Integrand size = 16, antiderivative size = 32 \[ \int \frac {x^5}{-4+x^2+3 x^4} \, dx=\frac {x^2}{6}+\frac {1}{14} \log \left (1-x^2\right )-\frac {8}{63} \log \left (4+3 x^2\right ) \]
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Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1128, 717, 646, 31} \[ \int \frac {x^5}{-4+x^2+3 x^4} \, dx=\frac {x^2}{6}+\frac {1}{14} \log \left (1-x^2\right )-\frac {8}{63} \log \left (3 x^2+4\right ) \]
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Rule 31
Rule 646
Rule 717
Rule 1128
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x^2}{-4+x+3 x^2} \, dx,x,x^2\right ) \\ & = \frac {x^2}{6}+\frac {1}{6} \text {Subst}\left (\int \frac {4-x}{-4+x+3 x^2} \, dx,x,x^2\right ) \\ & = \frac {x^2}{6}+\frac {3}{14} \text {Subst}\left (\int \frac {1}{-3+3 x} \, dx,x,x^2\right )-\frac {8}{21} \text {Subst}\left (\int \frac {1}{4+3 x} \, dx,x,x^2\right ) \\ & = \frac {x^2}{6}+\frac {1}{14} \log \left (1-x^2\right )-\frac {8}{63} \log \left (4+3 x^2\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {x^5}{-4+x^2+3 x^4} \, dx=\frac {x^2}{6}+\frac {1}{14} \log \left (1-x^2\right )-\frac {8}{63} \log \left (4+3 x^2\right ) \]
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Time = 0.05 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78
method | result | size |
default | \(\frac {x^{2}}{6}+\frac {\ln \left (x^{2}-1\right )}{14}-\frac {8 \ln \left (3 x^{2}+4\right )}{63}\) | \(25\) |
risch | \(\frac {x^{2}}{6}+\frac {\ln \left (x^{2}-1\right )}{14}-\frac {8 \ln \left (3 x^{2}+4\right )}{63}\) | \(25\) |
parallelrisch | \(\frac {x^{2}}{6}+\frac {\ln \left (-1+x \right )}{14}+\frac {\ln \left (1+x \right )}{14}-\frac {8 \ln \left (x^{2}+\frac {4}{3}\right )}{63}\) | \(27\) |
norman | \(\frac {x^{2}}{6}+\frac {\ln \left (-1+x \right )}{14}+\frac {\ln \left (1+x \right )}{14}-\frac {8 \ln \left (3 x^{2}+4\right )}{63}\) | \(29\) |
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Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75 \[ \int \frac {x^5}{-4+x^2+3 x^4} \, dx=\frac {1}{6} \, x^{2} - \frac {8}{63} \, \log \left (3 \, x^{2} + 4\right ) + \frac {1}{14} \, \log \left (x^{2} - 1\right ) \]
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Time = 0.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75 \[ \int \frac {x^5}{-4+x^2+3 x^4} \, dx=\frac {x^{2}}{6} + \frac {\log {\left (x^{2} - 1 \right )}}{14} - \frac {8 \log {\left (x^{2} + \frac {4}{3} \right )}}{63} \]
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Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75 \[ \int \frac {x^5}{-4+x^2+3 x^4} \, dx=\frac {1}{6} \, x^{2} - \frac {8}{63} \, \log \left (3 \, x^{2} + 4\right ) + \frac {1}{14} \, \log \left (x^{2} - 1\right ) \]
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Time = 0.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78 \[ \int \frac {x^5}{-4+x^2+3 x^4} \, dx=\frac {1}{6} \, x^{2} - \frac {8}{63} \, \log \left (3 \, x^{2} + 4\right ) + \frac {1}{14} \, \log \left ({\left | x^{2} - 1 \right |}\right ) \]
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Time = 0.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.69 \[ \int \frac {x^5}{-4+x^2+3 x^4} \, dx=\frac {\ln \left (x^2-1\right )}{14}-\frac {8\,\ln \left (x^2+\frac {4}{3}\right )}{63}+\frac {x^2}{6} \]
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