Optimal. Leaf size=27 \[ \frac {x^2}{2}+3 x-\log (1-x)+8 \log (2-x) \]
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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 = {701, 632, 31} \[ \frac {x^2}{2}+3 x-\log (1-x)+8 \log (2-x) \]
Antiderivative was successfully verified.
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Rule 31
Rule 632
Rule 701
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 \[ \frac {x^2}{2}+3 x-\log (1-x)+8 \log (2-x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 21, normalized size = 0.78 \[ \frac {1}{2} \, x^{2} + 3 \, x - \log \left (x - 1\right ) + 8 \, \log \left (x - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 23, normalized size = 0.85 \[ \frac {1}{2} \, x^{2} + 3 \, x - \log \left ({\left | x - 1 \right |}\right ) + 8 \, \log \left ({\left | x - 2 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 22, normalized size = 0.81 \[ \frac {x^{2}}{2}+3 x +8 \ln \left (x -2\right )-\ln \left (x -1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.87, size = 21, normalized size = 0.78 \[ \frac {1}{2} \, x^{2} + 3 \, x - \log \left (x - 1\right ) + 8 \, \log \left (x - 2\right ) \]
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 \[ 3\,x-\ln \left (x-1\right )+8\,\ln \left (x-2\right )+\frac {x^2}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 19, normalized size = 0.70 \[ \frac {x^{2}}{2} + 3 x + 8 \log {\left (x - 2 \right )} - \log {\left (x - 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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