Integrand size = 31, antiderivative size = 54 \[ \int \frac {x^5 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^2} \, dx=-\frac {27 x^2}{2}+\frac {5 x^4}{4}+\frac {102+103 x^2}{2 \left (2+3 x^2+x^4\right )}+3 \log \left (1+x^2\right )+46 \log \left (2+x^2\right ) \]
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Time = 0.07 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {1677, 1674, 1671, 646, 31} \[ \int \frac {x^5 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^2} \, dx=\frac {5 x^4}{4}-\frac {27 x^2}{2}+3 \log \left (x^2+1\right )+46 \log \left (x^2+2\right )+\frac {103 x^2+102}{2 \left (x^4+3 x^2+2\right )} \]
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Rule 31
Rule 646
Rule 1671
Rule 1674
Rule 1677
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x^2 \left (4+x+3 x^2+5 x^3\right )}{\left (2+3 x+x^2\right )^2} \, dx,x,x^2\right ) \\ & = \frac {102+103 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac {1}{2} \text {Subst}\left (\int \frac {-50-27 x+12 x^2-5 x^3}{2+3 x+x^2} \, dx,x,x^2\right ) \\ & = \frac {102+103 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac {1}{2} \text {Subst}\left (\int \left (27-5 x-\frac {2 (52+49 x)}{2+3 x+x^2}\right ) \, dx,x,x^2\right ) \\ & = -\frac {27 x^2}{2}+\frac {5 x^4}{4}+\frac {102+103 x^2}{2 \left (2+3 x^2+x^4\right )}+\text {Subst}\left (\int \frac {52+49 x}{2+3 x+x^2} \, dx,x,x^2\right ) \\ & = -\frac {27 x^2}{2}+\frac {5 x^4}{4}+\frac {102+103 x^2}{2 \left (2+3 x^2+x^4\right )}+3 \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,x^2\right )+46 \text {Subst}\left (\int \frac {1}{2+x} \, dx,x,x^2\right ) \\ & = -\frac {27 x^2}{2}+\frac {5 x^4}{4}+\frac {102+103 x^2}{2 \left (2+3 x^2+x^4\right )}+3 \log \left (1+x^2\right )+46 \log \left (2+x^2\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00 \[ \int \frac {x^5 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^2} \, dx=-\frac {27 x^2}{2}+\frac {5 x^4}{4}+\frac {102+103 x^2}{2 \left (2+3 x^2+x^4\right )}+3 \log \left (1+x^2\right )+46 \log \left (2+x^2\right ) \]
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Time = 0.09 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.85
method | result | size |
default | \(46 \ln \left (x^{2}+2\right )+\frac {52}{x^{2}+2}+\frac {5 x^{4}}{4}-\frac {27 x^{2}}{2}+3 \ln \left (x^{2}+1\right )-\frac {1}{2 \left (x^{2}+1\right )}\) | \(46\) |
norman | \(\frac {\frac {277}{2} x^{2}-\frac {39}{4} x^{6}+\frac {5}{4} x^{8}+127}{x^{4}+3 x^{2}+2}+3 \ln \left (x^{2}+1\right )+46 \ln \left (x^{2}+2\right )\) | \(48\) |
risch | \(\frac {5 x^{4}}{4}-\frac {27 x^{2}}{2}+\frac {729}{20}+\frac {\frac {103 x^{2}}{2}+51}{x^{4}+3 x^{2}+2}+3 \ln \left (x^{2}+1\right )+46 \ln \left (x^{2}+2\right )\) | \(49\) |
parallelrisch | \(\frac {5 x^{8}-39 x^{6}+12 \ln \left (x^{2}+1\right ) x^{4}+184 \ln \left (x^{2}+2\right ) x^{4}+508+36 \ln \left (x^{2}+1\right ) x^{2}+552 \ln \left (x^{2}+2\right ) x^{2}+554 x^{2}+24 \ln \left (x^{2}+1\right )+368 \ln \left (x^{2}+2\right )}{4 x^{4}+12 x^{2}+8}\) | \(92\) |
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Time = 0.26 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.33 \[ \int \frac {x^5 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^2} \, dx=\frac {5 \, x^{8} - 39 \, x^{6} - 152 \, x^{4} + 98 \, x^{2} + 184 \, {\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 2\right ) + 12 \, {\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 1\right ) + 204}{4 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} \]
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Time = 0.07 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.89 \[ \int \frac {x^5 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^2} \, dx=\frac {5 x^{4}}{4} - \frac {27 x^{2}}{2} + \frac {103 x^{2} + 102}{2 x^{4} + 6 x^{2} + 4} + 3 \log {\left (x^{2} + 1 \right )} + 46 \log {\left (x^{2} + 2 \right )} \]
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Time = 0.19 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.89 \[ \int \frac {x^5 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^2} \, dx=\frac {5}{4} \, x^{4} - \frac {27}{2} \, x^{2} + \frac {103 \, x^{2} + 102}{2 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} + 46 \, \log \left (x^{2} + 2\right ) + 3 \, \log \left (x^{2} + 1\right ) \]
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Time = 0.38 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.98 \[ \int \frac {x^5 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^2} \, dx=\frac {5}{4} \, x^{4} - \frac {27}{2} \, x^{2} - \frac {49 \, x^{4} + 44 \, x^{2} - 4}{2 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} + 46 \, \log \left (x^{2} + 2\right ) + 3 \, \log \left (x^{2} + 1\right ) \]
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Time = 0.02 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.87 \[ \int \frac {x^5 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^2} \, dx=3\,\ln \left (x^2+1\right )+46\,\ln \left (x^2+2\right )+\frac {\frac {103\,x^2}{2}+51}{x^4+3\,x^2+2}-\frac {27\,x^2}{2}+\frac {5\,x^4}{4} \]
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