Optimal. Leaf size=20 \[ 5-2 \log \left ((-5+x)^2\right )+\left (-2 x+x^2\right ) \log (x) \]
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Rubi [B] time = 0.15, antiderivative size = 43, normalized size of antiderivative = 2.15, number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6741, 6742, 698, 2313, 9} \begin {gather*} \frac {x^2}{2}-\left (2 x-x^2\right ) \log (x)-\frac {1}{2} (2-x)^2-2 x-4 \log (5-x) \end {gather*}
Antiderivative was successfully verified.
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Rule 9
Rule 698
Rule 2313
Rule 6741
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {(1-x) (-6+x-10 \log (x)+2 x \log (x))}{5-x} \, dx\\ &=\int \left (\frac {6-7 x+x^2}{-5+x}+2 (-1+x) \log (x)\right ) \, dx\\ &=2 \int (-1+x) \log (x) \, dx+\int \frac {6-7 x+x^2}{-5+x} \, dx\\ &=-\left (\left (2 x-x^2\right ) \log (x)\right )-2 \int \frac {1}{2} (-2+x) \, dx+\int \left (-2-\frac {4}{-5+x}+x\right ) \, dx\\ &=-\frac {1}{2} (2-x)^2-2 x+\frac {x^2}{2}-4 \log (5-x)-\left (2 x-x^2\right ) \log (x)\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.02, size = 20, normalized size = 1.00 \begin {gather*} -4 \log (5-x)-2 x \log (x)+x^2 \log (x) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 17, normalized size = 0.85 \begin {gather*} {\left (x^{2} - 2 \, x\right )} \log \relax (x) - 4 \, \log \left (x - 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 17, normalized size = 0.85 \begin {gather*} {\left (x^{2} - 2 \, x\right )} \log \relax (x) - 4 \, \log \left (x - 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 18, normalized size = 0.90
method | result | size |
risch | \(\left (x^{2}-2 x \right ) \ln \relax (x )-4 \ln \left (x -5\right )\) | \(18\) |
default | \(x^{2} \ln \relax (x )-2 x \ln \relax (x )-4 \ln \left (x -5\right )\) | \(19\) |
norman | \(x^{2} \ln \relax (x )-2 x \ln \relax (x )-4 \ln \left (x -5\right )\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 17, normalized size = 0.85 \begin {gather*} {\left (x^{2} - 2 \, x\right )} \log \relax (x) - 4 \, \log \left (x - 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.17, size = 18, normalized size = 0.90 \begin {gather*} x^2\,\ln \relax (x)-4\,\ln \left (x-5\right )-2\,x\,\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 15, normalized size = 0.75 \begin {gather*} \left (x^{2} - 2 x\right ) \log {\relax (x )} - 4 \log {\left (x - 5 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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