Optimal. Leaf size=18 \[ 7+x+\log \left (\left (3+e+\frac {2 x}{3}\right )^2 x^4\right ) \]
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Rubi [A]
time = 0.05, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6, 1607, 1834}
\begin {gather*} x+4 \log (x)+2 \log (2 x+3 (3+e)) \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 1607
Rule 1834
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {36+21 x+2 x^2+e (12+3 x)}{(9+3 e) x+2 x^2} \, dx\\ &=\int \frac {36+21 x+2 x^2+e (12+3 x)}{x (9+3 e+2 x)} \, dx\\ &=\int \left (1+\frac {4}{x}+\frac {4}{9+3 e+2 x}\right ) \, dx\\ &=x+4 \log (x)+2 \log (3 (3+e)+2 x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 17, normalized size = 0.94 \begin {gather*} x+4 \log (x)+2 \log (9+3 e+2 x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.36, size = 19, normalized size = 1.06
method | result | size |
default | \(x +4 \ln \left (x \right )+2 \ln \left (3 \,{\mathrm e}+2 x +9\right )\) | \(19\) |
norman | \(x +4 \ln \left (x \right )+2 \ln \left (3 \,{\mathrm e}+2 x +9\right )\) | \(19\) |
risch | \(x +4 \ln \left (x \right )+2 \ln \left (3 \,{\mathrm e}+2 x +9\right )\) | \(19\) |
meijerg | \(\frac {\left (3 \,{\mathrm e}+21\right ) \left (\frac {3 \,{\mathrm e}}{2}+\frac {9}{2}\right ) \ln \left (1+\frac {2 x}{3 \left (3+{\mathrm e}\right )}\right )}{3 \,{\mathrm e}+9}+\frac {2 \left (\frac {3 \,{\mathrm e}}{2}+\frac {9}{2}\right )^{2} \left (\frac {2 x}{3 \left (3+{\mathrm e}\right )}-\ln \left (1+\frac {2 x}{3 \left (3+{\mathrm e}\right )}\right )\right )}{3 \,{\mathrm e}+9}+\frac {8 \,{\mathrm e} \left (\frac {3 \,{\mathrm e}}{2}+\frac {9}{2}\right ) \left (-\ln \left (1+\frac {2 x}{3 \left (3+{\mathrm e}\right )}\right )+\ln \left (x \right )+\ln \left (2\right )-\ln \left (3\right )-\ln \left (3+{\mathrm e}\right )\right )}{\left (3 \,{\mathrm e}+9\right ) \left (3+{\mathrm e}\right )}+\frac {24 \left (\frac {3 \,{\mathrm e}}{2}+\frac {9}{2}\right ) \left (-\ln \left (1+\frac {2 x}{3 \left (3+{\mathrm e}\right )}\right )+\ln \left (x \right )+\ln \left (2\right )-\ln \left (3\right )-\ln \left (3+{\mathrm e}\right )\right )}{\left (3 \,{\mathrm e}+9\right ) \left (3+{\mathrm e}\right )}\) | \(183\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 18, normalized size = 1.00 \begin {gather*} x + 2 \, \log \left (2 \, x + 3 \, e + 9\right ) + 4 \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 18, normalized size = 1.00 \begin {gather*} x + 2 \, \log \left (2 \, x + 3 \, e + 9\right ) + 4 \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.29, size = 20, normalized size = 1.11 \begin {gather*} x + 4 \log {\left (x \right )} + 2 \log {\left (x + \frac {3 e}{2} + \frac {9}{2} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 20, normalized size = 1.11 \begin {gather*} x + 2 \, \log \left ({\left | 2 \, x + 3 \, e + 9 \right |}\right ) + 4 \, \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.12, size = 16, normalized size = 0.89 \begin {gather*} x+2\,\ln \left (x+\frac {3\,\mathrm {e}}{2}+\frac {9}{2}\right )+4\,\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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