Optimal. Leaf size=16 \[ \frac {25}{729} x \log ^4\left (5-x+x^2\right ) \]
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Rubi [A] time = 115.85, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 131, number of rules used = 23, integrand size = 57, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.404, Rules used = {6728, 2528, 2523, 773, 634, 618, 204, 628, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 2527, 12, 5057, 4920, 4854, 2402, 2315, 31} \begin {gather*} \frac {25}{729} x \log ^4\left (x^2-x+5\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 204
Rule 618
Rule 628
Rule 634
Rule 773
Rule 2301
Rule 2315
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2402
Rule 2418
Rule 2523
Rule 2524
Rule 2527
Rule 2528
Rule 4854
Rule 4920
Rule 5057
Rule 6728
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {100 x (-1+2 x) \log ^3\left (5-x+x^2\right )}{729 \left (5-x+x^2\right )}+\frac {25}{729} \log ^4\left (5-x+x^2\right )\right ) \, dx\\ &=\frac {25}{729} \int \log ^4\left (5-x+x^2\right ) \, dx+\frac {100}{729} \int \frac {x (-1+2 x) \log ^3\left (5-x+x^2\right )}{5-x+x^2} \, dx\\ &=\frac {25}{729} x \log ^4\left (5-x+x^2\right )-\frac {100}{729} \int \frac {x (-1+2 x) \log ^3\left (5-x+x^2\right )}{5-x+x^2} \, dx+\frac {100}{729} \int \left (2 \log ^3\left (5-x+x^2\right )-\frac {(10-x) \log ^3\left (5-x+x^2\right )}{5-x+x^2}\right ) \, dx\\ &=\frac {25}{729} x \log ^4\left (5-x+x^2\right )-\frac {100}{729} \int \frac {(10-x) \log ^3\left (5-x+x^2\right )}{5-x+x^2} \, dx-\frac {100}{729} \int \left (2 \log ^3\left (5-x+x^2\right )-\frac {(10-x) \log ^3\left (5-x+x^2\right )}{5-x+x^2}\right ) \, dx+\frac {200}{729} \int \log ^3\left (5-x+x^2\right ) \, dx\\ &=\frac {200}{729} x \log ^3\left (5-x+x^2\right )+\frac {25}{729} x \log ^4\left (5-x+x^2\right )+\frac {100}{729} \int \frac {(10-x) \log ^3\left (5-x+x^2\right )}{5-x+x^2} \, dx-\frac {100}{729} \int \left (\frac {\left (-1-i \sqrt {19}\right ) \log ^3\left (5-x+x^2\right )}{-1-i \sqrt {19}+2 x}+\frac {\left (-1+i \sqrt {19}\right ) \log ^3\left (5-x+x^2\right )}{-1+i \sqrt {19}+2 x}\right ) \, dx-\frac {200}{729} \int \log ^3\left (5-x+x^2\right ) \, dx-\frac {200}{243} \int \frac {x (-1+2 x) \log ^2\left (5-x+x^2\right )}{5-x+x^2} \, dx\\ &=\frac {25}{729} x \log ^4\left (5-x+x^2\right )+\frac {100}{729} \int \left (\frac {\left (-1-i \sqrt {19}\right ) \log ^3\left (5-x+x^2\right )}{-1-i \sqrt {19}+2 x}+\frac {\left (-1+i \sqrt {19}\right ) \log ^3\left (5-x+x^2\right )}{-1+i \sqrt {19}+2 x}\right ) \, dx+\frac {200}{243} \int \frac {x (-1+2 x) \log ^2\left (5-x+x^2\right )}{5-x+x^2} \, dx-\frac {200}{243} \int \left (2 \log ^2\left (5-x+x^2\right )-\frac {(10-x) \log ^2\left (5-x+x^2\right )}{5-x+x^2}\right ) \, dx+\frac {1}{729} \left (100 \left (1-i \sqrt {19}\right )\right ) \int \frac {\log ^3\left (5-x+x^2\right )}{-1+i \sqrt {19}+2 x} \, dx+\frac {1}{729} \left (100 \left (1+i \sqrt {19}\right )\right ) \int \frac {\log ^3\left (5-x+x^2\right )}{-1-i \sqrt {19}+2 x} \, dx\\ &=\frac {50}{729} \left (1+i \sqrt {19}\right ) \log \left (-1-i \sqrt {19}+2 x\right ) \log ^3\left (5-x+x^2\right )+\frac {50}{729} \left (1-i \sqrt {19}\right ) \log \left (-1+i \sqrt {19}+2 x\right ) \log ^3\left (5-x+x^2\right )+\frac {25}{729} x \log ^4\left (5-x+x^2\right )+\frac {200}{243} \int \frac {(10-x) \log ^2\left (5-x+x^2\right )}{5-x+x^2} \, dx+\frac {200}{243} \int \left (2 \log ^2\left (5-x+x^2\right )-\frac {(10-x) \log ^2\left (5-x+x^2\right )}{5-x+x^2}\right ) \, dx-\frac {400}{243} \int \log ^2\left (5-x+x^2\right ) \, dx-\frac {1}{729} \left (100 \left (1-i \sqrt {19}\right )\right ) \int \frac {\log ^3\left (5-x+x^2\right )}{-1+i \sqrt {19}+2 x} \, dx-\frac {1}{243} \left (50 \left (1-i \sqrt {19}\right )\right ) \int \frac {(-1+2 x) \log \left (-1+i \sqrt {19}+2 x\right ) \log ^2\left (5-x+x^2\right )}{5-x+x^2} \, dx-\frac {1}{729} \left (100 \left (1+i \sqrt {19}\right )\right ) \int \frac {\log ^3\left (5-x+x^2\right )}{-1-i \sqrt {19}+2 x} \, dx-\frac {1}{243} \left (50 \left (1+i \sqrt {19}\right )\right ) \int \frac {(-1+2 x) \log \left (-1-i \sqrt {19}+2 x\right ) \log ^2\left (5-x+x^2\right )}{5-x+x^2} \, dx\\ &=-\frac {400}{243} x \log ^2\left (5-x+x^2\right )+\frac {25}{729} x \log ^4\left (5-x+x^2\right )-\frac {200}{243} \int \frac {(10-x) \log ^2\left (5-x+x^2\right )}{5-x+x^2} \, dx+\frac {200}{243} \int \left (\frac {\left (-1-i \sqrt {19}\right ) \log ^2\left (5-x+x^2\right )}{-1-i \sqrt {19}+2 x}+\frac {\left (-1+i \sqrt {19}\right ) \log ^2\left (5-x+x^2\right )}{-1+i \sqrt {19}+2 x}\right ) \, dx+\frac {400}{243} \int \log ^2\left (5-x+x^2\right ) \, dx+\frac {800}{243} \int \frac {x (-1+2 x) \log \left (5-x+x^2\right )}{5-x+x^2} \, dx+\frac {1}{243} \left (50 \left (1-i \sqrt {19}\right )\right ) \int \frac {(-1+2 x) \log \left (-1+i \sqrt {19}+2 x\right ) \log ^2\left (5-x+x^2\right )}{5-x+x^2} \, dx-\frac {1}{243} \left (50 \left (1-i \sqrt {19}\right )\right ) \int \left (-\frac {\log \left (-1+i \sqrt {19}+2 x\right ) \log ^2\left (5-x+x^2\right )}{5-x+x^2}+\frac {2 x \log \left (-1+i \sqrt {19}+2 x\right ) \log ^2\left (5-x+x^2\right )}{5-x+x^2}\right ) \, dx+\frac {1}{243} \left (50 \left (1+i \sqrt {19}\right )\right ) \int \frac {(-1+2 x) \log \left (-1-i \sqrt {19}+2 x\right ) \log ^2\left (5-x+x^2\right )}{5-x+x^2} \, dx-\frac {1}{243} \left (50 \left (1+i \sqrt {19}\right )\right ) \int \left (-\frac {\log \left (-1-i \sqrt {19}+2 x\right ) \log ^2\left (5-x+x^2\right )}{5-x+x^2}+\frac {2 x \log \left (-1-i \sqrt {19}+2 x\right ) \log ^2\left (5-x+x^2\right )}{5-x+x^2}\right ) \, dx\\ &=\frac {25}{729} x \log ^4\left (5-x+x^2\right )-\frac {200}{243} \int \left (\frac {\left (-1-i \sqrt {19}\right ) \log ^2\left (5-x+x^2\right )}{-1-i \sqrt {19}+2 x}+\frac {\left (-1+i \sqrt {19}\right ) \log ^2\left (5-x+x^2\right )}{-1+i \sqrt {19}+2 x}\right ) \, dx-\frac {800}{243} \int \frac {x (-1+2 x) \log \left (5-x+x^2\right )}{5-x+x^2} \, dx+\frac {800}{243} \int \left (2 \log \left (5-x+x^2\right )-\frac {(10-x) \log \left (5-x+x^2\right )}{5-x+x^2}\right ) \, dx+\frac {1}{243} \left (50 \left (1-i \sqrt {19}\right )\right ) \int \frac {\log \left (-1+i \sqrt {19}+2 x\right ) \log ^2\left (5-x+x^2\right )}{5-x+x^2} \, dx+\frac {1}{243} \left (50 \left (1-i \sqrt {19}\right )\right ) \int \left (-\frac {\log \left (-1+i \sqrt {19}+2 x\right ) \log ^2\left (5-x+x^2\right )}{5-x+x^2}+\frac {2 x \log \left (-1+i \sqrt {19}+2 x\right ) \log ^2\left (5-x+x^2\right )}{5-x+x^2}\right ) \, dx-\frac {1}{243} \left (100 \left (1-i \sqrt {19}\right )\right ) \int \frac {x \log \left (-1+i \sqrt {19}+2 x\right ) \log ^2\left (5-x+x^2\right )}{5-x+x^2} \, dx-\frac {1}{243} \left (200 \left (1-i \sqrt {19}\right )\right ) \int \frac {\log ^2\left (5-x+x^2\right )}{-1+i \sqrt {19}+2 x} \, dx+\frac {1}{243} \left (50 \left (1+i \sqrt {19}\right )\right ) \int \frac {\log \left (-1-i \sqrt {19}+2 x\right ) \log ^2\left (5-x+x^2\right )}{5-x+x^2} \, dx+\frac {1}{243} \left (50 \left (1+i \sqrt {19}\right )\right ) \int \left (-\frac {\log \left (-1-i \sqrt {19}+2 x\right ) \log ^2\left (5-x+x^2\right )}{5-x+x^2}+\frac {2 x \log \left (-1-i \sqrt {19}+2 x\right ) \log ^2\left (5-x+x^2\right )}{5-x+x^2}\right ) \, dx-\frac {1}{243} \left (100 \left (1+i \sqrt {19}\right )\right ) \int \frac {x \log \left (-1-i \sqrt {19}+2 x\right ) \log ^2\left (5-x+x^2\right )}{5-x+x^2} \, dx-\frac {1}{243} \left (200 \left (1+i \sqrt {19}\right )\right ) \int \frac {\log ^2\left (5-x+x^2\right )}{-1-i \sqrt {19}+2 x} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A] time = 1.49, size = 16, normalized size = 1.00 \begin {gather*} \frac {25}{729} x \log ^4\left (5-x+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 14, normalized size = 0.88 \begin {gather*} \frac {25}{729} \, x \log \left (x^{2} - x + 5\right )^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.82, size = 14, normalized size = 0.88 \begin {gather*} \frac {25}{729} \, x \log \left (x^{2} - x + 5\right )^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.91, size = 15, normalized size = 0.94
method | result | size |
norman | \(\frac {25 x \ln \left (x^{2}-x +5\right )^{4}}{729}\) | \(15\) |
risch | \(\frac {25 x \ln \left (x^{2}-x +5\right )^{4}}{729}\) | \(15\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.25, size = 14, normalized size = 0.88 \begin {gather*} \frac {25}{729} \, x \log \left (x^{2} - x + 5\right )^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.89, size = 14, normalized size = 0.88 \begin {gather*} \frac {25\,x\,{\ln \left (x^2-x+5\right )}^4}{729} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 14, normalized size = 0.88 \begin {gather*} \frac {25 x \log {\left (x^{2} - x + 5 \right )}^{4}}{729} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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