Optimal. Leaf size=23 \[ (3+x) \left (4+x+\frac {x+\log (x)}{-3+x+\log (3+x)}\right )^2 \]
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Rubi [F] time = 4.12, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {216-900 x+630 x^2+72 x^3-92 x^4-x^5+3 x^6+\left (-18+204 x-52 x^2-20 x^3+2 x^4\right ) \log (x)+\left (-11 x-x^2\right ) \log ^2(x)+\left (-144+906 x-254 x^2-184 x^3+22 x^4+9 x^5+\left (6-108 x-8 x^2+6 x^3\right ) \log (x)+x \log ^2(x)\right ) \log (3+x)+\left (24-322 x-48 x^2+45 x^3+9 x^4+\left (14 x+4 x^2\right ) \log (x)\right ) \log ^2(3+x)+\left (40 x+22 x^2+3 x^3\right ) \log ^3(3+x)}{-27 x+27 x^2-9 x^3+x^4+\left (27 x-18 x^2+3 x^3\right ) \log (3+x)+\left (-9 x+3 x^2\right ) \log ^2(3+x)+x \log ^3(3+x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-216+900 x-630 x^2-72 x^3+92 x^4+x^5-3 x^6-\left (-18+204 x-52 x^2-20 x^3+2 x^4\right ) \log (x)-\left (-11 x-x^2\right ) \log ^2(x)-\left (-144+906 x-254 x^2-184 x^3+22 x^4+9 x^5+\left (6-108 x-8 x^2+6 x^3\right ) \log (x)+x \log ^2(x)\right ) \log (3+x)-\left (24-322 x-48 x^2+45 x^3+9 x^4+\left (14 x+4 x^2\right ) \log (x)\right ) \log ^2(3+x)-\left (40 x+22 x^2+3 x^3\right ) \log ^3(3+x)}{x (3-x-\log (3+x))^3} \, dx\\ &=\int \left (40+22 x+3 x^2-\frac {2 (4+x) (x+\log (x))^2}{(-3+x+\log (3+x))^3}+\frac {6 x-24 x^2-13 x^3-2 x^4+6 \log (x)-24 x \log (x)-12 x^2 \log (x)-2 x^3 \log (x)+x \log ^2(x)}{x (-3+x+\log (3+x))^2}+\frac {2 \left (12+19 x+15 x^2+3 x^3+7 x \log (x)+2 x^2 \log (x)\right )}{x (-3+x+\log (3+x))}\right ) \, dx\\ &=40 x+11 x^2+x^3-2 \int \frac {(4+x) (x+\log (x))^2}{(-3+x+\log (3+x))^3} \, dx+2 \int \frac {12+19 x+15 x^2+3 x^3+7 x \log (x)+2 x^2 \log (x)}{x (-3+x+\log (3+x))} \, dx+\int \frac {6 x-24 x^2-13 x^3-2 x^4+6 \log (x)-24 x \log (x)-12 x^2 \log (x)-2 x^3 \log (x)+x \log ^2(x)}{x (-3+x+\log (3+x))^2} \, dx\\ &=40 x+11 x^2+x^3-2 \int \left (\frac {4 (x+\log (x))^2}{(-3+x+\log (3+x))^3}+\frac {x (x+\log (x))^2}{(-3+x+\log (3+x))^3}\right ) \, dx+2 \int \left (\frac {19}{-3+x+\log (3+x)}+\frac {12}{x (-3+x+\log (3+x))}+\frac {15 x}{-3+x+\log (3+x)}+\frac {3 x^2}{-3+x+\log (3+x)}+\frac {7 \log (x)}{-3+x+\log (3+x)}+\frac {2 x \log (x)}{-3+x+\log (3+x)}\right ) \, dx+\int \left (\frac {6}{(-3+x+\log (3+x))^2}-\frac {24 x}{(-3+x+\log (3+x))^2}-\frac {13 x^2}{(-3+x+\log (3+x))^2}-\frac {2 x^3}{(-3+x+\log (3+x))^2}-\frac {24 \log (x)}{(-3+x+\log (3+x))^2}+\frac {6 \log (x)}{x (-3+x+\log (3+x))^2}-\frac {12 x \log (x)}{(-3+x+\log (3+x))^2}-\frac {2 x^2 \log (x)}{(-3+x+\log (3+x))^2}+\frac {\log ^2(x)}{(-3+x+\log (3+x))^2}\right ) \, dx\\ &=40 x+11 x^2+x^3-2 \int \frac {x (x+\log (x))^2}{(-3+x+\log (3+x))^3} \, dx-2 \int \frac {x^3}{(-3+x+\log (3+x))^2} \, dx-2 \int \frac {x^2 \log (x)}{(-3+x+\log (3+x))^2} \, dx+4 \int \frac {x \log (x)}{-3+x+\log (3+x)} \, dx+6 \int \frac {1}{(-3+x+\log (3+x))^2} \, dx+6 \int \frac {\log (x)}{x (-3+x+\log (3+x))^2} \, dx+6 \int \frac {x^2}{-3+x+\log (3+x)} \, dx-8 \int \frac {(x+\log (x))^2}{(-3+x+\log (3+x))^3} \, dx-12 \int \frac {x \log (x)}{(-3+x+\log (3+x))^2} \, dx-13 \int \frac {x^2}{(-3+x+\log (3+x))^2} \, dx+14 \int \frac {\log (x)}{-3+x+\log (3+x)} \, dx-24 \int \frac {x}{(-3+x+\log (3+x))^2} \, dx-24 \int \frac {\log (x)}{(-3+x+\log (3+x))^2} \, dx+24 \int \frac {1}{x (-3+x+\log (3+x))} \, dx+30 \int \frac {x}{-3+x+\log (3+x)} \, dx+38 \int \frac {1}{-3+x+\log (3+x)} \, dx+\int \frac {\log ^2(x)}{(-3+x+\log (3+x))^2} \, dx\\ &=40 x+11 x^2+x^3-2 \int \frac {x^3}{(-3+x+\log (3+x))^2} \, dx-2 \int \frac {x^2 \log (x)}{(-3+x+\log (3+x))^2} \, dx-2 \int \left (\frac {x^3}{(-3+x+\log (3+x))^3}+\frac {2 x^2 \log (x)}{(-3+x+\log (3+x))^3}+\frac {x \log ^2(x)}{(-3+x+\log (3+x))^3}\right ) \, dx+4 \int \frac {x \log (x)}{-3+x+\log (3+x)} \, dx+6 \int \frac {1}{(-3+x+\log (3+x))^2} \, dx+6 \int \frac {\log (x)}{x (-3+x+\log (3+x))^2} \, dx+6 \int \frac {x^2}{-3+x+\log (3+x)} \, dx-8 \int \left (\frac {x^2}{(-3+x+\log (3+x))^3}+\frac {2 x \log (x)}{(-3+x+\log (3+x))^3}+\frac {\log ^2(x)}{(-3+x+\log (3+x))^3}\right ) \, dx-12 \int \frac {x \log (x)}{(-3+x+\log (3+x))^2} \, dx-13 \int \frac {x^2}{(-3+x+\log (3+x))^2} \, dx+14 \int \frac {\log (x)}{-3+x+\log (3+x)} \, dx-24 \int \frac {x}{(-3+x+\log (3+x))^2} \, dx-24 \int \frac {\log (x)}{(-3+x+\log (3+x))^2} \, dx+24 \int \frac {1}{x (-3+x+\log (3+x))} \, dx+30 \int \frac {x}{-3+x+\log (3+x)} \, dx+38 \int \frac {1}{-3+x+\log (3+x)} \, dx+\int \frac {\log ^2(x)}{(-3+x+\log (3+x))^2} \, dx\\ &=40 x+11 x^2+x^3-2 \int \frac {x^3}{(-3+x+\log (3+x))^3} \, dx-2 \int \frac {x \log ^2(x)}{(-3+x+\log (3+x))^3} \, dx-2 \int \frac {x^3}{(-3+x+\log (3+x))^2} \, dx-2 \int \frac {x^2 \log (x)}{(-3+x+\log (3+x))^2} \, dx-4 \int \frac {x^2 \log (x)}{(-3+x+\log (3+x))^3} \, dx+4 \int \frac {x \log (x)}{-3+x+\log (3+x)} \, dx+6 \int \frac {1}{(-3+x+\log (3+x))^2} \, dx+6 \int \frac {\log (x)}{x (-3+x+\log (3+x))^2} \, dx+6 \int \frac {x^2}{-3+x+\log (3+x)} \, dx-8 \int \frac {x^2}{(-3+x+\log (3+x))^3} \, dx-8 \int \frac {\log ^2(x)}{(-3+x+\log (3+x))^3} \, dx-12 \int \frac {x \log (x)}{(-3+x+\log (3+x))^2} \, dx-13 \int \frac {x^2}{(-3+x+\log (3+x))^2} \, dx+14 \int \frac {\log (x)}{-3+x+\log (3+x)} \, dx-16 \int \frac {x \log (x)}{(-3+x+\log (3+x))^3} \, dx-24 \int \frac {x}{(-3+x+\log (3+x))^2} \, dx-24 \int \frac {\log (x)}{(-3+x+\log (3+x))^2} \, dx+24 \int \frac {1}{x (-3+x+\log (3+x))} \, dx+30 \int \frac {x}{-3+x+\log (3+x)} \, dx+38 \int \frac {1}{-3+x+\log (3+x)} \, dx+\int \frac {\log ^2(x)}{(-3+x+\log (3+x))^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.10, size = 52, normalized size = 2.26 \begin {gather*} 40 x+11 x^2+x^3+\frac {(3+x) (x+\log (x))^2}{(-3+x+\log (3+x))^2}+\frac {2 (3+x) (4+x) (x+\log (x))}{-3+x+\log (3+x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.93, size = 126, normalized size = 5.48 \begin {gather*} \frac {x^{5} + 7 \, x^{4} - 8 \, x^{3} + {\left (x^{3} + 11 \, x^{2} + 40 \, x\right )} \log \left (x + 3\right )^{2} + {\left (x + 3\right )} \log \relax (x)^{2} - 156 \, x^{2} + 2 \, {\left (x^{4} + 9 \, x^{3} + 14 \, x^{2} + {\left (x^{2} + 7 \, x + 12\right )} \log \relax (x) - 108 \, x\right )} \log \left (x + 3\right ) + 2 \, {\left (x^{3} + 5 \, x^{2} - 6 \, x - 36\right )} \log \relax (x) + 288 \, x}{x^{2} + 2 \, {\left (x - 3\right )} \log \left (x + 3\right ) + \log \left (x + 3\right )^{2} - 6 \, x + 9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 149, normalized size = 6.48 \begin {gather*} x^{3} + 11 \, x^{2} + 40 \, x + \frac {2 \, x^{4} + 2 \, x^{3} \log \left (x + 3\right ) + 2 \, x^{3} \log \relax (x) + 2 \, x^{2} \log \left (x + 3\right ) \log \relax (x) + 9 \, x^{3} + 14 \, x^{2} \log \left (x + 3\right ) + 10 \, x^{2} \log \relax (x) + 14 \, x \log \left (x + 3\right ) \log \relax (x) + x \log \relax (x)^{2} - 15 \, x^{2} + 24 \, x \log \left (x + 3\right ) - 12 \, x \log \relax (x) + 24 \, \log \left (x + 3\right ) \log \relax (x) + 3 \, \log \relax (x)^{2} - 72 \, x - 72 \, \log \relax (x)}{x^{2} + 2 \, x \log \left (x + 3\right ) + \log \left (x + 3\right )^{2} - 6 \, x - 6 \, \log \left (x + 3\right ) + 9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 93, normalized size = 4.04
method | result | size |
risch | \(x^{3}+11 x^{2}+40 x +\frac {\left (2 x^{3}+2 x^{2} \ln \relax (x )+2 \ln \left (3+x \right ) x^{2}+2 \ln \relax (x ) \ln \left (3+x \right ) x +3 x^{2}+4 x \ln \relax (x )+8 x \ln \left (3+x \right )+\ln \relax (x )^{2}+8 \ln \relax (x ) \ln \left (3+x \right )-24 x -24 \ln \relax (x )\right ) \left (3+x \right )}{\left (\ln \left (3+x \right )+x -3\right )^{2}}\) | \(93\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 126, normalized size = 5.48 \begin {gather*} \frac {x^{5} + 7 \, x^{4} - 8 \, x^{3} + {\left (x^{3} + 11 \, x^{2} + 40 \, x\right )} \log \left (x + 3\right )^{2} + {\left (x + 3\right )} \log \relax (x)^{2} - 156 \, x^{2} + 2 \, {\left (x^{4} + 9 \, x^{3} + 14 \, x^{2} + {\left (x^{2} + 7 \, x + 12\right )} \log \relax (x) - 108 \, x\right )} \log \left (x + 3\right ) + 2 \, {\left (x^{3} + 5 \, x^{2} - 6 \, x - 36\right )} \log \relax (x) + 288 \, x}{x^{2} + 2 \, {\left (x - 3\right )} \log \left (x + 3\right ) + \log \left (x + 3\right )^{2} - 6 \, x + 9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\ln \left (x+3\right )}^2\,\left (\ln \relax (x)\,\left (4\,x^2+14\,x\right )-322\,x-48\,x^2+45\,x^3+9\,x^4+24\right )-\ln \relax (x)\,\left (-2\,x^4+20\,x^3+52\,x^2-204\,x+18\right )-900\,x+{\ln \left (x+3\right )}^3\,\left (3\,x^3+22\,x^2+40\,x\right )-{\ln \relax (x)}^2\,\left (x^2+11\,x\right )+\ln \left (x+3\right )\,\left (906\,x+x\,{\ln \relax (x)}^2-254\,x^2-184\,x^3+22\,x^4+9\,x^5-\ln \relax (x)\,\left (-6\,x^3+8\,x^2+108\,x-6\right )-144\right )+630\,x^2+72\,x^3-92\,x^4-x^5+3\,x^6+216}{\ln \left (x+3\right )\,\left (3\,x^3-18\,x^2+27\,x\right )-27\,x-{\ln \left (x+3\right )}^2\,\left (9\,x-3\,x^2\right )+x\,{\ln \left (x+3\right )}^3+27\,x^2-9\,x^3+x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.46, size = 136, normalized size = 5.91 \begin {gather*} x^{3} + 11 x^{2} + 40 x + \frac {2 x^{4} + 2 x^{3} \log {\relax (x )} + 9 x^{3} + 10 x^{2} \log {\relax (x )} - 15 x^{2} + x \log {\relax (x )}^{2} - 12 x \log {\relax (x )} - 72 x + \left (2 x^{3} + 2 x^{2} \log {\relax (x )} + 14 x^{2} + 14 x \log {\relax (x )} + 24 x + 24 \log {\relax (x )}\right ) \log {\left (x + 3 \right )} + 3 \log {\relax (x )}^{2} - 72 \log {\relax (x )}}{x^{2} - 6 x + \left (2 x - 6\right ) \log {\left (x + 3 \right )} + \log {\left (x + 3 \right )}^{2} + 9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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