Integral Involving Dilogarithms I came across the identity
$$\int^x_0\frac{\ln(p+qt)}{r+st}{\rm d}t=\frac{1}{2s}\left[\ln^2{\left(\frac{q}{s}(r+sx)\right)}-\ln^2{\left(\frac{qr}{s}\right)}+2\mathrm{Li}_2\left(\frac{qr-ps}{q(r+sx)}\right)-2\mathrm{Li}_2\left(\frac{qr-ps}{qr}\right)\right]$$
in a book.  Unfortunately, as of now, I am not very adept at manipulating such integrals and thus I have little idea on how to proceed with proving this identity. For example, substituting $u=r+sx$ doesn't seem to help much. Hence, I would like to seek assistance as to how this integral can be evaluated. Help will be greatly appreciated. Thank you.
 A: Among various ways to do it, this one is simple :

A: $\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
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 \newcommand{\iff}{\Longleftrightarrow}
 \newcommand{\imp}{\Longrightarrow}
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\begin{align}&\overbrace{\color{#c00000}{\int_{0}^{x}%
{\ln\pars{p+qt} \over r + st}\,\dd t}}
^{\ds{\mbox{Set}\ p + qt\equiv \xi\ \imp\ t = {\xi - p \over q}}}\ =\
\int_{p}^{p + qx}{\ln\pars{\xi} \over r + s\pars{\xi - p}/q}\,{\dd\xi \over q}
=-\int_{p}^{p + qx}{\ln\pars{\xi} \over sp - rq - s\xi}\,\dd\xi
\\[5mm]&={1 \over rq - sp}\ \overbrace{\int_{p}^{p + qx}
{\ln\pars{\xi} \over 1 - s\xi/\pars{sp - rq}}\,\dd\xi}
^{\ds{\mbox{Set}\ {s \over sp - rq}\,\xi\equiv t\ \imp\ \xi = {sp - rq \over s}\,t}}
\\[5mm]&=
{1 \over rq - sp}\int_{sp/\pars{sp - rq}}^{s\pars{p + qx}/\pars{sp - rq}}
{\ln\pars{\bracks{sp - rq}t/s} \over 1 - t}\,{sp - rq \over s}\,\dd t
\\[3mm]&=-\,{1 \over s}\int_{sp/\pars{sp - rq}}^{s\pars{p + qx}/\pars{sp - rq}}
{\ln\pars{\bracks{sp - rq}t/s} \over 1 - t}\,\dd t
\\[3mm]&=\left.{1 \over s}\ln\pars{1 - t}\ln\pars{{sp - rq \over s}\,t}
\right\vert_{\,t\ =\ {sp \over sp\ -\ rq}}^{\, t\ =\ s\,{p\ +\ qx \over sp\ -\ rq}}
\ -\
{1 \over s}\int_{sp/\pars{sp - rq}}^{s\pars{p + qx}/\pars{sp - rq}}
{\ln\pars{1 - t} \over t}\,\dd t
\\[3mm]&={1 \over s}\bracks{\ln\pars{1 - s\,{p + qx \over sp - rq}}
\ln\pars{p + qx} - \ln\pars{1 - {sp \over sp - rq}}\ln\pars{p}}
\\[3mm]&\phantom{=}+{1 \over s}\bracks{%
{\rm Li}_{2}\pars{{p + qx \over sp - rq}\,s}
-{\rm Li}_{2}\pars{sp \over sp - rq}}
\end{align}

\begin{align}&\color{#66f}{\large\int_{0}^{x}{\ln\pars{p+qt} \over r + st}\,\dd t}
\\[3mm]&=\color{#66f}{\large{1 \over s}\bracks{%
\ln\pars{{r + sx \over rq - sp}\,q}\ln\pars{p + qx}
- \ln\pars{rq \over rq - sp}\ln\pars{p}}}
\\[3mm]&\color{#66f}{\large + {1 \over s}\bracks{%
{\rm Li}_{2}\pars{{p + qx \over sp - rq}\,s}
-{\rm Li}_{2}\pars{sp \over sp - rq}}}
\end{align}

Indeed, for particular values of the different parameters we should take care of possible $\color{#c00000}{\large\ds{\ln}}$ or/and
$\color{#c00000}{\large\ds{{\rm Li}_{2}}}$ branch cuts.
