Integrand size = 24, antiderivative size = 22 \[ \int \frac {-24+2 x^2+10 x^4}{-4 x+5 x^3} \, dx=-4+x^2+\log \left (\frac {25 x^4}{\left (-\frac {4}{5 x}+x\right )^2}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1607, 1265, 907} \[ \int \frac {-24+2 x^2+10 x^4}{-4 x+5 x^3} \, dx=x^2-2 \log \left (4-5 x^2\right )+6 \log (x) \]
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Rule 907
Rule 1265
Rule 1607
Rubi steps \begin{align*} \text {integral}& = \int \frac {-24+2 x^2+10 x^4}{x \left (-4+5 x^2\right )} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {-24+2 x+10 x^2}{x (-4+5 x)} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (2+\frac {6}{x}-\frac {20}{-4+5 x}\right ) \, dx,x,x^2\right ) \\ & = x^2+6 \log (x)-2 \log \left (4-5 x^2\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {-24+2 x^2+10 x^4}{-4 x+5 x^3} \, dx=2 \left (\frac {x^2}{2}+3 \log (x)-\log \left (4-5 x^2\right )\right ) \]
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Time = 1.83 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77
method | result | size |
parallelrisch | \(x^{2}+6 \ln \left (x \right )-2 \ln \left (x^{2}-\frac {4}{5}\right )\) | \(17\) |
default | \(x^{2}-2 \ln \left (5 x^{2}-4\right )+6 \ln \left (x \right )\) | \(19\) |
norman | \(x^{2}-2 \ln \left (5 x^{2}-4\right )+6 \ln \left (x \right )\) | \(19\) |
risch | \(x^{2}-2 \ln \left (5 x^{2}-4\right )+6 \ln \left (x \right )\) | \(19\) |
meijerg | \(6 \ln \left (x \right )+3 \ln \left (5\right )-6 \ln \left (2\right )+3 i \pi -2 \ln \left (1-\frac {5 x^{2}}{4}\right )+x^{2}\) | \(31\) |
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Time = 0.24 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {-24+2 x^2+10 x^4}{-4 x+5 x^3} \, dx=x^{2} - 2 \, \log \left (5 \, x^{2} - 4\right ) + 6 \, \log \left (x\right ) \]
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Time = 0.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {-24+2 x^2+10 x^4}{-4 x+5 x^3} \, dx=x^{2} + 6 \log {\left (x \right )} - 2 \log {\left (5 x^{2} - 4 \right )} \]
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Time = 0.20 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {-24+2 x^2+10 x^4}{-4 x+5 x^3} \, dx=x^{2} - 2 \, \log \left (5 \, x^{2} - 4\right ) + 6 \, \log \left (x\right ) \]
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Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {-24+2 x^2+10 x^4}{-4 x+5 x^3} \, dx=x^{2} + 3 \, \log \left (x^{2}\right ) - 2 \, \log \left ({\left | 5 \, x^{2} - 4 \right |}\right ) \]
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Time = 0.09 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.73 \[ \int \frac {-24+2 x^2+10 x^4}{-4 x+5 x^3} \, dx=6\,\ln \left (x\right )-2\,\ln \left (x^2-\frac {4}{5}\right )+x^2 \]
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