Optimal. Leaf size=20 \[ -5+4 \left (x+x^2\right )-x (4-\log (2+x)) \]
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Rubi [A] time = 0.05, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6688, 77, 2389, 2295} \begin {gather*} 4 x^2+(x+2) \log (x+2)-2 \log (x+2) \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 2295
Rule 2389
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {x (17+8 x)}{2+x}+\log (2+x)\right ) \, dx\\ &=\int \frac {x (17+8 x)}{2+x} \, dx+\int \log (2+x) \, dx\\ &=\int \left (1+8 x-\frac {2}{2+x}\right ) \, dx+\operatorname {Subst}(\int \log (x) \, dx,x,2+x)\\ &=4 x^2-2 \log (2+x)+(2+x) \log (2+x)\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 1.10 \begin {gather*} -x-15 (2+x)+4 (2+x)^2+x \log (2+x) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 12, normalized size = 0.60 \begin {gather*} 4 \, x^{2} + x \log \left (x + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 12, normalized size = 0.60 \begin {gather*} 4 \, x^{2} + x \log \left (x + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.46, size = 13, normalized size = 0.65
method | result | size |
norman | \(x \ln \left (2+x \right )+4 x^{2}\) | \(13\) |
risch | \(x \ln \left (2+x \right )+4 x^{2}\) | \(13\) |
derivativedivides | \(\left (2+x \right ) \ln \left (2+x \right )-32-16 x +4 \left (2+x \right )^{2}-2 \ln \left (2+x \right )\) | \(27\) |
default | \(\left (2+x \right ) \ln \left (2+x \right )-32-16 x +4 \left (2+x \right )^{2}-2 \ln \left (2+x \right )\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 27, normalized size = 1.35 \begin {gather*} 4 \, x^{2} + {\left (x - 2 \, \log \left (x + 2\right )\right )} \log \left (x + 2\right ) + 2 \, \log \left (x + 2\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.03, size = 10, normalized size = 0.50 \begin {gather*} x\,\left (4\,x+\ln \left (x+2\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 10, normalized size = 0.50 \begin {gather*} 4 x^{2} + x \log {\left (x + 2 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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