Integrand size = 18, antiderivative size = 12 \[ \int \frac {2+2 x+x^4}{x^4+x^5} \, dx=-\frac {2}{3 x^3}+\log (1+x) \]
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Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1607, 1634} \[ \int \frac {2+2 x+x^4}{x^4+x^5} \, dx=\log (x+1)-\frac {2}{3 x^3} \]
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Rule 1607
Rule 1634
Rubi steps \begin{align*} \text {integral}& = \int \frac {2+2 x+x^4}{x^4 (1+x)} \, dx \\ & = \int \left (\frac {2}{x^4}+\frac {1}{1+x}\right ) \, dx \\ & = -\frac {2}{3 x^3}+\log (1+x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {2+2 x+x^4}{x^4+x^5} \, dx=-\frac {2}{3 x^3}+\log (1+x) \]
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Time = 0.79 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.92
method | result | size |
default | \(-\frac {2}{3 x^{3}}+\ln \left (x +1\right )\) | \(11\) |
norman | \(-\frac {2}{3 x^{3}}+\ln \left (x +1\right )\) | \(11\) |
meijerg | \(-\frac {2}{3 x^{3}}+\ln \left (x +1\right )\) | \(11\) |
risch | \(-\frac {2}{3 x^{3}}+\ln \left (x +1\right )\) | \(11\) |
parallelrisch | \(\frac {3 \ln \left (x +1\right ) x^{3}-2}{3 x^{3}}\) | \(17\) |
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none
Time = 0.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.33 \[ \int \frac {2+2 x+x^4}{x^4+x^5} \, dx=\frac {3 \, x^{3} \log \left (x + 1\right ) - 2}{3 \, x^{3}} \]
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Time = 0.04 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {2+2 x+x^4}{x^4+x^5} \, dx=\log {\left (x + 1 \right )} - \frac {2}{3 x^{3}} \]
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none
Time = 0.18 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {2+2 x+x^4}{x^4+x^5} \, dx=-\frac {2}{3 \, x^{3}} + \log \left (x + 1\right ) \]
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none
Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.92 \[ \int \frac {2+2 x+x^4}{x^4+x^5} \, dx=-\frac {2}{3 \, x^{3}} + \log \left ({\left | x + 1 \right |}\right ) \]
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Time = 9.46 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {2+2 x+x^4}{x^4+x^5} \, dx=\ln \left (x+1\right )-\frac {2}{3\,x^3} \]
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