Integrand size = 125, antiderivative size = 28 \[ \int \frac {1}{4} \left (2500 x^3-1250 x^4+150 x^5+\left (-500 x^3+125 x^4\right ) \log (4)+25 x^3 \log ^2(4)+\left (15000 x^2-8000 x^3+1000 x^4+\left (-3000 x^2+800 x^3\right ) \log (4)+150 x^2 \log ^2(4)\right ) \log ^2(5)+\left (20000 x-12000 x^2+1600 x^3+\left (-4000 x+1200 x^2\right ) \log (4)+200 x \log ^2(4)\right ) \log ^4(5)\right ) \, dx=\frac {25}{16} x^2 (10-2 x-\log (4))^2 \left (x+4 \log ^2(5)\right )^2 \]
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Leaf count is larger than twice the leaf count of optimal. \(158\) vs. \(2(28)=56\).
Time = 0.05 (sec) , antiderivative size = 158, normalized size of antiderivative = 5.64, number of steps used = 10, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {6, 12} \[ \int \frac {1}{4} \left (2500 x^3-1250 x^4+150 x^5+\left (-500 x^3+125 x^4\right ) \log (4)+25 x^3 \log ^2(4)+\left (15000 x^2-8000 x^3+1000 x^4+\left (-3000 x^2+800 x^3\right ) \log (4)+150 x^2 \log ^2(4)\right ) \log ^2(5)+\left (20000 x-12000 x^2+1600 x^3+\left (-4000 x+1200 x^2\right ) \log (4)+200 x \log ^2(4)\right ) \log ^4(5)\right ) \, dx=\frac {25 x^6}{4}-\frac {125 x^5}{2}+50 x^5 \log ^2(5)+\frac {25}{4} x^5 \log (4)+100 x^4 \log ^4(5)+50 x^4 \log (4) \log ^2(5)-500 x^4 \log ^2(5)+\frac {25}{16} x^4 \left (100+\log ^2(4)\right )-\frac {125}{4} x^4 \log (4)+100 x^3 \log (4) \log ^4(5)-1000 x^3 \log ^4(5)+\frac {25}{2} x^3 \left (100+\log ^2(4)\right ) \log ^2(5)-250 x^3 \log (4) \log ^2(5)-500 x^2 \log (4) \log ^4(5)+25 x^2 \left (100+\log ^2(4)\right ) \log ^4(5) \]
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Rule 6
Rule 12
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{4} \left (-1250 x^4+150 x^5+\left (-500 x^3+125 x^4\right ) \log (4)+x^3 \left (2500+25 \log ^2(4)\right )+\left (15000 x^2-8000 x^3+1000 x^4+\left (-3000 x^2+800 x^3\right ) \log (4)+150 x^2 \log ^2(4)\right ) \log ^2(5)+\left (20000 x-12000 x^2+1600 x^3+\left (-4000 x+1200 x^2\right ) \log (4)+200 x \log ^2(4)\right ) \log ^4(5)\right ) \, dx \\ & = \frac {1}{4} \int \left (-1250 x^4+150 x^5+\left (-500 x^3+125 x^4\right ) \log (4)+x^3 \left (2500+25 \log ^2(4)\right )+\left (15000 x^2-8000 x^3+1000 x^4+\left (-3000 x^2+800 x^3\right ) \log (4)+150 x^2 \log ^2(4)\right ) \log ^2(5)+\left (20000 x-12000 x^2+1600 x^3+\left (-4000 x+1200 x^2\right ) \log (4)+200 x \log ^2(4)\right ) \log ^4(5)\right ) \, dx \\ & = -\frac {125 x^5}{2}+\frac {25 x^6}{4}+\frac {25}{16} x^4 \left (100+\log ^2(4)\right )+\frac {1}{4} \log (4) \int \left (-500 x^3+125 x^4\right ) \, dx+\frac {1}{4} \log ^2(5) \int \left (15000 x^2-8000 x^3+1000 x^4+\left (-3000 x^2+800 x^3\right ) \log (4)+150 x^2 \log ^2(4)\right ) \, dx+\frac {1}{4} \log ^4(5) \int \left (20000 x-12000 x^2+1600 x^3+\left (-4000 x+1200 x^2\right ) \log (4)+200 x \log ^2(4)\right ) \, dx \\ & = -\frac {125 x^5}{2}+\frac {25 x^6}{4}-\frac {125}{4} x^4 \log (4)+\frac {25}{4} x^5 \log (4)+\frac {25}{16} x^4 \left (100+\log ^2(4)\right )+\frac {1}{4} \log ^2(5) \int \left (-8000 x^3+1000 x^4+\left (-3000 x^2+800 x^3\right ) \log (4)+x^2 \left (15000+150 \log ^2(4)\right )\right ) \, dx+\frac {1}{4} \log ^4(5) \int \left (-12000 x^2+1600 x^3+\left (-4000 x+1200 x^2\right ) \log (4)+x \left (20000+200 \log ^2(4)\right )\right ) \, dx \\ & = -\frac {125 x^5}{2}+\frac {25 x^6}{4}-\frac {125}{4} x^4 \log (4)+\frac {25}{4} x^5 \log (4)+\frac {25}{16} x^4 \left (100+\log ^2(4)\right )-500 x^4 \log ^2(5)+50 x^5 \log ^2(5)+\frac {25}{2} x^3 \left (100+\log ^2(4)\right ) \log ^2(5)-1000 x^3 \log ^4(5)+100 x^4 \log ^4(5)+25 x^2 \left (100+\log ^2(4)\right ) \log ^4(5)+\frac {1}{4} \left (\log (4) \log ^2(5)\right ) \int \left (-3000 x^2+800 x^3\right ) \, dx+\frac {1}{4} \left (\log (4) \log ^4(5)\right ) \int \left (-4000 x+1200 x^2\right ) \, dx \\ & = -\frac {125 x^5}{2}+\frac {25 x^6}{4}-\frac {125}{4} x^4 \log (4)+\frac {25}{4} x^5 \log (4)+\frac {25}{16} x^4 \left (100+\log ^2(4)\right )-500 x^4 \log ^2(5)+50 x^5 \log ^2(5)-250 x^3 \log (4) \log ^2(5)+50 x^4 \log (4) \log ^2(5)+\frac {25}{2} x^3 \left (100+\log ^2(4)\right ) \log ^2(5)-1000 x^3 \log ^4(5)+100 x^4 \log ^4(5)-500 x^2 \log (4) \log ^4(5)+100 x^3 \log (4) \log ^4(5)+25 x^2 \left (100+\log ^2(4)\right ) \log ^4(5) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {1}{4} \left (2500 x^3-1250 x^4+150 x^5+\left (-500 x^3+125 x^4\right ) \log (4)+25 x^3 \log ^2(4)+\left (15000 x^2-8000 x^3+1000 x^4+\left (-3000 x^2+800 x^3\right ) \log (4)+150 x^2 \log ^2(4)\right ) \log ^2(5)+\left (20000 x-12000 x^2+1600 x^3+\left (-4000 x+1200 x^2\right ) \log (4)+200 x \log ^2(4)\right ) \log ^4(5)\right ) \, dx=\frac {25}{16} x^2 (-10+2 x+\log (4))^2 \left (x+4 \log ^2(5)\right )^2 \]
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Leaf count of result is larger than twice the leaf count of optimal. \(92\) vs. \(2(26)=52\).
Time = 0.30 (sec) , antiderivative size = 93, normalized size of antiderivative = 3.32
method | result | size |
gosper | \(\frac {25 \left (4 \ln \left (5\right )^{2}+x \right ) \left (4 \ln \left (2\right )^{2} \ln \left (5\right )^{2}+8 x \ln \left (2\right ) \ln \left (5\right )^{2}+4 x^{2} \ln \left (5\right )^{2}-40 \ln \left (5\right )^{2} \ln \left (2\right )-40 x \ln \left (5\right )^{2}+x \ln \left (2\right )^{2}+2 x^{2} \ln \left (2\right )+x^{3}+100 \ln \left (5\right )^{2}-10 x \ln \left (2\right )-10 x^{2}+25 x \right ) x^{2}}{4}\) | \(93\) |
norman | \(\left (50 \ln \left (5\right )^{2}+\frac {25 \ln \left (2\right )}{2}-\frac {125}{2}\right ) x^{5}+\left (100 \ln \left (5\right )^{4} \ln \left (2\right )^{2}-1000 \ln \left (5\right )^{4} \ln \left (2\right )+2500 \ln \left (5\right )^{4}\right ) x^{2}+\left (200 \ln \left (5\right )^{4} \ln \left (2\right )-1000 \ln \left (5\right )^{4}+50 \ln \left (2\right )^{2} \ln \left (5\right )^{2}-500 \ln \left (5\right )^{2} \ln \left (2\right )+1250 \ln \left (5\right )^{2}\right ) x^{3}+\left (100 \ln \left (5\right )^{4}+100 \ln \left (5\right )^{2} \ln \left (2\right )-500 \ln \left (5\right )^{2}+\frac {25 \ln \left (2\right )^{2}}{4}-\frac {125 \ln \left (2\right )}{2}+\frac {625}{4}\right ) x^{4}+\frac {25 x^{6}}{4}\) | \(131\) |
default | \(\frac {25 x^{6}}{4}+\frac {5 \left (20 \ln \left (5\right )^{2}+5 \ln \left (2\right )-25\right ) x^{5}}{2}+\frac {25 \left (12 \ln \left (5\right )^{2} \left (\ln \left (2\right )-5\right )+\left (4 \ln \left (5\right )^{2}+\ln \left (2\right )-5\right ) \left (8 \ln \left (5\right )^{2}+2 \ln \left (2\right )-10\right )+4 \ln \left (5\right )^{2} \ln \left (2\right )-20 \ln \left (5\right )^{2}\right ) x^{4}}{8}+\frac {25 \left (4 \ln \left (5\right )^{2} \left (\ln \left (2\right )-5\right ) \left (8 \ln \left (5\right )^{2}+2 \ln \left (2\right )-10\right )+\left (4 \ln \left (5\right )^{2}+\ln \left (2\right )-5\right ) \left (4 \ln \left (5\right )^{2} \ln \left (2\right )-20 \ln \left (5\right )^{2}\right )\right ) x^{3}}{6}+25 \ln \left (5\right )^{2} \left (\ln \left (2\right )-5\right ) \left (4 \ln \left (5\right )^{2} \ln \left (2\right )-20 \ln \left (5\right )^{2}\right ) x^{2}\) | \(159\) |
risch | \(100 x^{4} \ln \left (5\right )^{4}+200 \ln \left (5\right )^{4} x^{3} \ln \left (2\right )-1000 x^{3} \ln \left (5\right )^{4}+100 \ln \left (5\right )^{4} \ln \left (2\right )^{2} x^{2}-1000 \ln \left (5\right )^{4} \ln \left (2\right ) x^{2}+2500 \ln \left (5\right )^{4} x^{2}+50 x^{5} \ln \left (5\right )^{2}+100 \ln \left (5\right )^{2} x^{4} \ln \left (2\right )-500 x^{4} \ln \left (5\right )^{2}+50 \ln \left (5\right )^{2} \ln \left (2\right )^{2} x^{3}-500 \ln \left (5\right )^{2} \ln \left (2\right ) x^{3}+1250 x^{3} \ln \left (5\right )^{2}+\frac {25 x^{4} \ln \left (2\right )^{2}}{4}+\frac {25 x^{5} \ln \left (2\right )}{2}-\frac {125 x^{4} \ln \left (2\right )}{2}+\frac {25 x^{6}}{4}-\frac {125 x^{5}}{2}+\frac {625 x^{4}}{4}\) | \(164\) |
parallelrisch | \(100 x^{4} \ln \left (5\right )^{4}+200 \ln \left (5\right )^{4} x^{3} \ln \left (2\right )-1000 x^{3} \ln \left (5\right )^{4}+100 \ln \left (5\right )^{4} \ln \left (2\right )^{2} x^{2}-1000 \ln \left (5\right )^{4} \ln \left (2\right ) x^{2}+2500 \ln \left (5\right )^{4} x^{2}+50 x^{5} \ln \left (5\right )^{2}+100 \ln \left (5\right )^{2} x^{4} \ln \left (2\right )-500 x^{4} \ln \left (5\right )^{2}+50 \ln \left (5\right )^{2} \ln \left (2\right )^{2} x^{3}-500 \ln \left (5\right )^{2} \ln \left (2\right ) x^{3}+1250 x^{3} \ln \left (5\right )^{2}+\frac {25 x^{4} \ln \left (2\right )^{2}}{4}+\frac {25 x^{5} \ln \left (2\right )}{2}-\frac {125 x^{4} \ln \left (2\right )}{2}+\frac {25 x^{6}}{4}-\frac {125 x^{5}}{2}+\frac {625 x^{4}}{4}\) | \(164\) |
parts | \(100 x^{4} \ln \left (5\right )^{4}+200 \ln \left (5\right )^{4} x^{3} \ln \left (2\right )-1000 x^{3} \ln \left (5\right )^{4}+100 \ln \left (5\right )^{4} \ln \left (2\right )^{2} x^{2}-1000 \ln \left (5\right )^{4} \ln \left (2\right ) x^{2}+2500 \ln \left (5\right )^{4} x^{2}+50 x^{5} \ln \left (5\right )^{2}+100 \ln \left (5\right )^{2} x^{4} \ln \left (2\right )-500 x^{4} \ln \left (5\right )^{2}+50 \ln \left (5\right )^{2} \ln \left (2\right )^{2} x^{3}-500 \ln \left (5\right )^{2} \ln \left (2\right ) x^{3}+1250 x^{3} \ln \left (5\right )^{2}+\frac {25 x^{4} \ln \left (2\right )^{2}}{4}+\frac {25 x^{5} \ln \left (2\right )}{2}-\frac {125 x^{4} \ln \left (2\right )}{2}+\frac {25 x^{6}}{4}-\frac {125 x^{5}}{2}+\frac {625 x^{4}}{4}\) | \(164\) |
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Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (22) = 44\).
Time = 0.26 (sec) , antiderivative size = 120, normalized size of antiderivative = 4.29 \[ \int \frac {1}{4} \left (2500 x^3-1250 x^4+150 x^5+\left (-500 x^3+125 x^4\right ) \log (4)+25 x^3 \log ^2(4)+\left (15000 x^2-8000 x^3+1000 x^4+\left (-3000 x^2+800 x^3\right ) \log (4)+150 x^2 \log ^2(4)\right ) \log ^2(5)+\left (20000 x-12000 x^2+1600 x^3+\left (-4000 x+1200 x^2\right ) \log (4)+200 x \log ^2(4)\right ) \log ^4(5)\right ) \, dx=\frac {25}{4} \, x^{6} + \frac {25}{4} \, x^{4} \log \left (2\right )^{2} - \frac {125}{2} \, x^{5} + 100 \, {\left (x^{4} + x^{2} \log \left (2\right )^{2} - 10 \, x^{3} + 25 \, x^{2} + 2 \, {\left (x^{3} - 5 \, x^{2}\right )} \log \left (2\right )\right )} \log \left (5\right )^{4} + \frac {625}{4} \, x^{4} + 50 \, {\left (x^{5} + x^{3} \log \left (2\right )^{2} - 10 \, x^{4} + 25 \, x^{3} + 2 \, {\left (x^{4} - 5 \, x^{3}\right )} \log \left (2\right )\right )} \log \left (5\right )^{2} + \frac {25}{2} \, {\left (x^{5} - 5 \, x^{4}\right )} \log \left (2\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (27) = 54\).
Time = 0.04 (sec) , antiderivative size = 153, normalized size of antiderivative = 5.46 \[ \int \frac {1}{4} \left (2500 x^3-1250 x^4+150 x^5+\left (-500 x^3+125 x^4\right ) \log (4)+25 x^3 \log ^2(4)+\left (15000 x^2-8000 x^3+1000 x^4+\left (-3000 x^2+800 x^3\right ) \log (4)+150 x^2 \log ^2(4)\right ) \log ^2(5)+\left (20000 x-12000 x^2+1600 x^3+\left (-4000 x+1200 x^2\right ) \log (4)+200 x \log ^2(4)\right ) \log ^4(5)\right ) \, dx=\frac {25 x^{6}}{4} + x^{5} \left (- \frac {125}{2} + \frac {25 \log {\left (2 \right )}}{2} + 50 \log {\left (5 \right )}^{2}\right ) + x^{4} \left (- 500 \log {\left (5 \right )}^{2} - \frac {125 \log {\left (2 \right )}}{2} + \frac {25 \log {\left (2 \right )}^{2}}{4} + \frac {625}{4} + 100 \log {\left (2 \right )} \log {\left (5 \right )}^{2} + 100 \log {\left (5 \right )}^{4}\right ) + x^{3} \left (- 1000 \log {\left (5 \right )}^{4} - 500 \log {\left (2 \right )} \log {\left (5 \right )}^{2} + 50 \log {\left (2 \right )}^{2} \log {\left (5 \right )}^{2} + 200 \log {\left (2 \right )} \log {\left (5 \right )}^{4} + 1250 \log {\left (5 \right )}^{2}\right ) + x^{2} \left (- 1000 \log {\left (2 \right )} \log {\left (5 \right )}^{4} + 100 \log {\left (2 \right )}^{2} \log {\left (5 \right )}^{4} + 2500 \log {\left (5 \right )}^{4}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (22) = 44\).
Time = 0.20 (sec) , antiderivative size = 120, normalized size of antiderivative = 4.29 \[ \int \frac {1}{4} \left (2500 x^3-1250 x^4+150 x^5+\left (-500 x^3+125 x^4\right ) \log (4)+25 x^3 \log ^2(4)+\left (15000 x^2-8000 x^3+1000 x^4+\left (-3000 x^2+800 x^3\right ) \log (4)+150 x^2 \log ^2(4)\right ) \log ^2(5)+\left (20000 x-12000 x^2+1600 x^3+\left (-4000 x+1200 x^2\right ) \log (4)+200 x \log ^2(4)\right ) \log ^4(5)\right ) \, dx=\frac {25}{4} \, x^{6} + \frac {25}{4} \, x^{4} \log \left (2\right )^{2} - \frac {125}{2} \, x^{5} + 100 \, {\left (x^{4} + x^{2} \log \left (2\right )^{2} - 10 \, x^{3} + 25 \, x^{2} + 2 \, {\left (x^{3} - 5 \, x^{2}\right )} \log \left (2\right )\right )} \log \left (5\right )^{4} + \frac {625}{4} \, x^{4} + 50 \, {\left (x^{5} + x^{3} \log \left (2\right )^{2} - 10 \, x^{4} + 25 \, x^{3} + 2 \, {\left (x^{4} - 5 \, x^{3}\right )} \log \left (2\right )\right )} \log \left (5\right )^{2} + \frac {25}{2} \, {\left (x^{5} - 5 \, x^{4}\right )} \log \left (2\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (22) = 44\).
Time = 0.26 (sec) , antiderivative size = 120, normalized size of antiderivative = 4.29 \[ \int \frac {1}{4} \left (2500 x^3-1250 x^4+150 x^5+\left (-500 x^3+125 x^4\right ) \log (4)+25 x^3 \log ^2(4)+\left (15000 x^2-8000 x^3+1000 x^4+\left (-3000 x^2+800 x^3\right ) \log (4)+150 x^2 \log ^2(4)\right ) \log ^2(5)+\left (20000 x-12000 x^2+1600 x^3+\left (-4000 x+1200 x^2\right ) \log (4)+200 x \log ^2(4)\right ) \log ^4(5)\right ) \, dx=\frac {25}{4} \, x^{6} + \frac {25}{4} \, x^{4} \log \left (2\right )^{2} - \frac {125}{2} \, x^{5} + 100 \, {\left (x^{4} + x^{2} \log \left (2\right )^{2} - 10 \, x^{3} + 25 \, x^{2} + 2 \, {\left (x^{3} - 5 \, x^{2}\right )} \log \left (2\right )\right )} \log \left (5\right )^{4} + \frac {625}{4} \, x^{4} + 50 \, {\left (x^{5} + x^{3} \log \left (2\right )^{2} - 10 \, x^{4} + 25 \, x^{3} + 2 \, {\left (x^{4} - 5 \, x^{3}\right )} \log \left (2\right )\right )} \log \left (5\right )^{2} + \frac {25}{2} \, {\left (x^{5} - 5 \, x^{4}\right )} \log \left (2\right ) \]
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Time = 8.25 (sec) , antiderivative size = 96, normalized size of antiderivative = 3.43 \[ \int \frac {1}{4} \left (2500 x^3-1250 x^4+150 x^5+\left (-500 x^3+125 x^4\right ) \log (4)+25 x^3 \log ^2(4)+\left (15000 x^2-8000 x^3+1000 x^4+\left (-3000 x^2+800 x^3\right ) \log (4)+150 x^2 \log ^2(4)\right ) \log ^2(5)+\left (20000 x-12000 x^2+1600 x^3+\left (-4000 x+1200 x^2\right ) \log (4)+200 x \log ^2(4)\right ) \log ^4(5)\right ) \, dx=\frac {25\,x^6}{4}+\left (\frac {25\,\ln \left (2\right )}{2}+50\,{\ln \left (5\right )}^2-\frac {125}{2}\right )\,x^5+\left (100\,\ln \left (2\right )\,{\ln \left (5\right )}^2-\frac {125\,\ln \left (2\right )}{2}+\frac {25\,{\ln \left (2\right )}^2}{4}-500\,{\ln \left (5\right )}^2+100\,{\ln \left (5\right )}^4+\frac {625}{4}\right )\,x^4+50\,{\ln \left (5\right )}^2\,\left (\ln \left (2\right )-5\right )\,\left (\ln \left (2\right )+4\,{\ln \left (5\right )}^2-5\right )\,x^3+100\,{\ln \left (5\right )}^4\,{\left (\ln \left (2\right )-5\right )}^2\,x^2 \]
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