Optimal. Leaf size=23 \[ 3 x \left (-6+x+\frac {3}{3+x}+\frac {25}{\log ^2(3 x)}\right )^2 \]
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Rubi [F] time = 0.82, 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 {-202500-202500 x-67500 x^2-7500 x^3+\left (50625+50625 x+16875 x^2+1875 x^3\right ) \log (3 x)+\left (40500+35100 x+7200 x^2-900 x^3-300 x^4\right ) \log ^2(3 x)+\left (-20250-14850 x+1800 x^3+300 x^4\right ) \log ^3(3 x)+\left (2025+945 x-567 x^2-279 x^3+9 x^4+9 x^5\right ) \log ^5(3 x)}{\left (27+27 x+9 x^2+x^3\right ) \log ^5(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 \left (\frac {9 \left (225+105 x-63 x^2-31 x^3+x^4+x^5\right )}{(3+x)^3}-\frac {7500}{\log ^5(3 x)}+\frac {1875}{\log ^4(3 x)}-\frac {300 \left (-15-3 x+x^2\right )}{(3+x) \log ^3(3 x)}+\frac {150 \left (-45-18 x+6 x^2+2 x^3\right )}{(3+x)^2 \log ^2(3 x)}\right ) \, dx\\ &=9 \int \frac {225+105 x-63 x^2-31 x^3+x^4+x^5}{(3+x)^3} \, dx+150 \int \frac {-45-18 x+6 x^2+2 x^3}{(3+x)^2 \log ^2(3 x)} \, dx-300 \int \frac {-15-3 x+x^2}{(3+x) \log ^3(3 x)} \, dx+1875 \int \frac {1}{\log ^4(3 x)} \, dx-7500 \int \frac {1}{\log ^5(3 x)} \, dx\\ &=\frac {1875 x}{\log ^4(3 x)}-\frac {625 x}{\log ^3(3 x)}+9 \int \left (14-8 x+x^2+\frac {18}{(3+x)^3}-\frac {57}{(3+x)^2}\right ) \, dx+150 \int \frac {-45-18 x+6 x^2+2 x^3}{(3+x)^2 \log ^2(3 x)} \, dx-300 \int \frac {-15-3 x+x^2}{(3+x) \log ^3(3 x)} \, dx+625 \int \frac {1}{\log ^3(3 x)} \, dx-1875 \int \frac {1}{\log ^4(3 x)} \, dx\\ &=126 x-36 x^2+3 x^3-\frac {81}{(3+x)^2}+\frac {513}{3+x}+\frac {1875 x}{\log ^4(3 x)}-\frac {625 x}{2 \log ^2(3 x)}+150 \int \frac {-45-18 x+6 x^2+2 x^3}{(3+x)^2 \log ^2(3 x)} \, dx-300 \int \frac {-15-3 x+x^2}{(3+x) \log ^3(3 x)} \, dx+\frac {625}{2} \int \frac {1}{\log ^2(3 x)} \, dx-625 \int \frac {1}{\log ^3(3 x)} \, dx\\ &=126 x-36 x^2+3 x^3-\frac {81}{(3+x)^2}+\frac {513}{3+x}+\frac {1875 x}{\log ^4(3 x)}-\frac {625 x}{2 \log (3 x)}+150 \int \frac {-45-18 x+6 x^2+2 x^3}{(3+x)^2 \log ^2(3 x)} \, dx-300 \int \frac {-15-3 x+x^2}{(3+x) \log ^3(3 x)} \, dx-\frac {625}{2} \int \frac {1}{\log ^2(3 x)} \, dx+\frac {625}{2} \int \frac {1}{\log (3 x)} \, dx\\ &=126 x-36 x^2+3 x^3-\frac {81}{(3+x)^2}+\frac {513}{3+x}+\frac {1875 x}{\log ^4(3 x)}+\frac {625 \text {li}(3 x)}{6}+150 \int \frac {-45-18 x+6 x^2+2 x^3}{(3+x)^2 \log ^2(3 x)} \, dx-300 \int \frac {-15-3 x+x^2}{(3+x) \log ^3(3 x)} \, dx-\frac {625}{2} \int \frac {1}{\log (3 x)} \, dx\\ &=126 x-36 x^2+3 x^3-\frac {81}{(3+x)^2}+\frac {513}{3+x}+\frac {1875 x}{\log ^4(3 x)}+150 \int \frac {-45-18 x+6 x^2+2 x^3}{(3+x)^2 \log ^2(3 x)} \, dx-300 \int \frac {-15-3 x+x^2}{(3+x) \log ^3(3 x)} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 1.45, size = 59, normalized size = 2.57 \begin {gather*} 126 x-36 x^2+3 x^3-\frac {81}{(3+x)^2}+\frac {513}{3+x}+\frac {1875 x}{\log ^4(3 x)}+\frac {150 x \left (-15-3 x+x^2\right )}{(3+x) \log ^2(3 x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.95, size = 82, normalized size = 3.57 \begin {gather*} \frac {3 \, {\left ({\left (x^{5} - 6 \, x^{4} - 21 \, x^{3} + 144 \, x^{2} + 549 \, x + 486\right )} \log \left (3 \, x\right )^{4} + 625 \, x^{3} + 50 \, {\left (x^{4} - 24 \, x^{2} - 45 \, x\right )} \log \left (3 \, x\right )^{2} + 3750 \, x^{2} + 5625 \, x\right )}}{{\left (x^{2} + 6 \, x + 9\right )} \log \left (3 \, x\right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 92, normalized size = 4.00 \begin {gather*} 3 \, x^{3} - 36 \, x^{2} + 126 \, x + \frac {75 \, {\left (2 \, x^{3} \log \left (3 \, x\right )^{2} - 6 \, x^{2} \log \left (3 \, x\right )^{2} - 30 \, x \log \left (3 \, x\right )^{2} + 25 \, x^{2} + 75 \, x\right )}}{x \log \left (3 \, x\right )^{4} + 3 \, \log \left (3 \, x\right )^{4}} + \frac {27 \, {\left (19 \, x + 54\right )}}{x^{2} + 6 \, x + 9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.11, size = 84, normalized size = 3.65
method | result | size |
risch | \(\frac {3 x^{5}-18 x^{4}-63 x^{3}+432 x^{2}+1647 x +1458}{x^{2}+6 x +9}+\frac {75 x \left (2 x^{2} \ln \left (3 x \right )^{2}-6 x \ln \left (3 x \right )^{2}-30 \ln \left (3 x \right )^{2}+25 x +75\right )}{\left (3+x \right ) \ln \left (3 x \right )^{4}}\) | \(84\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.50, size = 410, normalized size = 17.83 \begin {gather*} \frac {3 \, {\left (x^{5} \log \relax (3)^{4} - 2 \, {\left (3 \, \log \relax (3)^{4} - 25 \, \log \relax (3)^{2}\right )} x^{4} + {\left (x^{5} - 6 \, x^{4} - 21 \, x^{3} + 144 \, x^{2} + 549 \, x + 486\right )} \log \relax (x)^{4} - {\left (21 \, \log \relax (3)^{4} - 625\right )} x^{3} + 486 \, \log \relax (3)^{4} + 4 \, {\left (x^{5} \log \relax (3) - 6 \, x^{4} \log \relax (3) - 21 \, x^{3} \log \relax (3) + 144 \, x^{2} \log \relax (3) + 549 \, x \log \relax (3) + 486 \, \log \relax (3)\right )} \log \relax (x)^{3} + 6 \, {\left (24 \, \log \relax (3)^{4} - 200 \, \log \relax (3)^{2} + 625\right )} x^{2} + 2 \, {\left (3 \, x^{5} \log \relax (3)^{2} - {\left (18 \, \log \relax (3)^{2} - 25\right )} x^{4} - 63 \, x^{3} \log \relax (3)^{2} + 24 \, {\left (18 \, \log \relax (3)^{2} - 25\right )} x^{2} + 9 \, {\left (183 \, \log \relax (3)^{2} - 125\right )} x + 1458 \, \log \relax (3)^{2}\right )} \log \relax (x)^{2} + 9 \, {\left (61 \, \log \relax (3)^{4} - 250 \, \log \relax (3)^{2} + 625\right )} x + 4 \, {\left (x^{5} \log \relax (3)^{3} - 21 \, x^{3} \log \relax (3)^{3} - {\left (6 \, \log \relax (3)^{3} - 25 \, \log \relax (3)\right )} x^{4} + 24 \, {\left (6 \, \log \relax (3)^{3} - 25 \, \log \relax (3)\right )} x^{2} + 486 \, \log \relax (3)^{3} + 9 \, {\left (61 \, \log \relax (3)^{3} - 125 \, \log \relax (3)\right )} x\right )} \log \relax (x)\right )}}{x^{2} \log \relax (3)^{4} + 6 \, x \log \relax (3)^{4} + {\left (x^{2} + 6 \, x + 9\right )} \log \relax (x)^{4} + 9 \, \log \relax (3)^{4} + 4 \, {\left (x^{2} \log \relax (3) + 6 \, x \log \relax (3) + 9 \, \log \relax (3)\right )} \log \relax (x)^{3} + 6 \, {\left (x^{2} \log \relax (3)^{2} + 6 \, x \log \relax (3)^{2} + 9 \, \log \relax (3)^{2}\right )} \log \relax (x)^{2} + 4 \, {\left (x^{2} \log \relax (3)^{3} + 6 \, x \log \relax (3)^{3} + 9 \, \log \relax (3)^{3}\right )} \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.79, size = 68, normalized size = 2.96 \begin {gather*} \frac {3\,x\,{\left (25\,x+75\right )}^2-6\,x\,{\ln \left (3\,x\right )}^2\,\left (25\,x+75\right )\,\left (-x^2+3\,x+15\right )}{{\ln \left (3\,x\right )}^4\,{\left (x+3\right )}^2}+\frac {3\,x\,{\left (-x^2+3\,x+15\right )}^2}{{\left (x+3\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.44, size = 65, normalized size = 2.83 \begin {gather*} 3 x^{3} - 36 x^{2} + 126 x + \frac {513 x + 1458}{x^{2} + 6 x + 9} + \frac {1875 x^{2} + 5625 x + \left (150 x^{3} - 450 x^{2} - 2250 x\right ) \log {\left (3 x \right )}^{2}}{\left (x + 3\right ) \log {\left (3 x \right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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