Optimal. Leaf size=81 \[ -\frac {a \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b^2}-\frac {a \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{b^2}+\frac {(c+d x) \log (c+d x)}{b d}-\frac {x}{b} \]
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Rubi [A] time = 0.10, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {43, 2416, 2389, 2295, 2394, 2393, 2391} \[ -\frac {a \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{b^2}-\frac {a \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{b^2}+\frac {(c+d x) \log (c+d x)}{b d}-\frac {x}{b} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2295
Rule 2389
Rule 2391
Rule 2393
Rule 2394
Rule 2416
Rubi steps
\begin {align*} \int \frac {x \log (c+d x)}{a+b x} \, dx &=\int \left (\frac {\log (c+d x)}{b}-\frac {a \log (c+d x)}{b (a+b x)}\right ) \, dx\\ &=\frac {\int \log (c+d x) \, dx}{b}-\frac {a \int \frac {\log (c+d x)}{a+b x} \, dx}{b}\\ &=-\frac {a \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2}+\frac {\operatorname {Subst}(\int \log (x) \, dx,x,c+d x)}{b d}+\frac {(a d) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^2}\\ &=-\frac {x}{b}+\frac {(c+d x) \log (c+d x)}{b d}-\frac {a \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2}+\frac {a \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b^2}\\ &=-\frac {x}{b}+\frac {(c+d x) \log (c+d x)}{b d}-\frac {a \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2}-\frac {a \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 73, normalized size = 0.90 \[ \frac {-a d \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (-a d \log \left (\frac {d (a+b x)}{a d-b c}\right )+b c+b d x\right )-b d x}{b^2 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x \log \left (d x + c\right )}{b x + a}, x\right ) \]
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 \[ \int \frac {x \log \left (d x + c\right )}{b x + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 114, normalized size = 1.41 \[ -\frac {a \ln \left (\frac {a d -b c +\left (d x +c \right ) b}{a d -b c}\right ) \ln \left (d x +c \right )}{b^{2}}+\frac {x \ln \left (d x +c \right )}{b}-\frac {a \dilog \left (\frac {a d -b c +\left (d x +c \right ) b}{a d -b c}\right )}{b^{2}}+\frac {c \ln \left (d x +c \right )}{b d}-\frac {x}{b}-\frac {c}{b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.68, size = 111, normalized size = 1.37 \[ d {\left (\frac {{\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )} a}{b^{2} d} - \frac {x}{b d} + \frac {c \log \left (d x + c\right )}{b d^{2}}\right )} + {\left (\frac {x}{b} - \frac {a \log \left (b x + a\right )}{b^{2}}\right )} \log \left (d x + c\right ) \]
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 \[ \int \frac {x\,\ln \left (c+d\,x\right )}{a+b\,x} \,d x \]
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 \[ \int \frac {x \log {\left (c + d x \right )}}{a + b x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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