Optimal. Leaf size=55 \[ -6 x+\frac {6 (a+b x) \log (a+b x)}{b}-\frac {3 (a+b x) \log ^2(a+b x)}{b}+\frac {(a+b x) \log ^3(a+b x)}{b} \]
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Rubi [A]
time = 0.01, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2436, 2333,
2332} \begin {gather*} \frac {(a+b x) \log ^3(a+b x)}{b}-\frac {3 (a+b x) \log ^2(a+b x)}{b}+\frac {6 (a+b x) \log (a+b x)}{b}-6 x \end {gather*}
Antiderivative was successfully verified.
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Rule 2332
Rule 2333
Rule 2436
Rubi steps
\begin {align*} \int \log ^3(a+b x) \, dx &=\frac {\text {Subst}\left (\int \log ^3(x) \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \log ^3(a+b x)}{b}-\frac {3 \text {Subst}\left (\int \log ^2(x) \, dx,x,a+b x\right )}{b}\\ &=-\frac {3 (a+b x) \log ^2(a+b x)}{b}+\frac {(a+b x) \log ^3(a+b x)}{b}+\frac {6 \text {Subst}(\int \log (x) \, dx,x,a+b x)}{b}\\ &=-6 x+\frac {6 (a+b x) \log (a+b x)}{b}-\frac {3 (a+b x) \log ^2(a+b x)}{b}+\frac {(a+b x) \log ^3(a+b x)}{b}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 51, normalized size = 0.93 \begin {gather*} \frac {-6 b x+6 (a+b x) \log (a+b x)-3 (a+b x) \log ^2(a+b x)+(a+b x) \log ^3(a+b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 55, normalized size = 1.00
method | result | size |
derivativedivides | \(\frac {\ln \left (b x +a \right )^{3} \left (b x +a \right )-3 \ln \left (b x +a \right )^{2} \left (b x +a \right )+6 \left (b x +a \right ) \ln \left (b x +a \right )-6 b x -6 a}{b}\) | \(55\) |
default | \(\frac {\ln \left (b x +a \right )^{3} \left (b x +a \right )-3 \ln \left (b x +a \right )^{2} \left (b x +a \right )+6 \left (b x +a \right ) \ln \left (b x +a \right )-6 b x -6 a}{b}\) | \(55\) |
risch | \(\frac {\left (b x +a \right ) \ln \left (b x +a \right )^{3}}{b}-\frac {3 \left (b x +a \right ) \ln \left (b x +a \right )^{2}}{b}+6 x \ln \left (b x +a \right )-6 x +\frac {6 a \ln \left (b x +a \right )}{b}\) | \(61\) |
norman | \(x \ln \left (b x +a \right )^{3}+\frac {a \ln \left (b x +a \right )^{3}}{b}-6 x +6 x \ln \left (b x +a \right )-3 x \ln \left (b x +a \right )^{2}+\frac {6 a \ln \left (b x +a \right )}{b}-\frac {3 a \ln \left (b x +a \right )^{2}}{b}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 37, normalized size = 0.67 \begin {gather*} \frac {{\left (\log \left (b x + a\right )^{3} - 3 \, \log \left (b x + a\right )^{2} + 6 \, \log \left (b x + a\right ) - 6\right )} {\left (b x + a\right )}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 51, normalized size = 0.93 \begin {gather*} \frac {{\left (b x + a\right )} \log \left (b x + a\right )^{3} - 3 \, {\left (b x + a\right )} \log \left (b x + a\right )^{2} - 6 \, b x + 6 \, {\left (b x + a\right )} \log \left (b x + a\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 63, normalized size = 1.15 \begin {gather*} - 6 b \left (- \frac {a \log {\left (a + b x \right )}}{b^{2}} + \frac {x}{b}\right ) + 6 x \log {\left (a + b x \right )} + \frac {\left (- 3 a - 3 b x\right ) \log {\left (a + b x \right )}^{2}}{b} + \frac {\left (a + b x\right ) \log {\left (a + b x \right )}^{3}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.49, size = 62, normalized size = 1.13 \begin {gather*} \frac {{\left (b x + a\right )} \log \left (b x + a\right )^{3}}{b} - \frac {3 \, {\left (b x + a\right )} \log \left (b x + a\right )^{2}}{b} + \frac {6 \, {\left (b x + a\right )} \log \left (b x + a\right )}{b} - \frac {6 \, {\left (b x + a\right )}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.25, size = 73, normalized size = 1.33 \begin {gather*} 6\,x\,\ln \left (a+b\,x\right )-6\,x-3\,x\,{\ln \left (a+b\,x\right )}^2+x\,{\ln \left (a+b\,x\right )}^3-\frac {3\,a\,{\ln \left (a+b\,x\right )}^2}{b}+\frac {a\,{\ln \left (a+b\,x\right )}^3}{b}+\frac {6\,a\,\ln \left (a+b\,x\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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