Optimal. Leaf size=23 \[ x-\frac {1}{4} x \left (-5-2 x^2-\frac {2}{(2+x)^2}+\log (4)\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 35, normalized size of antiderivative = 1.52, number of steps
used = 2, number of rules used = 1, integrand size = 61, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {2099}
\begin {gather*} \frac {x^3}{2}+\frac {1}{2 (x+2)}-\frac {1}{(x+2)^2}+\frac {1}{4} x (9-\log (4)) \end {gather*}
Antiderivative was successfully verified.
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Rule 2099
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {3 x^2}{2}+\frac {2}{(2+x)^3}-\frac {1}{2 (2+x)^2}+\frac {1}{4} (9-\log (4))\right ) \, dx\\ &=\frac {x^3}{2}-\frac {1}{(2+x)^2}+\frac {1}{2 (2+x)}+\frac {1}{4} x (9-\log (4))\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.03, size = 37, normalized size = 1.61 \begin {gather*} \frac {1}{4} \left (2 x^3+\frac {2 \left (x+17 (2+x)^2\right )}{(2+x)^2}-x (-9+\log (4))-\log (16)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 29, normalized size = 1.26
method | result | size |
risch | \(\frac {x^{3}}{2}-\frac {x \ln \left (2\right )}{2}+\frac {9 x}{4}+\frac {x}{2 x^{2}+8 x +8}\) | \(28\) |
default | \(\frac {x^{3}}{2}+\frac {9 x}{4}-\frac {x \ln \left (2\right )}{2}+\frac {1}{2 x +4}-\frac {1}{\left (2+x \right )^{2}}\) | \(29\) |
norman | \(\frac {\left (-\frac {\ln \left (2\right )}{2}+\frac {17}{4}\right ) x^{3}+\left (-\frac {53}{2}+6 \ln \left (2\right )\right ) x +2 x^{4}+\frac {x^{5}}{2}-36+8 \ln \left (2\right )}{\left (2+x \right )^{2}}\) | \(41\) |
gosper | \(-\frac {-2 x^{5}+2 x^{3} \ln \left (2\right )-8 x^{4}-17 x^{3}-24 x \ln \left (2\right )-32 \ln \left (2\right )+106 x +144}{4 \left (x^{2}+4 x +4\right )}\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 28, normalized size = 1.22 \begin {gather*} \frac {1}{2} \, x^{3} - \frac {1}{4} \, x {\left (2 \, \log \left (2\right ) - 9\right )} + \frac {x}{2 \, {\left (x^{2} + 4 \, x + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 52 vs.
\(2 (23) = 46\).
time = 0.33, size = 52, normalized size = 2.26 \begin {gather*} \frac {2 \, x^{5} + 8 \, x^{4} + 17 \, x^{3} + 36 \, x^{2} - 2 \, {\left (x^{3} + 4 \, x^{2} + 4 \, x\right )} \log \left (2\right ) + 38 \, x}{4 \, {\left (x^{2} + 4 \, x + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 26, normalized size = 1.13 \begin {gather*} \frac {x^{3}}{2} + x \left (\frac {9}{4} - \frac {\log {\left (2 \right )}}{2}\right ) + \frac {x}{2 x^{2} + 8 x + 8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.38, size = 22, normalized size = 0.96 \begin {gather*} \frac {1}{2} \, x^{3} - \frac {1}{2} \, x \log \left (2\right ) + \frac {9}{4} \, x + \frac {x}{2 \, {\left (x + 2\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.62, size = 29, normalized size = 1.26 \begin {gather*} \frac {x}{2\,\left (x^2+4\,x+4\right )}-x\,\left (\frac {\ln \left (4\right )}{4}-\frac {9}{4}\right )+\frac {x^3}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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