Optimal. Leaf size=16 \[ -21+x+\frac {\log (x)}{8 (3-x)} \]
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Rubi [A] time = 0.21, antiderivative size = 22, normalized size of antiderivative = 1.38, number of steps used = 13, number of rules used = 8, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1594, 27, 12, 6742, 44, 43, 2314, 31} \begin {gather*} x+\frac {x \log (x)}{24 (3-x)}+\frac {\log (x)}{24} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 31
Rule 43
Rule 44
Rule 1594
Rule 2314
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3+71 x-48 x^2+8 x^3+x \log (x)}{x \left (72-48 x+8 x^2\right )} \, dx\\ &=\int \frac {3+71 x-48 x^2+8 x^3+x \log (x)}{8 (-3+x)^2 x} \, dx\\ &=\frac {1}{8} \int \frac {3+71 x-48 x^2+8 x^3+x \log (x)}{(-3+x)^2 x} \, dx\\ &=\frac {1}{8} \int \left (\frac {71}{(-3+x)^2}+\frac {3}{(-3+x)^2 x}-\frac {48 x}{(-3+x)^2}+\frac {8 x^2}{(-3+x)^2}+\frac {\log (x)}{(-3+x)^2}\right ) \, dx\\ &=\frac {71}{8 (3-x)}+\frac {1}{8} \int \frac {\log (x)}{(-3+x)^2} \, dx+\frac {3}{8} \int \frac {1}{(-3+x)^2 x} \, dx-6 \int \frac {x}{(-3+x)^2} \, dx+\int \frac {x^2}{(-3+x)^2} \, dx\\ &=\frac {71}{8 (3-x)}+\frac {x \log (x)}{24 (3-x)}+\frac {1}{24} \int \frac {1}{-3+x} \, dx+\frac {3}{8} \int \left (\frac {1}{3 (-3+x)^2}-\frac {1}{9 (-3+x)}+\frac {1}{9 x}\right ) \, dx-6 \int \left (\frac {3}{(-3+x)^2}+\frac {1}{-3+x}\right ) \, dx+\int \left (1+\frac {9}{(-3+x)^2}+\frac {6}{-3+x}\right ) \, dx\\ &=x+\frac {\log (x)}{24}+\frac {x \log (x)}{24 (3-x)}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.04, size = 17, normalized size = 1.06 \begin {gather*} \frac {1}{8} \left (8 x-\frac {\log (x)}{-3+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 20, normalized size = 1.25 \begin {gather*} \frac {8 \, x^{2} - 24 \, x - \log \relax (x)}{8 \, {\left (x - 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 11, normalized size = 0.69 \begin {gather*} x - \frac {\log \relax (x)}{8 \, {\left (x - 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 12, normalized size = 0.75
method | result | size |
risch | \(-\frac {\ln \relax (x )}{8 \left (x -3\right )}+x\) | \(12\) |
norman | \(\frac {x^{2}-\frac {\ln \relax (x )}{8}-9}{x -3}\) | \(16\) |
default | \(-\frac {\ln \relax (x ) x}{24 \left (x -3\right )}+x +\frac {\ln \relax (x )}{24}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 11, normalized size = 0.69 \begin {gather*} x - \frac {\log \relax (x)}{8 \, {\left (x - 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.31, size = 13, normalized size = 0.81 \begin {gather*} x-\frac {\ln \relax (x)}{8\,\left (x-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 8, normalized size = 0.50 \begin {gather*} x - \frac {\log {\relax (x )}}{8 x - 24} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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