Integrand size = 25, antiderivative size = 17 \[ \int \frac {10+4 x^2+5 x^3}{5 x+x^3+x^4} \, dx=\log \left (x \left (5+x^2+x^3\right ) (x+x \log (4))\right ) \]
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Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {1608, 6874, 1601} \[ \int \frac {10+4 x^2+5 x^3}{5 x+x^3+x^4} \, dx=\log \left (x^3+x^2+5\right )+2 \log (x) \]
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Rule 1601
Rule 1608
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {10+4 x^2+5 x^3}{x \left (5+x^2+x^3\right )} \, dx \\ & = \int \left (\frac {2}{x}+\frac {x (2+3 x)}{5+x^2+x^3}\right ) \, dx \\ & = 2 \log (x)+\int \frac {x (2+3 x)}{5+x^2+x^3} \, dx \\ & = 2 \log (x)+\log \left (5+x^2+x^3\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \frac {10+4 x^2+5 x^3}{5 x+x^3+x^4} \, dx=2 \log (x)+\log \left (5+x^2+x^3\right ) \]
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Time = 0.05 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88
method | result | size |
default | \(2 \ln \left (x \right )+\ln \left (x^{3}+x^{2}+5\right )\) | \(15\) |
norman | \(2 \ln \left (x \right )+\ln \left (x^{3}+x^{2}+5\right )\) | \(15\) |
risch | \(2 \ln \left (x \right )+\ln \left (x^{3}+x^{2}+5\right )\) | \(15\) |
parallelrisch | \(2 \ln \left (x \right )+\ln \left (x^{3}+x^{2}+5\right )\) | \(15\) |
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none
Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \frac {10+4 x^2+5 x^3}{5 x+x^3+x^4} \, dx=\log \left (x^{3} + x^{2} + 5\right ) + 2 \, \log \left (x\right ) \]
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Time = 0.06 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \frac {10+4 x^2+5 x^3}{5 x+x^3+x^4} \, dx=2 \log {\left (x \right )} + \log {\left (x^{3} + x^{2} + 5 \right )} \]
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none
Time = 0.19 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \frac {10+4 x^2+5 x^3}{5 x+x^3+x^4} \, dx=\log \left (x^{3} + x^{2} + 5\right ) + 2 \, \log \left (x\right ) \]
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none
Time = 0.29 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int \frac {10+4 x^2+5 x^3}{5 x+x^3+x^4} \, dx=\log \left ({\left | x^{3} + x^{2} + 5 \right |}\right ) + 2 \, \log \left ({\left | x \right |}\right ) \]
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Time = 12.30 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \frac {10+4 x^2+5 x^3}{5 x+x^3+x^4} \, dx=\ln \left (x^3+x^2+5\right )+2\,\ln \left (x\right ) \]
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