Optimal. Leaf size=27 \[ 2 x-x^2-3 (1+2 x) \left (-1-x^2+\log (6+x)\right ) \]
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Rubi [A] time = 0.10, antiderivative size = 29, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {6742, 1850, 2389, 2295} \begin {gather*} 6 x^3+2 x^2+8 x-6 (x+6) \log (x+6)+33 \log (x+6) \end {gather*}
Antiderivative was successfully verified.
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Rule 1850
Rule 2295
Rule 2389
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {45+26 x+112 x^2+18 x^3}{6+x}-6 \log (6+x)\right ) \, dx\\ &=-(6 \int \log (6+x) \, dx)+\int \frac {45+26 x+112 x^2+18 x^3}{6+x} \, dx\\ &=-(6 \operatorname {Subst}(\int \log (x) \, dx,x,6+x))+\int \left (2+4 x+18 x^2+\frac {33}{6+x}\right ) \, dx\\ &=8 x+2 x^2+6 x^3+33 \log (6+x)-6 (6+x) \log (6+x)\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.03, size = 27, normalized size = 1.00 \begin {gather*} 8 x+2 x^2+6 x^3-3 \log (6+x)-6 x \log (6+x) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 25, normalized size = 0.93 \begin {gather*} 6 \, x^{3} + 2 \, x^{2} - 3 \, {\left (2 \, x + 1\right )} \log \left (x + 6\right ) + 8 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 27, normalized size = 1.00 \begin {gather*} 6 \, x^{3} + 2 \, x^{2} - 6 \, x \log \left (x + 6\right ) + 8 \, x - 3 \, \log \left (x + 6\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 28, normalized size = 1.04
method | result | size |
norman | \(-3 \ln \left (x +6\right )+8 x +2 x^{2}+6 x^{3}-6 x \ln \left (x +6\right )\) | \(28\) |
risch | \(-3 \ln \left (x +6\right )+8 x +2 x^{2}+6 x^{3}-6 x \ln \left (x +6\right )\) | \(28\) |
derivativedivides | \(6 \left (x +6\right )^{3}-6 \left (x +6\right ) \ln \left (x +6\right )+632 x +3792-106 \left (x +6\right )^{2}+33 \ln \left (x +6\right )\) | \(35\) |
default | \(6 \left (x +6\right )^{3}-6 \left (x +6\right ) \ln \left (x +6\right )+632 x +3792-106 \left (x +6\right )^{2}+33 \ln \left (x +6\right )\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 42, normalized size = 1.56 \begin {gather*} 6 \, x^{3} + 2 \, x^{2} - 6 \, {\left (x - 6 \, \log \left (x + 6\right )\right )} \log \left (x + 6\right ) - 36 \, \log \left (x + 6\right )^{2} + 8 \, x - 3 \, \log \left (x + 6\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 28, normalized size = 1.04 \begin {gather*} 2\,x^2-3\,\ln \left (x+6\right )+6\,x^3-x\,\left (6\,\ln \left (x+6\right )-8\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 27, normalized size = 1.00 \begin {gather*} 6 x^{3} + 2 x^{2} - 6 x \log {\left (x + 6 \right )} + 8 x - 3 \log {\left (x + 6 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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