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 = 1.00, 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] Leaf count is larger than twice the leaf count of optimal. \(59\) vs. \(2(23)=46\).
time = 1.52, 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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(183\) vs.
\(2(23)=46\).
time = 12.12, size = 184, normalized size = 8.00
method | result | size |
risch | \(\frac {3 x^{5}-18 x^{4}-63 x^{3}+432 x^{2}+1647 x +1458}{x^{2}+6 x +9}+\frac {75 x \left (2 x^{2} \ln \left (3 x \right )^{2}-6 x \ln \left (3 x \right )^{2}-30 \ln \left (3 x \right )^{2}+25 x +75\right )}{\left (3+x \right ) \ln \left (3 x \right )^{4}}\) | \(84\) |
default | \(\frac {450}{\ln \left (x \right )^{2}+2 \ln \left (3\right ) \ln \left (x \right )+\ln \left (3\right )^{2}}+3 x^{3}-\frac {6 \left (6 \ln \left (3\right )^{2}+12 \ln \left (3\right ) \ln \left (x \right )+6 \ln \left (x \right )^{2}-25\right ) x^{2}}{\left (\ln \left (x \right )+\ln \left (3\right )\right )^{2}}+\frac {3 \left (42 \ln \left (3\right )^{4}+168 \ln \left (x \right ) \ln \left (3\right )^{3}+252 \ln \left (x \right )^{2} \ln \left (3\right )^{2}+168 \ln \left (3\right ) \ln \left (x \right )^{3}+42 \ln \left (x \right )^{4}-300 \ln \left (3\right )^{2}-600 \ln \left (3\right ) \ln \left (x \right )-300 \ln \left (x \right )^{2}+625\right ) x}{\left (\ln \left (x \right )+\ln \left (3\right )\right )^{4}}+\frac {513 x \ln \left (3\right )^{2}+1026 x \ln \left (3\right ) \ln \left (x \right )+513 x \ln \left (x \right )^{2}+1458 \ln \left (3\right )^{2}+2916 \ln \left (3\right ) \ln \left (x \right )+1458 \ln \left (x \right )^{2}-1350 x -4050}{\left (3+x \right )^{2} \left (\ln \left (x \right )+\ln \left (3\right )\right )^{2}}\) | \(184\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 410 vs.
\(2 (23) = 46\).
time = 0.54, size = 410, normalized size = 17.83 \begin {gather*} \frac {3 \, {\left (x^{5} \log \left (3\right )^{4} - 2 \, {\left (3 \, \log \left (3\right )^{4} - 25 \, \log \left (3\right )^{2}\right )} x^{4} + {\left (x^{5} - 6 \, x^{4} - 21 \, x^{3} + 144 \, x^{2} + 549 \, x + 486\right )} \log \left (x\right )^{4} - {\left (21 \, \log \left (3\right )^{4} - 625\right )} x^{3} + 486 \, \log \left (3\right )^{4} + 4 \, {\left (x^{5} \log \left (3\right ) - 6 \, x^{4} \log \left (3\right ) - 21 \, x^{3} \log \left (3\right ) + 144 \, x^{2} \log \left (3\right ) + 549 \, x \log \left (3\right ) + 486 \, \log \left (3\right )\right )} \log \left (x\right )^{3} + 6 \, {\left (24 \, \log \left (3\right )^{4} - 200 \, \log \left (3\right )^{2} + 625\right )} x^{2} + 2 \, {\left (3 \, x^{5} \log \left (3\right )^{2} - {\left (18 \, \log \left (3\right )^{2} - 25\right )} x^{4} - 63 \, x^{3} \log \left (3\right )^{2} + 24 \, {\left (18 \, \log \left (3\right )^{2} - 25\right )} x^{2} + 9 \, {\left (183 \, \log \left (3\right )^{2} - 125\right )} x + 1458 \, \log \left (3\right )^{2}\right )} \log \left (x\right )^{2} + 9 \, {\left (61 \, \log \left (3\right )^{4} - 250 \, \log \left (3\right )^{2} + 625\right )} x + 4 \, {\left (x^{5} \log \left (3\right )^{3} - 21 \, x^{3} \log \left (3\right )^{3} - {\left (6 \, \log \left (3\right )^{3} - 25 \, \log \left (3\right )\right )} x^{4} + 24 \, {\left (6 \, \log \left (3\right )^{3} - 25 \, \log \left (3\right )\right )} x^{2} + 486 \, \log \left (3\right )^{3} + 9 \, {\left (61 \, \log \left (3\right )^{3} - 125 \, \log \left (3\right )\right )} x\right )} \log \left (x\right )\right )}}{x^{2} \log \left (3\right )^{4} + 6 \, x \log \left (3\right )^{4} + {\left (x^{2} + 6 \, x + 9\right )} \log \left (x\right )^{4} + 9 \, \log \left (3\right )^{4} + 4 \, {\left (x^{2} \log \left (3\right ) + 6 \, x \log \left (3\right ) + 9 \, \log \left (3\right )\right )} \log \left (x\right )^{3} + 6 \, {\left (x^{2} \log \left (3\right )^{2} + 6 \, x \log \left (3\right )^{2} + 9 \, \log \left (3\right )^{2}\right )} \log \left (x\right )^{2} + 4 \, {\left (x^{2} \log \left (3\right )^{3} + 6 \, x \log \left (3\right )^{3} + 9 \, \log \left (3\right )^{3}\right )} \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 82 vs.
\(2 (23) = 46\).
time = 0.37, 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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 65 vs.
\(2 (20) = 40\).
time = 0.24, 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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 92 vs.
\(2 (23) = 46\).
time = 0.43, 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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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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