Optimal. Leaf size=22 \[ \frac {x^2}{2}-2 x+\frac {1}{x+1}+3 \log (x+1) \]
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Rubi [A] time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {27, 43} \begin {gather*} \frac {x^2}{2}-2 x+\frac {1}{x+1}+3 \log (x+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int \frac {x^3}{1+2 x+x^2} \, dx &=\int \frac {x^3}{(1+x)^2} \, dx\\ &=\int \left (-2+x-\frac {1}{(1+x)^2}+\frac {3}{1+x}\right ) \, dx\\ &=-2 x+\frac {x^2}{2}+\frac {1}{1+x}+3 \log (1+x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 1.18 \begin {gather*} \frac {1}{2} (x+1)^2-3 (x+1)+\frac {1}{x+1}+3 \log (x+1) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3}{1+2 x+x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 29, normalized size = 1.32 \begin {gather*} \frac {x^{3} - 3 \, x^{2} + 6 \, {\left (x + 1\right )} \log \left (x + 1\right ) - 4 \, x + 2}{2 \, {\left (x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 21, normalized size = 0.95 \begin {gather*} \frac {1}{2} \, x^{2} - 2 \, x + \frac {1}{x + 1} + 3 \, \log \left ({\left | x + 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 21, normalized size = 0.95 \begin {gather*} \frac {x^{2}}{2}-2 x +3 \ln \left (x +1\right )+\frac {1}{x +1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.76, size = 20, normalized size = 0.91 \begin {gather*} \frac {1}{2} \, x^{2} - 2 \, x + \frac {1}{x + 1} + 3 \, \log \left (x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 20, normalized size = 0.91 \begin {gather*} 3\,\ln \left (x+1\right )-2\,x+\frac {1}{x+1}+\frac {x^2}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 19, normalized size = 0.86 \begin {gather*} \frac {x^{2}}{2} - 2 x + 3 \log {\left (x + 1 \right )} + \frac {1}{x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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