Optimal. Leaf size=105 \[ -\frac {\text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{d}+\frac {\log \left (\frac {a}{x}+b\right ) \log (c+d x)}{d}-\frac {\log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{d}+\frac {\text {Li}_2\left (\frac {d x}{c}+1\right )}{d}+\frac {\log \left (-\frac {d x}{c}\right ) \log (c+d x)}{d} \]
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Rubi [A] time = 0.17, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {2465, 2462, 260, 2416, 2394, 2315, 2393, 2391} \[ -\frac {\text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{d}+\frac {\text {PolyLog}\left (2,\frac {d x}{c}+1\right )}{d}+\frac {\log \left (\frac {a}{x}+b\right ) \log (c+d x)}{d}-\frac {\log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{d}+\frac {\log \left (-\frac {d x}{c}\right ) \log (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 260
Rule 2315
Rule 2391
Rule 2393
Rule 2394
Rule 2416
Rule 2462
Rule 2465
Rubi steps
\begin {align*} \int \frac {\log \left (\frac {a+b x}{x}\right )}{c+d x} \, dx &=\int \frac {\log \left (b+\frac {a}{x}\right )}{c+d x} \, dx\\ &=\frac {\log \left (b+\frac {a}{x}\right ) \log (c+d x)}{d}+\frac {a \int \frac {\log (c+d x)}{\left (b+\frac {a}{x}\right ) x^2} \, dx}{d}\\ &=\frac {\log \left (b+\frac {a}{x}\right ) \log (c+d x)}{d}+\frac {a \int \left (\frac {\log (c+d x)}{a x}-\frac {b \log (c+d x)}{a (a+b x)}\right ) \, dx}{d}\\ &=\frac {\log \left (b+\frac {a}{x}\right ) \log (c+d x)}{d}+\frac {\int \frac {\log (c+d x)}{x} \, dx}{d}-\frac {b \int \frac {\log (c+d x)}{a+b x} \, dx}{d}\\ &=\frac {\log \left (b+\frac {a}{x}\right ) \log (c+d x)}{d}+\frac {\log \left (-\frac {d x}{c}\right ) \log (c+d x)}{d}-\frac {\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d}-\int \frac {\log \left (-\frac {d x}{c}\right )}{c+d x} \, dx+\int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx\\ &=\frac {\log \left (b+\frac {a}{x}\right ) \log (c+d x)}{d}+\frac {\log \left (-\frac {d x}{c}\right ) \log (c+d x)}{d}-\frac {\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d}+\frac {\text {Li}_2\left (1+\frac {d x}{c}\right )}{d}+\frac {\operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{d}\\ &=\frac {\log \left (b+\frac {a}{x}\right ) \log (c+d x)}{d}+\frac {\log \left (-\frac {d x}{c}\right ) \log (c+d x)}{d}-\frac {\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d}-\frac {\text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{d}+\frac {\text {Li}_2\left (1+\frac {d x}{c}\right )}{d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 80, normalized size = 0.76 \[ \frac {-\text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (-\log \left (\frac {d (a+b x)}{a d-b c}\right )+\log \left (\frac {a}{x}+b\right )+\log \left (-\frac {d x}{c}\right )\right )+\text {Li}_2\left (\frac {d x}{c}+1\right )}{d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left (\frac {b x + a}{x}\right )}{d x + c}, x\right ) \]
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 \[ \int \frac {\log \left (\frac {b x + a}{x}\right )}{d x + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 114, normalized size = 1.09 \[ \frac {\ln \left (\frac {a d -b c +\left (b +\frac {a}{x}\right ) c}{a d -b c}\right ) \ln \left (b +\frac {a}{x}\right )}{d}-\frac {\ln \left (-\frac {a}{b x}\right ) \ln \left (b +\frac {a}{x}\right )}{d}+\frac {\dilog \left (\frac {a d -b c +\left (b +\frac {a}{x}\right ) c}{a d -b c}\right )}{d}-\frac {\dilog \left (-\frac {a}{b x}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 124, normalized size = 1.18 \[ -\frac {{\left (\log \left (b x + a\right ) - \log \relax (x)\right )} \log \left (d x + c\right )}{d} + \frac {\log \left (d x + c\right ) \log \left (\frac {b x + a}{x}\right )}{d} - \frac {\log \left (\frac {d x}{c} + 1\right ) \log \relax (x) + {\rm Li}_2\left (-\frac {d x}{c}\right )}{d} + \frac {\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\ln \left (\frac {a+b\,x}{x}\right )}{c+d\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log {\left (\frac {a}{x} + b \right )}}{c + d x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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