Integrand size = 16, antiderivative size = 364 \[ \int \frac {\log (a+b x) \log (c+d x)}{x} \, dx=\log \left (-\frac {b x}{a}\right ) \log (a+b x) \log (c+d x)+\frac {1}{2} \left (\log \left (-\frac {b x}{a}\right )+\log \left (\frac {b c-a d}{b (c+d x)}\right )-\log \left (-\frac {(b c-a d) x}{a (c+d x)}\right )\right ) \log ^2\left (\frac {a (c+d x)}{c (a+b x)}\right )-\frac {1}{2} \left (\log \left (-\frac {b x}{a}\right )-\log \left (-\frac {d x}{c}\right )\right ) \left (\log (a+b x)+\log \left (\frac {a (c+d x)}{c (a+b x)}\right )\right )^2+\left (\log (c+d x)-\log \left (\frac {a (c+d x)}{c (a+b x)}\right )\right ) \operatorname {PolyLog}\left (2,1+\frac {b x}{a}\right )+\log \left (\frac {a (c+d x)}{c (a+b x)}\right ) \operatorname {PolyLog}\left (2,\frac {c (a+b x)}{a (c+d x)}\right )-\log \left (\frac {a (c+d x)}{c (a+b x)}\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )+\left (\log (a+b x)+\log \left (\frac {a (c+d x)}{c (a+b x)}\right )\right ) \operatorname {PolyLog}\left (2,1+\frac {d x}{c}\right )-\operatorname {PolyLog}\left (3,1+\frac {b x}{a}\right )+\operatorname {PolyLog}\left (3,\frac {c (a+b x)}{a (c+d x)}\right )-\operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )-\operatorname {PolyLog}\left (3,1+\frac {d x}{c}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2485} \[ \int \frac {\log (a+b x) \log (c+d x)}{x} \, dx=\operatorname {PolyLog}\left (3,\frac {c (a+b x)}{a (c+d x)}\right )-\operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )+\operatorname {PolyLog}\left (2,\frac {c (a+b x)}{a (c+d x)}\right ) \log \left (\frac {a (c+d x)}{c (a+b x)}\right )-\operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right ) \log \left (\frac {a (c+d x)}{c (a+b x)}\right )+\operatorname {PolyLog}\left (2,\frac {b x}{a}+1\right ) \left (\log (c+d x)-\log \left (\frac {a (c+d x)}{c (a+b x)}\right )\right )+\operatorname {PolyLog}\left (2,\frac {d x}{c}+1\right ) \left (\log \left (\frac {a (c+d x)}{c (a+b x)}\right )+\log (a+b x)\right )+\frac {1}{2} \left (\log \left (\frac {b c-a d}{b (c+d x)}\right )-\log \left (-\frac {x (b c-a d)}{a (c+d x)}\right )+\log \left (-\frac {b x}{a}\right )\right ) \log ^2\left (\frac {a (c+d x)}{c (a+b x)}\right )-\frac {1}{2} \left (\log \left (-\frac {b x}{a}\right )-\log \left (-\frac {d x}{c}\right )\right ) \left (\log \left (\frac {a (c+d x)}{c (a+b x)}\right )+\log (a+b x)\right )^2+\log \left (-\frac {b x}{a}\right ) \log (a+b x) \log (c+d x)-\operatorname {PolyLog}\left (3,\frac {b x}{a}+1\right )-\operatorname {PolyLog}\left (3,\frac {d x}{c}+1\right ) \]
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Rule 2485
Rubi steps \begin{align*} \text {integral}& = \log \left (-\frac {b x}{a}\right ) \log (a+b x) \log (c+d x)+\frac {1}{2} \left (\log \left (-\frac {b x}{a}\right )+\log \left (\frac {b c-a d}{b (c+d x)}\right )-\log \left (-\frac {(b c-a d) x}{a (c+d x)}\right )\right ) \log ^2\left (\frac {a (c+d x)}{c (a+b x)}\right )-\frac {1}{2} \left (\log \left (-\frac {b x}{a}\right )-\log \left (-\frac {d x}{c}\right )\right ) \left (\log (a+b x)+\log \left (\frac {a (c+d x)}{c (a+b x)}\right )\right )^2+\left (\log (c+d x)-\log \left (\frac {a (c+d x)}{c (a+b x)}\right )\right ) \text {Li}_2\left (1+\frac {b x}{a}\right )+\log \left (\frac {a (c+d x)}{c (a+b x)}\right ) \text {Li}_2\left (\frac {c (a+b x)}{a (c+d x)}\right )-\log \left (\frac {a (c+d x)}{c (a+b x)}\right ) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )+\left (\log (a+b x)+\log \left (\frac {a (c+d x)}{c (a+b x)}\right )\right ) \text {Li}_2\left (1+\frac {d x}{c}\right )-\text {Li}_3\left (1+\frac {b x}{a}\right )+\text {Li}_3\left (\frac {c (a+b x)}{a (c+d x)}\right )-\text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )-\text {Li}_3\left (1+\frac {d x}{c}\right ) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.08 \[ \int \frac {\log (a+b x) \log (c+d x)}{x} \, dx=\log \left (-\frac {b x}{a}\right ) \log (a+b x) \log (c+d x)+\frac {1}{2} \log ^2\left (\frac {a (c+d x)}{c (a+b x)}\right ) \left (\log \left (-\frac {b x}{a}\right )+\log \left (\frac {-b c+a d}{d (a+b x)}\right )-\log \left (\frac {b c x-a d x}{a c+b c x}\right )\right )+\left (-\log \left (-\frac {b x}{a}\right )+\log \left (-\frac {d x}{c}\right )\right ) \log \left (\frac {a (c+d x)}{c (a+b x)}\right ) \log \left (1+\frac {d x}{c}\right )+\frac {1}{2} \left (\log \left (-\frac {b x}{a}\right )-\log \left (-\frac {d x}{c}\right )\right ) \log \left (1+\frac {d x}{c}\right ) \left (-2 \log (a+b x)+\log \left (1+\frac {d x}{c}\right )\right )+\left (\log (c+d x)-\log \left (\frac {a (c+d x)}{c (a+b x)}\right )\right ) \operatorname {PolyLog}\left (2,1+\frac {b x}{a}\right )+\log \left (\frac {a (c+d x)}{c (a+b x)}\right ) \left (-\operatorname {PolyLog}\left (2,\frac {a (c+d x)}{c (a+b x)}\right )+\operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )\right )+\left (\log (a+b x)+\log \left (\frac {a (c+d x)}{c (a+b x)}\right )\right ) \operatorname {PolyLog}\left (2,1+\frac {d x}{c}\right )-\operatorname {PolyLog}\left (3,1+\frac {b x}{a}\right )+\operatorname {PolyLog}\left (3,\frac {a (c+d x)}{c (a+b x)}\right )-\operatorname {PolyLog}\left (3,\frac {b (c+d x)}{d (a+b x)}\right )-\operatorname {PolyLog}\left (3,1+\frac {d x}{c}\right ) \]
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\[\int \frac {\ln \left (b x +a \right ) \ln \left (d x +c \right )}{x}d x\]
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\[ \int \frac {\log (a+b x) \log (c+d x)}{x} \, dx=\int { \frac {\log \left (b x + a\right ) \log \left (d x + c\right )}{x} \,d x } \]
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Timed out. \[ \int \frac {\log (a+b x) \log (c+d x)}{x} \, dx=\text {Timed out} \]
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\[ \int \frac {\log (a+b x) \log (c+d x)}{x} \, dx=\int { \frac {\log \left (b x + a\right ) \log \left (d x + c\right )}{x} \,d x } \]
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\[ \int \frac {\log (a+b x) \log (c+d x)}{x} \, dx=\int { \frac {\log \left (b x + a\right ) \log \left (d x + c\right )}{x} \,d x } \]
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Timed out. \[ \int \frac {\log (a+b x) \log (c+d x)}{x} \, dx=\int \frac {\ln \left (a+b\,x\right )\,\ln \left (c+d\,x\right )}{x} \,d x \]
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