Integrand size = 20, antiderivative size = 15 \[ \int \frac {2 x+x^2}{4+3 x^2+x^3} \, dx=\frac {1}{3} \log \left (4+3 x^2+x^3\right ) \]
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Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1601} \[ \int \frac {2 x+x^2}{4+3 x^2+x^3} \, dx=\frac {1}{3} \log \left (x^3+3 x^2+4\right ) \]
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Rule 1601
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \log \left (4+3 x^2+x^3\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {2 x+x^2}{4+3 x^2+x^3} \, dx=\frac {1}{3} \log \left (4+3 x^2+x^3\right ) \]
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Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93
method | result | size |
default | \(\frac {\ln \left (x^{3}+3 x^{2}+4\right )}{3}\) | \(14\) |
norman | \(\frac {\ln \left (x^{3}+3 x^{2}+4\right )}{3}\) | \(14\) |
risch | \(\frac {\ln \left (x^{3}+3 x^{2}+4\right )}{3}\) | \(14\) |
parallelrisch | \(\frac {\ln \left (x^{3}+3 x^{2}+4\right )}{3}\) | \(14\) |
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none
Time = 0.23 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {2 x+x^2}{4+3 x^2+x^3} \, dx=\frac {1}{3} \, \log \left (x^{3} + 3 \, x^{2} + 4\right ) \]
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Time = 0.04 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int \frac {2 x+x^2}{4+3 x^2+x^3} \, dx=\frac {\log {\left (x^{3} + 3 x^{2} + 4 \right )}}{3} \]
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none
Time = 0.18 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {2 x+x^2}{4+3 x^2+x^3} \, dx=\frac {1}{3} \, \log \left (x^{3} + 3 \, x^{2} + 4\right ) \]
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none
Time = 0.29 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {2 x+x^2}{4+3 x^2+x^3} \, dx=\frac {1}{3} \, \log \left ({\left | x^{3} + 3 \, x^{2} + 4 \right |}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {2 x+x^2}{4+3 x^2+x^3} \, dx=\frac {\ln \left (x^3+3\,x^2+4\right )}{3} \]
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