∫ −48−470x−378x2+30x3+50x4+(−48−16x+216x2−240x3−200x4) log(x)_______ −1728−33264x−181611x2−200033x3+5697x4+79731x5+10255x6−11475x7−1125x8+625x9 dx [9375]

Optimal. Leaf size=28 \[ \frac {x \log (x)}{4 (-3+x)^2 \left (1+\frac {1}{8} x (9+5 x)^2\right )} \]

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Rubi [F]
time = 65.32, 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 {-48-470 x-378 x^2+30 x^3+50 x^4+\left (-48-16 x+216 x^2-240 x^3-200 x^4\right ) \log (x)}{-1728-33264 x-181611 x^2-200033 x^3+5697 x^4+79731 x^5+10255 x^6-11475 x^7-1125 x^8+625 x^9} \, dx \end {gather*}

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {48+470 x+378 x^2-30 x^3-50 x^4-\left (-48-16 x+216 x^2-240 x^3-200 x^4\right ) \log (x)}{(3-x)^3 \left (8+81 x+90 x^2+25 x^3\right )^2} \, dx\\ &=\int \left (-\frac {48}{(-3+x)^3 \left (8+81 x+90 x^2+25 x^3\right )^2}-\frac {470 x}{(-3+x)^3 \left (8+81 x+90 x^2+25 x^3\right )^2}-\frac {378 x^2}{(-3+x)^3 \left (8+81 x+90 x^2+25 x^3\right )^2}+\frac {30 x^3}{(-3+x)^3 \left (8+81 x+90 x^2+25 x^3\right )^2}+\frac {50 x^4}{(-3+x)^3 \left (8+81 x+90 x^2+25 x^3\right )^2}-\frac {8 \left (6+2 x-27 x^2+30 x^3+25 x^4\right ) \log (x)}{(-3+x)^3 \left (8+81 x+90 x^2+25 x^3\right )^2}\right ) \, dx\\ &=-\left (8 \int \frac {\left (6+2 x-27 x^2+30 x^3+25 x^4\right ) \log (x)}{(-3+x)^3 \left (8+81 x+90 x^2+25 x^3\right )^2} \, dx\right )+30 \int \frac {x^3}{(-3+x)^3 \left (8+81 x+90 x^2+25 x^3\right )^2} \, dx-48 \int \frac {1}{(-3+x)^3 \left (8+81 x+90 x^2+25 x^3\right )^2} \, dx+50 \int \frac {x^4}{(-3+x)^3 \left (8+81 x+90 x^2+25 x^3\right )^2} \, dx-378 \int \frac {x^2}{(-3+x)^3 \left (8+81 x+90 x^2+25 x^3\right )^2} \, dx-470 \int \frac {x}{(-3+x)^3 \left (8+81 x+90 x^2+25 x^3\right )^2} \, dx\\ &=-\left (8 \int \left (\frac {3 \log (x)}{3472 (-3+x)^3}-\frac {269 \log (x)}{1506848 (-3+x)^2}-\frac {3 \left (97656+497915 x+256275 x^2\right ) \log (x)}{753424 \left (8+81 x+90 x^2+25 x^3\right )^2}+\frac {5 (6402+1345 x) \log (x)}{1506848 \left (8+81 x+90 x^2+25 x^3\right )}\right ) \, dx\right )+30 \int \left (\frac {27}{3013696 (-3+x)^3}-\frac {2889}{653972032 (-3+x)^2}+\frac {759411}{567647723776 (-3+x)}+\frac {-290456-1792395 x+9493425 x^2}{653972032 \left (8+81 x+90 x^2+25 x^3\right )^2}-\frac {9 \left (-4141716+6956835 x+2109475 x^2\right )}{567647723776 \left (8+81 x+90 x^2+25 x^3\right )}\right ) \, dx-48 \text {Subst}\left (\int \frac {1}{\left (-\frac {21}{5}+x\right )^3 \left (-\frac {14}{5}-27 x+25 x^3\right )^2} \, dx,x,\frac {6}{5}+x\right )+50 \int \left (\frac {81}{3013696 (-3+x)^3}-\frac {351}{81746504 (-3+x)^2}-\frac {229419}{567647723776 (-3+x)}+\frac {-3037896-31049153 x-35968725 x^2}{653972032 \left (8+81 x+90 x^2+25 x^3\right )^2}+\frac {27 \left (12432852+3658805 x+212425 x^2\right )}{567647723776 \left (8+81 x+90 x^2+25 x^3\right )}\right ) \, dx-378 \int \left (\frac {9}{3013696 (-3+x)^3}-\frac {807}{326986016 (-3+x)^2}+\frac {720121}{567647723776 (-3+x)}+\frac {1148472+12761055 x+907675 x^2}{653972032 \left (8+81 x+90 x^2+25 x^3\right )^2}+\frac {-120941316-83796165 x-18003025 x^2}{567647723776 \left (8+81 x+90 x^2+25 x^3\right )}\right ) \, dx-470 \int \left (\frac {3}{3013696 (-3+x)^3}-\frac {755}{653972032 (-3+x)^2}+\frac {458487}{567647723776 (-3+x)}+\frac {1132776-12012635 x-3588975 x^2}{653972032 \left (8+81 x+90 x^2+25 x^3\right )^2}+\frac {-120933612-59266855 x-11462175 x^2}{567647723776 \left (8+81 x+90 x^2+25 x^3\right )}\right ) \, dx\\ &=-\frac {3}{376712 (3-x)^2}+\frac {46117}{40873252 (3-x)}-\frac {29773953 \log (3-x)}{35477982736}-\frac {135 \int \frac {-4141716+6956835 x+2109475 x^2}{8+81 x+90 x^2+25 x^3} \, dx}{283823861888}-\frac {27 \int \frac {-120941316-83796165 x-18003025 x^2}{8+81 x+90 x^2+25 x^3} \, dx}{40546265984}-\frac {235 \int \frac {-120933612-59266855 x-11462175 x^2}{8+81 x+90 x^2+25 x^3} \, dx}{283823861888}+\frac {675 \int \frac {12432852+3658805 x+212425 x^2}{8+81 x+90 x^2+25 x^3} \, dx}{283823861888}+\frac {15 \int \frac {-290456-1792395 x+9493425 x^2}{\left (8+81 x+90 x^2+25 x^3\right )^2} \, dx}{326986016}+\frac {25 \int \frac {-3037896-31049153 x-35968725 x^2}{\left (8+81 x+90 x^2+25 x^3\right )^2} \, dx}{326986016}-\frac {27 \int \frac {1148472+12761055 x+907675 x^2}{\left (8+81 x+90 x^2+25 x^3\right )^2} \, dx}{46712288}-\frac {235 \int \frac {1132776-12012635 x-3588975 x^2}{\left (8+81 x+90 x^2+25 x^3\right )^2} \, dx}{326986016}-\frac {5 \int \frac {(6402+1345 x) \log (x)}{8+81 x+90 x^2+25 x^3} \, dx}{188356}+\frac {3 \int \frac {\left (97656+497915 x+256275 x^2\right ) \log (x)}{\left (8+81 x+90 x^2+25 x^3\right )^2} \, dx}{94178}+\frac {269 \int \frac {\log (x)}{(-3+x)^2} \, dx}{188356}-\frac {3}{434} \int \frac {\log (x)}{(-3+x)^3} \, dx-18750000 \text {Subst}\left (\int \frac {1}{\left (-\frac {21}{5}+x\right )^3 \left (-\frac {5 \left (9+\left (7+2 i \sqrt {170}\right )^{2/3}\right )}{\sqrt [3]{7+2 i \sqrt {170}}}+25 x\right )^2 \left (-25 \left (9-\frac {81}{\left (7+2 i \sqrt {170}\right )^{2/3}}-\left (7+2 i \sqrt {170}\right )^{2/3}\right )+\frac {125 \left (9+\left (7+2 i \sqrt {170}\right )^{2/3}\right ) x}{\sqrt [3]{7+2 i \sqrt {170}}}+625 x^2\right )^2} \, dx,x,\frac {6}{5}+x\right )\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.14, size = 27, normalized size = 0.96 \begin {gather*} \frac {2 x \log (x)}{(-3+x)^2 \left (8+81 x+90 x^2+25 x^3\right )} \end {gather*}

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.72, size = 157, normalized size = 5.61


method	result	size

norman	\(\frac {2 x \ln \left (x \right )}{\left (25 x^{3}+90 x^{2}+81 x +8\right ) \left (x -3\right )^{2}}\)	\(28\)
risch	\(\frac {2 x \ln \left (x \right )}{25 x^{5}-60 x^{4}-234 x^{3}+332 x^{2}+681 x +72}\)	\(33\)
default	\(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (25 \textit {\_Z}^{3}+90 \textit {\_Z}^{2}+81 \textit {\_Z} +8\right )}{\sum }\frac {\left (4050 \textit {\_R}^{2}+21305 \textit {\_R} +41232\right ) \ln \left (x -\textit {\_R} \right )}{25 \textit {\_R}^{2}+60 \textit {\_R} +27}\right )}{565068}-\frac {269 \ln \left (x \right ) x}{565068 \left (x -3\right )}-\frac {\ln \left (x \right ) x \left (x -6\right )}{2604 \left (x -3\right )^{2}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (25 \textit {\_Z}^{3}+90 \textit {\_Z}^{2}+81 \textit {\_Z} +8\right )}{\sum }\frac {\left (-4050 \textit {\_R}^{2}-21305 \textit {\_R} -41232\right ) \ln \left (x -\textit {\_R} \right )}{25 \textit {\_R}^{2}+60 \textit {\_R} +27}\right )}{565068}+\frac {\ln \left (x \right ) x \left (4050 x^{2}+21305 x +41232\right )}{4708900 x^{3}+16952040 x^{2}+15256836 x +1506848}\)	\(157\)

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Maxima [A]
time = 0.27, size = 32, normalized size = 1.14 \begin {gather*} \frac {2 \, x \log \left (x\right )}{25 \, x^{5} - 60 \, x^{4} - 234 \, x^{3} + 332 \, x^{2} + 681 \, x + 72} \end {gather*}

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Fricas [A]
time = 0.37, size = 32, normalized size = 1.14 \begin {gather*} \frac {2 \, x \log \left (x\right )}{25 \, x^{5} - 60 \, x^{4} - 234 \, x^{3} + 332 \, x^{2} + 681 \, x + 72} \end {gather*}

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Sympy [A]
time = 0.12, size = 31, normalized size = 1.11 \begin {gather*} \frac {2 x \log {\left (x \right )}}{25 x^{5} - 60 x^{4} - 234 x^{3} + 332 x^{2} + 681 x + 72} \end {gather*}

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (23) = 46\).
time = 0.42, size = 50, normalized size = 1.79 \begin {gather*} \frac {1}{188356} \, {\left (\frac {6725 \, x^{2} + 28110 \, x - 1296}{25 \, x^{3} + 90 \, x^{2} + 81 \, x + 8} - \frac {269 \, x - 1458}{x^{2} - 6 \, x + 9}\right )} \log \left (x\right ) \end {gather*}

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Mupad [B]
time = 9.21, size = 32, normalized size = 1.14 \begin {gather*} \frac {2\,x\,\ln \left (x\right )}{25\,\left (x^5-\frac {12\,x^4}{5}-\frac {234\,x^3}{25}+\frac {332\,x^2}{25}+\frac {681\,x}{25}+\frac {72}{25}\right )} \end {gather*}

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3.94.75 \(\int \frac {-48-470 x-378 x^2+30 x^3+50 x^4+(-48-16 x+216 x^2-240 x^3-200 x^4) \log (x)}{-1728-33264 x-181611 x^2-200033 x^3+5697 x^4+79731 x^5+10255 x^6-11475 x^7-1125 x^8+625 x^9} \, dx\) [9375]