Optimal. Leaf size=23 \[ \frac {x^2}{2}+2 x+9 \log (3-x)+\log (x+1) \]
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Rubi [A] time = 0.03, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1593, 1628, 632, 31} \[ \frac {x^2}{2}+2 x+9 \log (3-x)+\log (x+1) \]
Antiderivative was successfully verified.
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Rule 31
Rule 632
Rule 1593
Rule 1628
Rubi steps
\begin {align*} \int \frac {3 x+x^3}{-3-2 x+x^2} \, dx &=\int \frac {x \left (3+x^2\right )}{-3-2 x+x^2} \, dx\\ &=\int \left (2+x+\frac {2 (3+5 x)}{-3-2 x+x^2}\right ) \, dx\\ &=2 x+\frac {x^2}{2}+2 \int \frac {3+5 x}{-3-2 x+x^2} \, dx\\ &=2 x+\frac {x^2}{2}+9 \int \frac {1}{-3+x} \, dx+\int \frac {1}{1+x} \, dx\\ &=2 x+\frac {x^2}{2}+9 \log (3-x)+\log (1+x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 23, normalized size = 1.00 \[ \frac {x^2}{2}+2 x+9 \log (3-x)+\log (x+1) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 19, normalized size = 0.83 \[ \frac {1}{2} \, x^{2} + 2 \, x + \log \left (x + 1\right ) + 9 \, \log \left (x - 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.01, size = 21, normalized size = 0.91 \[ \frac {1}{2} \, x^{2} + 2 \, x + \log \left ({\left | x + 1 \right |}\right ) + 9 \, \log \left ({\left | x - 3 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 20, normalized size = 0.87 \[ \frac {x^{2}}{2}+2 x +9 \ln \left (x -3\right )+\ln \left (x +1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 19, normalized size = 0.83 \[ \frac {1}{2} \, x^{2} + 2 \, x + \log \left (x + 1\right ) + 9 \, \log \left (x - 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 19, normalized size = 0.83 \[ 2\,x+\ln \left (x+1\right )+9\,\ln \left (x-3\right )+\frac {x^2}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 19, normalized size = 0.83 \[ \frac {x^{2}}{2} + 2 x + 9 \log {\left (x - 3 \right )} + \log {\left (x + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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