Integrand size = 67, antiderivative size = 29 \[ \int \frac {-900+2370 x-1836 x^2+541 x^3-165 x^4+97 x^5+4 x^7}{900 x-1440 x^2+516 x^3-72 x^4+97 x^5+4 x^6+4 x^7} \, dx=x-\frac {1}{x-\frac {6 (5-4 x)}{x (1+2 x)}}-\log (x) \]
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Leaf count is larger than twice the leaf count of optimal. \(1356\) vs. \(2(29)=58\).
Time = 4.22 (sec) , antiderivative size = 1356, normalized size of antiderivative = 46.76, number of steps used = 20, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.149, Rules used = {2099, 2126, 2106, 2104, 836, 814, 648, 632, 210, 642} \[ \int \frac {-900+2370 x-1836 x^2+541 x^3-165 x^4+97 x^5+4 x^7}{900 x-1440 x^2+516 x^3-72 x^4+97 x^5+4 x^6+4 x^7} \, dx =\text {Too large to display} \]
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 814
Rule 836
Rule 2099
Rule 2104
Rule 2106
Rule 2126
Rubi steps \begin{align*} \text {integral}& = \int \left (1-\frac {1}{x}-\frac {12 \left (-5-13 x+8 x^2\right )}{\left (-30+24 x+x^2+2 x^3\right )^2}+\frac {1+2 x}{-30+24 x+x^2+2 x^3}\right ) \, dx \\ & = x-\log (x)-12 \int \frac {-5-13 x+8 x^2}{\left (-30+24 x+x^2+2 x^3\right )^2} \, dx+\int \frac {1+2 x}{-30+24 x+x^2+2 x^3} \, dx \\ & = x-\frac {16}{30-24 x-x^2-2 x^3}-\log (x)-2 \int \frac {-222-94 x}{\left (-30+24 x+x^2+2 x^3\right )^2} \, dx+\text {Subst}\left (\int \frac {\frac {2}{3}+2 x}{-\frac {1835}{54}+\frac {143 x}{6}+2 x^3} \, dx,x,\frac {1}{6}+x\right ) \\ & = x-\frac {16}{30-24 x-x^2-2 x^3}-\log (x)-2 \text {Subst}\left (\int \frac {-\frac {619}{3}-94 x}{\left (-\frac {1835}{54}+\frac {143 x}{6}+2 x^3\right )^2} \, dx,x,\frac {1}{6}+x\right )+4 \text {Subst}\left (\int \frac {\frac {2}{3}+2 x}{\left (\frac {143-\left (1835+18 \sqrt {19418}\right )^{2/3}}{3 \sqrt [3]{1835+18 \sqrt {19418}}}+2 x\right ) \left (\frac {1}{9} \left (143+\frac {20449}{\left (1835+18 \sqrt {19418}\right )^{2/3}}+\left (1835+18 \sqrt {19418}\right )^{2/3}\right )-\frac {2 \left (143-\left (1835+18 \sqrt {19418}\right )^{2/3}\right ) x}{3 \sqrt [3]{1835+18 \sqrt {19418}}}+4 x^2\right )} \, dx,x,\frac {1}{6}+x\right ) \\ & = x-\frac {16}{30-24 x-x^2-2 x^3}-\log (x)+4 \text {Subst}\left (\int \left (\frac {3 \left (1835+18 \sqrt {19418}\right )^{2/3} \left (-143+2 \sqrt [3]{1835+18 \sqrt {19418}}+\left (1835+18 \sqrt {19418}\right )^{2/3}\right )}{\left (20449-143 \left (1835+18 \sqrt {19418}\right )^{2/3}+\left (1835+18 \sqrt {19418}\right )^{4/3}\right ) \left (143-\left (1835+18 \sqrt {19418}\right )^{2/3}+6 \sqrt [3]{1835+18 \sqrt {19418}} x\right )}+\frac {3 \left (1835+18 \sqrt {19418}\right )^{2/3} \left (13109-72 \sqrt {19418}+143 \left (1835+18 \sqrt {19418}\right )^{2/3}+\sqrt [3]{1835+18 \sqrt {19418}} \left (2407+18 \sqrt {19418}\right )-6 \left (1835+18 \sqrt {19418}-143 \sqrt [3]{1835+18 \sqrt {19418}}+2 \left (1835+18 \sqrt {19418}\right )^{2/3}\right ) x\right )}{\left (20449-143 \left (1835+18 \sqrt {19418}\right )^{2/3}+\left (1835+18 \sqrt {19418}\right )^{4/3}\right ) \left (20449+143 \left (1835+18 \sqrt {19418}\right )^{2/3}+\left (1835+18 \sqrt {19418}\right )^{4/3}+6 \left (1835+18 \sqrt {19418}-143 \sqrt [3]{1835+18 \sqrt {19418}}\right ) x+36 \left (1835+18 \sqrt {19418}\right )^{2/3} x^2\right )}\right ) \, dx,x,\frac {1}{6}+x\right )-32 \text {Subst}\left (\int \frac {-\frac {619}{3}-94 x}{\left (\frac {143-\left (1835+18 \sqrt {19418}\right )^{2/3}}{3 \sqrt [3]{1835+18 \sqrt {19418}}}+2 x\right )^2 \left (\frac {1}{9} \left (143+\frac {20449}{\left (1835+18 \sqrt {19418}\right )^{2/3}}+\left (1835+18 \sqrt {19418}\right )^{2/3}\right )-\frac {2 \left (143-\left (1835+18 \sqrt {19418}\right )^{2/3}\right ) x}{3 \sqrt [3]{1835+18 \sqrt {19418}}}+4 x^2\right )^2} \, dx,x,\frac {1}{6}+x\right ) \\ & = x-\frac {16}{30-24 x-x^2-2 x^3}+\frac {108 \sqrt [3]{2 \left (349524+1835 \sqrt {19418}\right )} \left (19418-\frac {\left (9709+619 \sqrt {19418}-\sqrt [3]{1835+18 \sqrt {19418}} \left (9709+47 \sqrt {19418}\right )\right ) (1+6 x)}{\left (1835+18 \sqrt {19418}\right )^{2/3}}\right )}{9709^{2/3} \left (143+\left (1835+18 \sqrt {19418}\right )^{2/3}\right ) \left (1+\frac {143}{\sqrt [3]{1835+18 \sqrt {19418}}}-\sqrt [3]{1835+18 \sqrt {19418}}+6 x\right ) \left (143+\frac {20449}{\left (1835+18 \sqrt {19418}\right )^{2/3}}+\left (1835+18 \sqrt {19418}\right )^{2/3}-\frac {\left (143-\left (1835+18 \sqrt {19418}\right )^{2/3}\right ) (1+6 x)}{\sqrt [3]{1835+18 \sqrt {19418}}}+(1+6 x)^2\right )}-\log (x)-\frac {2 \sqrt [3]{1835+18 \sqrt {19418}} \left (11-\sqrt [3]{1835+18 \sqrt {19418}}\right ) \left (13+\sqrt [3]{1835+18 \sqrt {19418}}\right ) \log \left (143-\left (1835+18 \sqrt {19418}\right )^{2/3}+\sqrt [3]{1835+18 \sqrt {19418}} (1+6 x)\right )}{20449-143 \left (1835+18 \sqrt {19418}\right )^{2/3}+\left (1835+18 \sqrt {19418}\right )^{4/3}}+\frac {\left (12 \left (1835+18 \sqrt {19418}\right )^{2/3}\right ) \text {Subst}\left (\int \frac {13109-72 \sqrt {19418}+143 \left (1835+18 \sqrt {19418}\right )^{2/3}+\sqrt [3]{1835+18 \sqrt {19418}} \left (2407+18 \sqrt {19418}\right )-6 \left (1835+18 \sqrt {19418}-143 \sqrt [3]{1835+18 \sqrt {19418}}+2 \left (1835+18 \sqrt {19418}\right )^{2/3}\right ) x}{20449+143 \left (1835+18 \sqrt {19418}\right )^{2/3}+\left (1835+18 \sqrt {19418}\right )^{4/3}+6 \left (1835+18 \sqrt {19418}-143 \sqrt [3]{1835+18 \sqrt {19418}}\right ) x+36 \left (1835+18 \sqrt {19418}\right )^{2/3} x^2} \, dx,x,\frac {1}{6}+x\right )}{20449-143 \left (1835+18 \sqrt {19418}\right )^{2/3}+\left (1835+18 \sqrt {19418}\right )^{4/3}}+\frac {32 \text {Subst}\left (\int \frac {-64 \left (9709+\frac {143 \left (9709+47 \sqrt {19418}\right )}{\left (1835+18 \sqrt {19418}\right )^{2/3}}+\frac {9709+619 \sqrt {19418}}{\sqrt [3]{1835+18 \sqrt {19418}}}\right )+\frac {384 \left (9709+619 \sqrt {19418}-\sqrt [3]{1835+18 \sqrt {19418}} \left (9709+47 \sqrt {19418}\right )\right ) x}{\left (1835+18 \sqrt {19418}\right )^{2/3}}}{\left (\frac {143-\left (1835+18 \sqrt {19418}\right )^{2/3}}{3 \sqrt [3]{1835+18 \sqrt {19418}}}+2 x\right )^2 \left (\frac {1}{9} \left (143+\frac {20449}{\left (1835+18 \sqrt {19418}\right )^{2/3}}+\left (1835+18 \sqrt {19418}\right )^{2/3}\right )-\frac {2 \left (143-\left (1835+18 \sqrt {19418}\right )^{2/3}\right ) x}{3 \sqrt [3]{1835+18 \sqrt {19418}}}+4 x^2\right )} \, dx,x,\frac {1}{6}+x\right )}{\left (\frac {4 \left (143-\left (1835+18 \sqrt {19418}\right )^{2/3}\right )^2}{9 \left (1835+18 \sqrt {19418}\right )^{2/3}}-\frac {16}{9} \left (143+\frac {20449}{\left (1835+18 \sqrt {19418}\right )^{2/3}}+\left (1835+18 \sqrt {19418}\right )^{2/3}\right )\right ) \left (\frac {8 \left (143-\left (1835+18 \sqrt {19418}\right )^{2/3}\right )^2}{9 \left (1835+18 \sqrt {19418}\right )^{2/3}}+\frac {4}{9} \left (143+\frac {20449}{\left (1835+18 \sqrt {19418}\right )^{2/3}}+\left (1835+18 \sqrt {19418}\right )^{2/3}\right )\right )} \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {-900+2370 x-1836 x^2+541 x^3-165 x^4+97 x^5+4 x^7}{900 x-1440 x^2+516 x^3-72 x^4+97 x^5+4 x^6+4 x^7} \, dx=x+\frac {-x-2 x^2}{-30+24 x+x^2+2 x^3}-\log (x) \]
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Time = 0.11 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10
method | result | size |
default | \(x -\ln \left (x \right )+\frac {-x^{2}-\frac {1}{2} x}{x^{3}+\frac {1}{2} x^{2}+12 x -15}\) | \(32\) |
risch | \(x -\ln \left (x \right )+\frac {-x^{2}-\frac {1}{2} x}{x^{3}+\frac {1}{2} x^{2}+12 x -15}\) | \(32\) |
norman | \(\frac {-43 x +\frac {43}{2} x^{2}+2 x^{4}+15}{2 x^{3}+x^{2}+24 x -30}-\ln \left (x \right )\) | \(37\) |
parallelrisch | \(-\frac {4 x^{3} \ln \left (x \right )-4 x^{4}-30+2 x^{2} \ln \left (x \right )+48 x \ln \left (x \right )-43 x^{2}-60 \ln \left (x \right )+86 x}{2 \left (2 x^{3}+x^{2}+24 x -30\right )}\) | \(56\) |
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Time = 0.24 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.72 \[ \int \frac {-900+2370 x-1836 x^2+541 x^3-165 x^4+97 x^5+4 x^7}{900 x-1440 x^2+516 x^3-72 x^4+97 x^5+4 x^6+4 x^7} \, dx=\frac {2 \, x^{4} + x^{3} + 22 \, x^{2} - {\left (2 \, x^{3} + x^{2} + 24 \, x - 30\right )} \log \left (x\right ) - 31 \, x}{2 \, x^{3} + x^{2} + 24 \, x - 30} \]
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Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {-900+2370 x-1836 x^2+541 x^3-165 x^4+97 x^5+4 x^7}{900 x-1440 x^2+516 x^3-72 x^4+97 x^5+4 x^6+4 x^7} \, dx=x + \frac {- 2 x^{2} - x}{2 x^{3} + x^{2} + 24 x - 30} - \log {\left (x \right )} \]
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Time = 0.20 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03 \[ \int \frac {-900+2370 x-1836 x^2+541 x^3-165 x^4+97 x^5+4 x^7}{900 x-1440 x^2+516 x^3-72 x^4+97 x^5+4 x^6+4 x^7} \, dx=x - \frac {2 \, x^{2} + x}{2 \, x^{3} + x^{2} + 24 \, x - 30} - \log \left (x\right ) \]
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Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {-900+2370 x-1836 x^2+541 x^3-165 x^4+97 x^5+4 x^7}{900 x-1440 x^2+516 x^3-72 x^4+97 x^5+4 x^6+4 x^7} \, dx=x - \frac {2 \, x^{2} + x}{2 \, x^{3} + x^{2} + 24 \, x - 30} - \log \left ({\left | x \right |}\right ) \]
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Time = 0.08 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03 \[ \int \frac {-900+2370 x-1836 x^2+541 x^3-165 x^4+97 x^5+4 x^7}{900 x-1440 x^2+516 x^3-72 x^4+97 x^5+4 x^6+4 x^7} \, dx=x-\ln \left (x\right )-\frac {x^2+\frac {x}{2}}{x^3+\frac {x^2}{2}+12\,x-15} \]
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