Integrand size = 52, antiderivative size = 26 \[ \int \frac {-100+\left (-20 x^2+50 x^3\right ) \log ^2(3)+\left (-10 x^5+54 x^6-30 x^7+4 x^8\right ) \log ^4(3)}{x^5 \log ^4(3)} \, dx=\log \left (3 e^{\left (-3+(-2+x)^2-x+\frac {5}{x^2 \log ^2(3)}\right )^2}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.69, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {12, 14} \[ \int \frac {-100+\left (-20 x^2+50 x^3\right ) \log ^2(3)+\left (-10 x^5+54 x^6-30 x^7+4 x^8\right ) \log ^4(3)}{x^5 \log ^4(3)} \, dx=x^4+\frac {25}{x^4 \log ^4(3)}-10 x^3+27 x^2+\frac {10}{x^2 \log ^2(3)}-10 x-\frac {50}{x \log ^2(3)} \]
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Rule 12
Rule 14
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {-100+\left (-20 x^2+50 x^3\right ) \log ^2(3)+\left (-10 x^5+54 x^6-30 x^7+4 x^8\right ) \log ^4(3)}{x^5} \, dx}{\log ^4(3)} \\ & = \frac {\int \left (-\frac {100}{x^5}-\frac {20 \log ^2(3)}{x^3}+\frac {50 \log ^2(3)}{x^2}-10 \log ^4(3)+54 x \log ^4(3)-30 x^2 \log ^4(3)+4 x^3 \log ^4(3)\right ) \, dx}{\log ^4(3)} \\ & = -10 x+27 x^2-10 x^3+x^4+\frac {25}{x^4 \log ^4(3)}+\frac {10}{x^2 \log ^2(3)}-\frac {50}{x \log ^2(3)} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.69 \[ \int \frac {-100+\left (-20 x^2+50 x^3\right ) \log ^2(3)+\left (-10 x^5+54 x^6-30 x^7+4 x^8\right ) \log ^4(3)}{x^5 \log ^4(3)} \, dx=-10 x+27 x^2-10 x^3+x^4+\frac {25}{x^4 \log ^4(3)}+\frac {10}{x^2 \log ^2(3)}-\frac {50}{x \log ^2(3)} \]
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Time = 0.12 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.54
method | result | size |
risch | \(\left (x^{2}-5 x +1\right )^{2}+\frac {-50 x^{3} \ln \left (3\right )^{2}+10 x^{2} \ln \left (3\right )^{2}+25}{\ln \left (3\right )^{4} x^{4}}\) | \(40\) |
default | \(\frac {x^{4} \ln \left (3\right )^{4}-10 \ln \left (3\right )^{4} x^{3}+27 x^{2} \ln \left (3\right )^{4}-10 x \ln \left (3\right )^{4}+\frac {25}{x^{4}}-\frac {50 \ln \left (3\right )^{2}}{x}+\frac {10 \ln \left (3\right )^{2}}{x^{2}}}{\ln \left (3\right )^{4}}\) | \(63\) |
gosper | \(\frac {x^{8} \ln \left (3\right )^{4}-10 \ln \left (3\right )^{4} x^{7}+27 x^{6} \ln \left (3\right )^{4}-10 x^{5} \ln \left (3\right )^{4}-50 x^{3} \ln \left (3\right )^{2}+10 x^{2} \ln \left (3\right )^{2}+25}{\ln \left (3\right )^{4} x^{4}}\) | \(64\) |
parallelrisch | \(\frac {x^{8} \ln \left (3\right )^{4}-10 \ln \left (3\right )^{4} x^{7}+27 x^{6} \ln \left (3\right )^{4}-10 x^{5} \ln \left (3\right )^{4}-50 x^{3} \ln \left (3\right )^{2}+10 x^{2} \ln \left (3\right )^{2}+25}{\ln \left (3\right )^{4} x^{4}}\) | \(64\) |
norman | \(\frac {\ln \left (3\right )^{3} x^{8}+\frac {25}{\ln \left (3\right )}+10 x^{2} \ln \left (3\right )-50 x^{3} \ln \left (3\right )-10 \ln \left (3\right )^{3} x^{5}+27 \ln \left (3\right )^{3} x^{6}-10 \ln \left (3\right )^{3} x^{7}}{x^{4} \ln \left (3\right )^{3}}\) | \(65\) |
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Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (25) = 50\).
Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.96 \[ \int \frac {-100+\left (-20 x^2+50 x^3\right ) \log ^2(3)+\left (-10 x^5+54 x^6-30 x^7+4 x^8\right ) \log ^4(3)}{x^5 \log ^4(3)} \, dx=\frac {{\left (x^{8} - 10 \, x^{7} + 27 \, x^{6} - 10 \, x^{5}\right )} \log \left (3\right )^{4} - 10 \, {\left (5 \, x^{3} - x^{2}\right )} \log \left (3\right )^{2} + 25}{x^{4} \log \left (3\right )^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (24) = 48\).
Time = 0.13 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.54 \[ \int \frac {-100+\left (-20 x^2+50 x^3\right ) \log ^2(3)+\left (-10 x^5+54 x^6-30 x^7+4 x^8\right ) \log ^4(3)}{x^5 \log ^4(3)} \, dx=\frac {x^{4} \log {\left (3 \right )}^{4} - 10 x^{3} \log {\left (3 \right )}^{4} + 27 x^{2} \log {\left (3 \right )}^{4} - 10 x \log {\left (3 \right )}^{4} + \frac {- 50 x^{3} \log {\left (3 \right )}^{2} + 10 x^{2} \log {\left (3 \right )}^{2} + 25}{x^{4}}}{\log {\left (3 \right )}^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (25) = 50\).
Time = 0.18 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.46 \[ \int \frac {-100+\left (-20 x^2+50 x^3\right ) \log ^2(3)+\left (-10 x^5+54 x^6-30 x^7+4 x^8\right ) \log ^4(3)}{x^5 \log ^4(3)} \, dx=\frac {x^{4} \log \left (3\right )^{4} - 10 \, x^{3} \log \left (3\right )^{4} + 27 \, x^{2} \log \left (3\right )^{4} - 10 \, x \log \left (3\right )^{4} - \frac {5 \, {\left (10 \, x^{3} \log \left (3\right )^{2} - 2 \, x^{2} \log \left (3\right )^{2} - 5\right )}}{x^{4}}}{\log \left (3\right )^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (25) = 50\).
Time = 0.27 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.46 \[ \int \frac {-100+\left (-20 x^2+50 x^3\right ) \log ^2(3)+\left (-10 x^5+54 x^6-30 x^7+4 x^8\right ) \log ^4(3)}{x^5 \log ^4(3)} \, dx=\frac {x^{4} \log \left (3\right )^{4} - 10 \, x^{3} \log \left (3\right )^{4} + 27 \, x^{2} \log \left (3\right )^{4} - 10 \, x \log \left (3\right )^{4} - \frac {5 \, {\left (10 \, x^{3} \log \left (3\right )^{2} - 2 \, x^{2} \log \left (3\right )^{2} - 5\right )}}{x^{4}}}{\log \left (3\right )^{4}} \]
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Time = 0.07 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.73 \[ \int \frac {-100+\left (-20 x^2+50 x^3\right ) \log ^2(3)+\left (-10 x^5+54 x^6-30 x^7+4 x^8\right ) \log ^4(3)}{x^5 \log ^4(3)} \, dx=27\,x^2-10\,x-10\,x^3+x^4+\frac {-50\,{\ln \left (3\right )}^2\,x^3+10\,{\ln \left (3\right )}^2\,x^2+25}{x^4\,{\ln \left (3\right )}^4} \]
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