly, obtained series converges badly (5 digs by 200 terms).
Alternative way is based on the orthogonal polynomials technic.
Let $c=p^2,\; z=py^2,\; t=\dfrac{1+p^2}{2p},$ then
$$(1-y^2)(1-p^2y^2) = 1-2 t z+z^2,$$
and from the generating function representation it follows
$$\dfrac1{\sqrt{1-2tz+z^2}} = \sum\limits_{k=0}^\infty \operatorname P_k(t) z^k =\sum\limits_{k=0}^\infty c^{k}\operatorname P_k\left(\dfrac{1+c}{2\sqrt c}\right) y^{2k},\tag4$$
where
$$\operatorname P_k(t)= \sum_{m=0}^n \dbinom nm \dbinom{-n-1}m \left(\dfrac{1-x}2\right)^m\tag5$$
are the Legendre orthogonal polynomials.
Taking in account definite integration of
$$\int\limits_0^1 x^k\ln x dx = -\dfrac1{(k+1)^2},\qquad (k>-1),\tag6$$
finally we can obtain
$$\color{brown}{\mathrm{I(c) = -4\,\sum_{k=0}^\infty \binom{k-\frac12}k \dfrac1{(k+1)^2} \operatorname{_1F_2}\left(\dfrac12, -k, -\dfrac12-k, \dfrac1c\right)}}.\tag7$$
However, this series converges similarly bad, as the previous one.