Optimal. Leaf size=27 \[ 3 x+\frac {x^2}{2}-\log (1-x)+8 \log (2-x) \]
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Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {715, 646, 31}
\begin {gather*} \frac {x^2}{2}+3 x-\log (1-x)+8 \log (2-x) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 646
Rule 715
Rubi steps
\begin {align*} \int \frac {x^3}{2-3 x+x^2} \, dx &=\int \left (3+x-\frac {6-7 x}{2-3 x+x^2}\right ) \, dx\\ &=3 x+\frac {x^2}{2}-\int \frac {6-7 x}{2-3 x+x^2} \, dx\\ &=3 x+\frac {x^2}{2}+8 \int \frac {1}{-2+x} \, dx-\int \frac {1}{-1+x} \, dx\\ &=3 x+\frac {x^2}{2}-\log (1-x)+8 \log (2-x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 27, normalized size = 1.00 \begin {gather*} 3 x+\frac {x^2}{2}-\log (1-x)+8 \log (2-x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.67, size = 22, normalized size = 0.81
method | result | size |
default | \(3 x +\frac {x^{2}}{2}-\ln \left (x -1\right )+8 \ln \left (x -2\right )\) | \(22\) |
norman | \(3 x +\frac {x^{2}}{2}-\ln \left (x -1\right )+8 \ln \left (x -2\right )\) | \(22\) |
risch | \(3 x +\frac {x^{2}}{2}-\ln \left (x -1\right )+8 \ln \left (x -2\right )\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 21, normalized size = 0.78 \begin {gather*} \frac {1}{2} \, x^{2} + 3 \, x - \log \left (x - 1\right ) + 8 \, \log \left (x - 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.86, size = 21, normalized size = 0.78 \begin {gather*} \frac {1}{2} \, x^{2} + 3 \, x - \log \left (x - 1\right ) + 8 \, \log \left (x - 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 19, normalized size = 0.70 \begin {gather*} \frac {x^{2}}{2} + 3 x + 8 \log {\left (x - 2 \right )} - \log {\left (x - 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.81, size = 23, normalized size = 0.85 \begin {gather*} \frac {1}{2} \, x^{2} + 3 \, x - \log \left ({\left | x - 1 \right |}\right ) + 8 \, \log \left ({\left | x - 2 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 21, normalized size = 0.78 \begin {gather*} 3\,x-\ln \left (x-1\right )+8\,\ln \left (x-2\right )+\frac {x^2}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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