Integrand size = 63, antiderivative size = 18 \[ \int \frac {-1030725+20952000 x-144180000 x^2+350400000 x^3-100000000 x^4+\left (-202500+4050000 x-27000000 x^2+60000000 x^3\right ) \log (x)}{14641 x^5+6050 x^5 \log (x)+625 x^5 \log ^2(x)} \, dx=\frac {\left (-20+\frac {3}{x}\right )^4}{\frac {121}{25}+\log (x)} \]
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Leaf count is larger than twice the leaf count of optimal. \(63\) vs. \(2(18)=36\).
Time = 0.57 (sec) , antiderivative size = 63, normalized size of antiderivative = 3.50, number of steps used = 30, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6820, 12, 6874, 2395, 2343, 2346, 2209, 2339, 30} \[ \int \frac {-1030725+20952000 x-144180000 x^2+350400000 x^3-100000000 x^4+\left (-202500+4050000 x-27000000 x^2+60000000 x^3\right ) \log (x)}{14641 x^5+6050 x^5 \log (x)+625 x^5 \log ^2(x)} \, dx=\frac {2025}{x^4 (25 \log (x)+121)}-\frac {54000}{x^3 (25 \log (x)+121)}+\frac {540000}{x^2 (25 \log (x)+121)}-\frac {2400000}{x (25 \log (x)+121)}+\frac {4000000}{25 \log (x)+121} \]
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Rule 12
Rule 30
Rule 2209
Rule 2339
Rule 2343
Rule 2346
Rule 2395
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {25 (3-20 x)^3 (-1527+500 x-300 \log (x))}{x^5 (121+25 \log (x))^2} \, dx \\ & = 25 \int \frac {(3-20 x)^3 (-1527+500 x-300 \log (x))}{x^5 (121+25 \log (x))^2} \, dx \\ & = 25 \int \left (-\frac {25 (-3+20 x)^4}{x^5 (121+25 \log (x))^2}+\frac {12 (-3+20 x)^3}{x^5 (121+25 \log (x))}\right ) \, dx \\ & = 300 \int \frac {(-3+20 x)^3}{x^5 (121+25 \log (x))} \, dx-625 \int \frac {(-3+20 x)^4}{x^5 (121+25 \log (x))^2} \, dx \\ & = 300 \int \left (-\frac {27}{x^5 (121+25 \log (x))}+\frac {540}{x^4 (121+25 \log (x))}-\frac {3600}{x^3 (121+25 \log (x))}+\frac {8000}{x^2 (121+25 \log (x))}\right ) \, dx-625 \int \left (\frac {81}{x^5 (121+25 \log (x))^2}-\frac {2160}{x^4 (121+25 \log (x))^2}+\frac {21600}{x^3 (121+25 \log (x))^2}-\frac {96000}{x^2 (121+25 \log (x))^2}+\frac {160000}{x (121+25 \log (x))^2}\right ) \, dx \\ & = -\left (8100 \int \frac {1}{x^5 (121+25 \log (x))} \, dx\right )-50625 \int \frac {1}{x^5 (121+25 \log (x))^2} \, dx+162000 \int \frac {1}{x^4 (121+25 \log (x))} \, dx-1080000 \int \frac {1}{x^3 (121+25 \log (x))} \, dx+1350000 \int \frac {1}{x^4 (121+25 \log (x))^2} \, dx+2400000 \int \frac {1}{x^2 (121+25 \log (x))} \, dx-13500000 \int \frac {1}{x^3 (121+25 \log (x))^2} \, dx+60000000 \int \frac {1}{x^2 (121+25 \log (x))^2} \, dx-100000000 \int \frac {1}{x (121+25 \log (x))^2} \, dx \\ & = \frac {2025}{x^4 (121+25 \log (x))}-\frac {54000}{x^3 (121+25 \log (x))}+\frac {540000}{x^2 (121+25 \log (x))}-\frac {2400000}{x (121+25 \log (x))}+8100 \int \frac {1}{x^5 (121+25 \log (x))} \, dx-8100 \text {Subst}\left (\int \frac {e^{-4 x}}{121+25 x} \, dx,x,\log (x)\right )-162000 \int \frac {1}{x^4 (121+25 \log (x))} \, dx+162000 \text {Subst}\left (\int \frac {e^{-3 x}}{121+25 x} \, dx,x,\log (x)\right )+1080000 \int \frac {1}{x^3 (121+25 \log (x))} \, dx-1080000 \text {Subst}\left (\int \frac {e^{-2 x}}{121+25 x} \, dx,x,\log (x)\right )-2400000 \int \frac {1}{x^2 (121+25 \log (x))} \, dx+2400000 \text {Subst}\left (\int \frac {e^{-x}}{121+25 x} \, dx,x,\log (x)\right )-4000000 \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,121+25 \log (x)\right ) \\ & = 96000 e^{121/25} \operatorname {ExpIntegralEi}\left (\frac {1}{25} (-121-25 \log (x))\right )-324 e^{484/25} \operatorname {ExpIntegralEi}\left (-\frac {4}{25} (121+25 \log (x))\right )+6480 e^{363/25} \operatorname {ExpIntegralEi}\left (-\frac {3}{25} (121+25 \log (x))\right )-43200 e^{242/25} \operatorname {ExpIntegralEi}\left (-\frac {2}{25} (121+25 \log (x))\right )+\frac {4000000}{121+25 \log (x)}+\frac {2025}{x^4 (121+25 \log (x))}-\frac {54000}{x^3 (121+25 \log (x))}+\frac {540000}{x^2 (121+25 \log (x))}-\frac {2400000}{x (121+25 \log (x))}+8100 \text {Subst}\left (\int \frac {e^{-4 x}}{121+25 x} \, dx,x,\log (x)\right )-162000 \text {Subst}\left (\int \frac {e^{-3 x}}{121+25 x} \, dx,x,\log (x)\right )+1080000 \text {Subst}\left (\int \frac {e^{-2 x}}{121+25 x} \, dx,x,\log (x)\right )-2400000 \text {Subst}\left (\int \frac {e^{-x}}{121+25 x} \, dx,x,\log (x)\right ) \\ & = \frac {4000000}{121+25 \log (x)}+\frac {2025}{x^4 (121+25 \log (x))}-\frac {54000}{x^3 (121+25 \log (x))}+\frac {540000}{x^2 (121+25 \log (x))}-\frac {2400000}{x (121+25 \log (x))} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {-1030725+20952000 x-144180000 x^2+350400000 x^3-100000000 x^4+\left (-202500+4050000 x-27000000 x^2+60000000 x^3\right ) \log (x)}{14641 x^5+6050 x^5 \log (x)+625 x^5 \log ^2(x)} \, dx=\frac {25 (-3+20 x)^4}{x^4 (121+25 \log (x))} \]
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Time = 1.12 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.83
method | result | size |
norman | \(\frac {4000000 x^{4}-2400000 x^{3}+540000 x^{2}-54000 x +2025}{x^{4} \left (25 \ln \left (x \right )+121\right )}\) | \(33\) |
risch | \(\frac {4000000 x^{4}-2400000 x^{3}+540000 x^{2}-54000 x +2025}{x^{4} \left (25 \ln \left (x \right )+121\right )}\) | \(34\) |
parallelrisch | \(\frac {100000000 x^{4}-60000000 x^{3}+13500000 x^{2}-1350000 x +50625}{25 x^{4} \left (25 \ln \left (x \right )+121\right )}\) | \(34\) |
default | \(\frac {4000000}{25 \ln \left (x \right )+121}+\frac {2025}{x^{4} \left (25 \ln \left (x \right )+121\right )}-\frac {54000}{x^{3} \left (25 \ln \left (x \right )+121\right )}+\frac {540000}{x^{2} \left (25 \ln \left (x \right )+121\right )}-\frac {2400000}{x \left (25 \ln \left (x \right )+121\right )}\) | \(64\) |
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Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 2.06 \[ \int \frac {-1030725+20952000 x-144180000 x^2+350400000 x^3-100000000 x^4+\left (-202500+4050000 x-27000000 x^2+60000000 x^3\right ) \log (x)}{14641 x^5+6050 x^5 \log (x)+625 x^5 \log ^2(x)} \, dx=\frac {25 \, {\left (160000 \, x^{4} - 96000 \, x^{3} + 21600 \, x^{2} - 2160 \, x + 81\right )}}{25 \, x^{4} \log \left (x\right ) + 121 \, x^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (12) = 24\).
Time = 0.06 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.78 \[ \int \frac {-1030725+20952000 x-144180000 x^2+350400000 x^3-100000000 x^4+\left (-202500+4050000 x-27000000 x^2+60000000 x^3\right ) \log (x)}{14641 x^5+6050 x^5 \log (x)+625 x^5 \log ^2(x)} \, dx=\frac {4000000 x^{4} - 2400000 x^{3} + 540000 x^{2} - 54000 x + 2025}{25 x^{4} \log {\left (x \right )} + 121 x^{4}} \]
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Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 2.06 \[ \int \frac {-1030725+20952000 x-144180000 x^2+350400000 x^3-100000000 x^4+\left (-202500+4050000 x-27000000 x^2+60000000 x^3\right ) \log (x)}{14641 x^5+6050 x^5 \log (x)+625 x^5 \log ^2(x)} \, dx=\frac {25 \, {\left (160000 \, x^{4} - 96000 \, x^{3} + 21600 \, x^{2} - 2160 \, x + 81\right )}}{25 \, x^{4} \log \left (x\right ) + 121 \, x^{4}} \]
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Time = 0.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 2.06 \[ \int \frac {-1030725+20952000 x-144180000 x^2+350400000 x^3-100000000 x^4+\left (-202500+4050000 x-27000000 x^2+60000000 x^3\right ) \log (x)}{14641 x^5+6050 x^5 \log (x)+625 x^5 \log ^2(x)} \, dx=\frac {25 \, {\left (160000 \, x^{4} - 96000 \, x^{3} + 21600 \, x^{2} - 2160 \, x + 81\right )}}{25 \, x^{4} \log \left (x\right ) + 121 \, x^{4}} \]
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Time = 12.56 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.78 \[ \int \frac {-1030725+20952000 x-144180000 x^2+350400000 x^3-100000000 x^4+\left (-202500+4050000 x-27000000 x^2+60000000 x^3\right ) \log (x)}{14641 x^5+6050 x^5 \log (x)+625 x^5 \log ^2(x)} \, dx=\frac {4000000\,x^4-2400000\,x^3+540000\,x^2-54000\,x+2025}{x^4\,\left (25\,\ln \left (x\right )+121\right )} \]
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