Optimal. Leaf size=25 \[ \frac {1}{10} \log (1-2 x)+\frac {\log (x)}{2}-\frac {1}{10} \log (2+x) \]
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Rubi [A]
time = 0.03, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1608, 1642}
\begin {gather*} \frac {1}{10} \log (1-2 x)+\frac {\log (x)}{2}-\frac {1}{10} \log (x+2) \end {gather*}
Antiderivative was successfully verified.
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Rule 1608
Rule 1642
Rubi steps
\begin {align*} \int \frac {-1+2 x+x^2}{-2 x+3 x^2+2 x^3} \, dx &=\int \frac {-1+2 x+x^2}{x \left (-2+3 x+2 x^2\right )} \, dx\\ &=\int \left (\frac {1}{2 x}-\frac {1}{10 (2+x)}+\frac {1}{5 (-1+2 x)}\right ) \, dx\\ &=\frac {1}{10} \log (1-2 x)+\frac {\log (x)}{2}-\frac {1}{10} \log (2+x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 25, normalized size = 1.00 \begin {gather*} \frac {1}{10} \log (1-2 x)+\frac {\log (x)}{2}-\frac {1}{10} \log (2+x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 20, normalized size = 0.80
method | result | size |
default | \(\frac {\ln \left (x \right )}{2}+\frac {\ln \left (2 x -1\right )}{10}-\frac {\ln \left (2+x \right )}{10}\) | \(20\) |
norman | \(\frac {\ln \left (x \right )}{2}+\frac {\ln \left (2 x -1\right )}{10}-\frac {\ln \left (2+x \right )}{10}\) | \(20\) |
risch | \(\frac {\ln \left (x \right )}{2}+\frac {\ln \left (2 x -1\right )}{10}-\frac {\ln \left (2+x \right )}{10}\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.98, size = 19, normalized size = 0.76 \begin {gather*} \frac {1}{10} \, \log \left (2 \, x - 1\right ) - \frac {1}{10} \, \log \left (x + 2\right ) + \frac {1}{2} \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 19, normalized size = 0.76 \begin {gather*} \frac {1}{10} \, \log \left (2 \, x - 1\right ) - \frac {1}{10} \, \log \left (x + 2\right ) + \frac {1}{2} \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 19, normalized size = 0.76 \begin {gather*} \frac {\log {\left (x \right )}}{2} + \frac {\log {\left (x - \frac {1}{2} \right )}}{10} - \frac {\log {\left (x + 2 \right )}}{10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.47, size = 22, normalized size = 0.88 \begin {gather*} \frac {1}{10} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) - \frac {1}{10} \, \log \left ({\left | x + 2 \right |}\right ) + \frac {1}{2} \, \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.19, size = 19, normalized size = 0.76 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {24}{145\,\left (\frac {29\,x}{100}-\frac {11}{50}\right )}+\frac {35}{29}\right )}{5}+\frac {\ln \left (x\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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