Optimal. Leaf size=364 \[ \text {Li}_3\left (\frac {c (a+b x)}{a (c+d x)}\right )-\text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )+\text {Li}_2\left (\frac {c (a+b x)}{a (c+d x)}\right ) \log \left (\frac {a (c+d x)}{c (a+b x)}\right )-\text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right ) \log \left (\frac {a (c+d x)}{c (a+b x)}\right )+\text {Li}_2\left (\frac {b x}{a}+1\right ) \left (\log (c+d x)-\log \left (\frac {a (c+d x)}{c (a+b x)}\right )\right )+\text {Li}_2\left (\frac {d x}{c}+1\right ) \left (\log \left (\frac {a (c+d x)}{c (a+b x)}\right )+\log (a+b x)\right )+\frac {1}{2} \left (\log \left (\frac {b c-a d}{b (c+d x)}\right )-\log \left (-\frac {x (b c-a d)}{a (c+d x)}\right )+\log \left (-\frac {b x}{a}\right )\right ) \log ^2\left (\frac {a (c+d x)}{c (a+b x)}\right )-\frac {1}{2} \left (\log \left (-\frac {b x}{a}\right )-\log \left (-\frac {d x}{c}\right )\right ) \left (\log \left (\frac {a (c+d x)}{c (a+b x)}\right )+\log (a+b x)\right )^2+\log \left (-\frac {b x}{a}\right ) \log (a+b x) \log (c+d x)-\text {Li}_3\left (\frac {b x}{a}+1\right )-\text {Li}_3\left (\frac {d x}{c}+1\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 364, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2435} \[ \text {PolyLog}\left (3,\frac {c (a+b x)}{a (c+d x)}\right )-\text {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )+\log \left (\frac {a (c+d x)}{c (a+b x)}\right ) \text {PolyLog}\left (2,\frac {c (a+b x)}{a (c+d x)}\right )-\log \left (\frac {a (c+d x)}{c (a+b x)}\right ) \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )+\text {PolyLog}\left (2,\frac {b x}{a}+1\right ) \left (\log (c+d x)-\log \left (\frac {a (c+d x)}{c (a+b x)}\right )\right )+\text {PolyLog}\left (2,\frac {d x}{c}+1\right ) \left (\log \left (\frac {a (c+d x)}{c (a+b x)}\right )+\log (a+b x)\right )-\text {PolyLog}\left (3,\frac {b x}{a}+1\right )-\text {PolyLog}\left (3,\frac {d x}{c}+1\right )+\frac {1}{2} \left (\log \left (\frac {b c-a d}{b (c+d x)}\right )-\log \left (-\frac {x (b c-a d)}{a (c+d x)}\right )+\log \left (-\frac {b x}{a}\right )\right ) \log ^2\left (\frac {a (c+d x)}{c (a+b x)}\right )-\frac {1}{2} \left (\log \left (-\frac {b x}{a}\right )-\log \left (-\frac {d x}{c}\right )\right ) \left (\log \left (\frac {a (c+d x)}{c (a+b x)}\right )+\log (a+b x)\right )^2+\log \left (-\frac {b x}{a}\right ) \log (a+b x) \log (c+d x) \]
Antiderivative was successfully verified.
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Rule 2435
Rubi steps
\begin {align*} \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 ) \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*}
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Mathematica [A] time = 0.06, size = 394, normalized size = 1.08 \[ \text {Li}_3\left (\frac {a (c+d x)}{c (a+b x)}\right )-\text {Li}_3\left (\frac {b (c+d x)}{d (a+b x)}\right )+\left (\text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )-\text {Li}_2\left (\frac {a (c+d x)}{c (a+b x)}\right )\right ) \log \left (\frac {a (c+d x)}{c (a+b x)}\right )+\text {Li}_2\left (\frac {b x}{a}+1\right ) \left (\log (c+d x)-\log \left (\frac {a (c+d x)}{c (a+b x)}\right )\right )+\text {Li}_2\left (\frac {d x}{c}+1\right ) \left (\log \left (\frac {a (c+d x)}{c (a+b x)}\right )+\log (a+b x)\right )+\frac {1}{2} \left (\log \left (\frac {a d-b c}{d (a+b x)}\right )-\log \left (\frac {b c x-a d x}{a c+b c x}\right )+\log \left (-\frac {b x}{a}\right )\right ) \log ^2\left (\frac {a (c+d x)}{c (a+b x)}\right )+\log \left (\frac {d x}{c}+1\right ) \left (\log \left (-\frac {d x}{c}\right )-\log \left (-\frac {b x}{a}\right )\right ) \log \left (\frac {a (c+d x)}{c (a+b x)}\right )+\log \left (-\frac {b x}{a}\right ) \log (a+b x) \log (c+d x)+\frac {1}{2} \log \left (\frac {d x}{c}+1\right ) \left (\log \left (-\frac {b x}{a}\right )-\log \left (-\frac {d x}{c}\right )\right ) \left (\log \left (\frac {d x}{c}+1\right )-2 \log (a+b x)\right )-\text {Li}_3\left (\frac {b x}{a}+1\right )-\text {Li}_3\left (\frac {d x}{c}+1\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left (b x + a\right ) \log \left (d x + c\right )}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (b x + a\right ) \log \left (d x + c\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (b x +a \right ) \ln \left (d x +c \right )}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (b x + a\right ) \log \left (d x + c\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\ln \left (a+b\,x\right )\,\ln \left (c+d\,x\right )}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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