Integrand size = 84, antiderivative size = 24 \[ \int \frac {1}{25} \left (25+200 x-60 x^2+4 x^3+500 x^4-60 x^5+200 x^7+\left (-200 x+20 x^2-100 x^4\right ) \log (x)+\left (-200 x+30 x^2-250 x^4\right ) \log ^2(x)+100 x \log ^3(x)+50 x \log ^4(x)\right ) \, dx=x+\left (2 x+x^4-\frac {1}{5} x \left (x+5 \log ^2(x)\right )\right )^2 \]
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Leaf count is larger than twice the leaf count of optimal. \(73\) vs. \(2(24)=48\).
Time = 0.13 (sec) , antiderivative size = 73, normalized size of antiderivative = 3.04, number of steps used = 24, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.060, Rules used = {12, 1608, 2403, 2341, 2342} \[ \int \frac {1}{25} \left (25+200 x-60 x^2+4 x^3+500 x^4-60 x^5+200 x^7+\left (-200 x+20 x^2-100 x^4\right ) \log (x)+\left (-200 x+30 x^2-250 x^4\right ) \log ^2(x)+100 x \log ^3(x)+50 x \log ^4(x)\right ) \, dx=x^8-\frac {2 x^6}{5}+4 x^5-2 x^5 \log ^2(x)+\frac {x^4}{25}-\frac {4 x^3}{5}+\frac {2}{5} x^3 \log ^2(x)+4 x^2+x^2 \log ^4(x)-4 x^2 \log ^2(x)+x \]
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Rule 12
Rule 1608
Rule 2341
Rule 2342
Rule 2403
Rubi steps \begin{align*} \text {integral}& = \frac {1}{25} \int \left (25+200 x-60 x^2+4 x^3+500 x^4-60 x^5+200 x^7+\left (-200 x+20 x^2-100 x^4\right ) \log (x)+\left (-200 x+30 x^2-250 x^4\right ) \log ^2(x)+100 x \log ^3(x)+50 x \log ^4(x)\right ) \, dx \\ & = x+4 x^2-\frac {4 x^3}{5}+\frac {x^4}{25}+4 x^5-\frac {2 x^6}{5}+x^8+\frac {1}{25} \int \left (-200 x+20 x^2-100 x^4\right ) \log (x) \, dx+\frac {1}{25} \int \left (-200 x+30 x^2-250 x^4\right ) \log ^2(x) \, dx+2 \int x \log ^4(x) \, dx+4 \int x \log ^3(x) \, dx \\ & = x+4 x^2-\frac {4 x^3}{5}+\frac {x^4}{25}+4 x^5-\frac {2 x^6}{5}+x^8+2 x^2 \log ^3(x)+x^2 \log ^4(x)+\frac {1}{25} \int x \left (-200+20 x-100 x^3\right ) \log (x) \, dx+\frac {1}{25} \int x \left (-200+30 x-250 x^3\right ) \log ^2(x) \, dx-4 \int x \log ^3(x) \, dx-6 \int x \log ^2(x) \, dx \\ & = x+4 x^2-\frac {4 x^3}{5}+\frac {x^4}{25}+4 x^5-\frac {2 x^6}{5}+x^8-3 x^2 \log ^2(x)+x^2 \log ^4(x)+\frac {1}{25} \int \left (-200 x \log (x)+20 x^2 \log (x)-100 x^4 \log (x)\right ) \, dx+\frac {1}{25} \int \left (-200 x \log ^2(x)+30 x^2 \log ^2(x)-250 x^4 \log ^2(x)\right ) \, dx+6 \int x \log (x) \, dx+6 \int x \log ^2(x) \, dx \\ & = x+\frac {5 x^2}{2}-\frac {4 x^3}{5}+\frac {x^4}{25}+4 x^5-\frac {2 x^6}{5}+x^8+3 x^2 \log (x)+x^2 \log ^4(x)+\frac {4}{5} \int x^2 \log (x) \, dx+\frac {6}{5} \int x^2 \log ^2(x) \, dx-4 \int x^4 \log (x) \, dx-6 \int x \log (x) \, dx-8 \int x \log (x) \, dx-8 \int x \log ^2(x) \, dx-10 \int x^4 \log ^2(x) \, dx \\ & = x+6 x^2-\frac {8 x^3}{9}+\frac {x^4}{25}+\frac {104 x^5}{25}-\frac {2 x^6}{5}+x^8-4 x^2 \log (x)+\frac {4}{15} x^3 \log (x)-\frac {4}{5} x^5 \log (x)-4 x^2 \log ^2(x)+\frac {2}{5} x^3 \log ^2(x)-2 x^5 \log ^2(x)+x^2 \log ^4(x)-\frac {4}{5} \int x^2 \log (x) \, dx+4 \int x^4 \log (x) \, dx+8 \int x \log (x) \, dx \\ & = x+4 x^2-\frac {4 x^3}{5}+\frac {x^4}{25}+4 x^5-\frac {2 x^6}{5}+x^8-4 x^2 \log ^2(x)+\frac {2}{5} x^3 \log ^2(x)-2 x^5 \log ^2(x)+x^2 \log ^4(x) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(73\) vs. \(2(24)=48\).
Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 3.04 \[ \int \frac {1}{25} \left (25+200 x-60 x^2+4 x^3+500 x^4-60 x^5+200 x^7+\left (-200 x+20 x^2-100 x^4\right ) \log (x)+\left (-200 x+30 x^2-250 x^4\right ) \log ^2(x)+100 x \log ^3(x)+50 x \log ^4(x)\right ) \, dx=x+4 x^2-\frac {4 x^3}{5}+\frac {x^4}{25}+4 x^5-\frac {2 x^6}{5}+x^8-4 x^2 \log ^2(x)+\frac {2}{5} x^3 \log ^2(x)-2 x^5 \log ^2(x)+x^2 \log ^4(x) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(65\) vs. \(2(22)=44\).
Time = 0.06 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.75
method | result | size |
default | \(x -2 x^{5} \ln \left (x \right )^{2}+\frac {2 x^{3} \ln \left (x \right )^{2}}{5}-4 x^{2} \ln \left (x \right )^{2}+4 x^{2}-\frac {4 x^{3}}{5}+\frac {x^{4}}{25}+4 x^{5}-\frac {2 x^{6}}{5}+x^{8}+x^{2} \ln \left (x \right )^{4}\) | \(66\) |
risch | \(x -2 x^{5} \ln \left (x \right )^{2}+\frac {2 x^{3} \ln \left (x \right )^{2}}{5}-4 x^{2} \ln \left (x \right )^{2}+4 x^{2}-\frac {4 x^{3}}{5}+\frac {x^{4}}{25}+4 x^{5}-\frac {2 x^{6}}{5}+x^{8}+x^{2} \ln \left (x \right )^{4}\) | \(66\) |
parallelrisch | \(x -2 x^{5} \ln \left (x \right )^{2}+\frac {2 x^{3} \ln \left (x \right )^{2}}{5}-4 x^{2} \ln \left (x \right )^{2}+4 x^{2}-\frac {4 x^{3}}{5}+\frac {x^{4}}{25}+4 x^{5}-\frac {2 x^{6}}{5}+x^{8}+x^{2} \ln \left (x \right )^{4}\) | \(66\) |
parts | \(x -2 x^{5} \ln \left (x \right )^{2}+\frac {2 x^{3} \ln \left (x \right )^{2}}{5}-4 x^{2} \ln \left (x \right )^{2}+4 x^{2}-\frac {4 x^{3}}{5}+\frac {x^{4}}{25}+4 x^{5}-\frac {2 x^{6}}{5}+x^{8}+x^{2} \ln \left (x \right )^{4}\) | \(66\) |
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Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (26) = 52\).
Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.50 \[ \int \frac {1}{25} \left (25+200 x-60 x^2+4 x^3+500 x^4-60 x^5+200 x^7+\left (-200 x+20 x^2-100 x^4\right ) \log (x)+\left (-200 x+30 x^2-250 x^4\right ) \log ^2(x)+100 x \log ^3(x)+50 x \log ^4(x)\right ) \, dx=x^{8} - \frac {2}{5} \, x^{6} + x^{2} \log \left (x\right )^{4} + 4 \, x^{5} + \frac {1}{25} \, x^{4} - \frac {4}{5} \, x^{3} - \frac {2}{5} \, {\left (5 \, x^{5} - x^{3} + 10 \, x^{2}\right )} \log \left (x\right )^{2} + 4 \, x^{2} + x \]
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Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (20) = 40\).
Time = 0.08 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.62 \[ \int \frac {1}{25} \left (25+200 x-60 x^2+4 x^3+500 x^4-60 x^5+200 x^7+\left (-200 x+20 x^2-100 x^4\right ) \log (x)+\left (-200 x+30 x^2-250 x^4\right ) \log ^2(x)+100 x \log ^3(x)+50 x \log ^4(x)\right ) \, dx=x^{8} - \frac {2 x^{6}}{5} + 4 x^{5} + \frac {x^{4}}{25} - \frac {4 x^{3}}{5} + x^{2} \log {\left (x \right )}^{4} + 4 x^{2} + x + \left (- 2 x^{5} + \frac {2 x^{3}}{5} - 4 x^{2}\right ) \log {\left (x \right )}^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (26) = 52\).
Time = 0.21 (sec) , antiderivative size = 153, normalized size of antiderivative = 6.38 \[ \int \frac {1}{25} \left (25+200 x-60 x^2+4 x^3+500 x^4-60 x^5+200 x^7+\left (-200 x+20 x^2-100 x^4\right ) \log (x)+\left (-200 x+30 x^2-250 x^4\right ) \log ^2(x)+100 x \log ^3(x)+50 x \log ^4(x)\right ) \, dx=x^{8} - \frac {2}{25} \, {\left (25 \, \log \left (x\right )^{2} - 10 \, \log \left (x\right ) + 2\right )} x^{5} - \frac {2}{5} \, x^{6} + \frac {104}{25} \, x^{5} + \frac {2}{45} \, {\left (9 \, \log \left (x\right )^{2} - 6 \, \log \left (x\right ) + 2\right )} x^{3} + \frac {1}{25} \, x^{4} + \frac {1}{2} \, {\left (2 \, \log \left (x\right )^{4} - 4 \, \log \left (x\right )^{3} + 6 \, \log \left (x\right )^{2} - 6 \, \log \left (x\right ) + 3\right )} x^{2} + \frac {1}{2} \, {\left (4 \, \log \left (x\right )^{3} - 6 \, \log \left (x\right )^{2} + 6 \, \log \left (x\right ) - 3\right )} x^{2} - 2 \, {\left (2 \, \log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 1\right )} x^{2} - \frac {8}{9} \, x^{3} + 6 \, x^{2} - \frac {4}{15} \, {\left (3 \, x^{5} - x^{3} + 15 \, x^{2}\right )} \log \left (x\right ) + x \]
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Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (26) = 52\).
Time = 0.27 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.71 \[ \int \frac {1}{25} \left (25+200 x-60 x^2+4 x^3+500 x^4-60 x^5+200 x^7+\left (-200 x+20 x^2-100 x^4\right ) \log (x)+\left (-200 x+30 x^2-250 x^4\right ) \log ^2(x)+100 x \log ^3(x)+50 x \log ^4(x)\right ) \, dx=x^{8} - 2 \, x^{5} \log \left (x\right )^{2} - \frac {2}{5} \, x^{6} + x^{2} \log \left (x\right )^{4} + 4 \, x^{5} + \frac {2}{5} \, x^{3} \log \left (x\right )^{2} + \frac {1}{25} \, x^{4} - 4 \, x^{2} \log \left (x\right )^{2} - \frac {4}{5} \, x^{3} + 4 \, x^{2} + x \]
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Time = 13.61 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.71 \[ \int \frac {1}{25} \left (25+200 x-60 x^2+4 x^3+500 x^4-60 x^5+200 x^7+\left (-200 x+20 x^2-100 x^4\right ) \log (x)+\left (-200 x+30 x^2-250 x^4\right ) \log ^2(x)+100 x \log ^3(x)+50 x \log ^4(x)\right ) \, dx=x^8-\frac {2\,x^6}{5}-2\,x^5\,{\ln \left (x\right )}^2+4\,x^5+\frac {x^4}{25}+\frac {2\,x^3\,{\ln \left (x\right )}^2}{5}-\frac {4\,x^3}{5}+x^2\,{\ln \left (x\right )}^4-4\,x^2\,{\ln \left (x\right )}^2+4\,x^2+x \]
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