Optimal. Leaf size=48 \[ \frac {1}{9} \sqrt {4+12 x+9 x^2}-\frac {2 (2+3 x) \log (2+3 x)}{9 \sqrt {4+12 x+9 x^2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {654, 622, 31}
\begin {gather*} \frac {1}{9} \sqrt {9 x^2+12 x+4}-\frac {2 (3 x+2) \log (3 x+2)}{9 \sqrt {9 x^2+12 x+4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 622
Rule 654
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {4+12 x+9 x^2}} \, dx &=\frac {1}{9} \sqrt {4+12 x+9 x^2}-\frac {2}{3} \int \frac {1}{\sqrt {4+12 x+9 x^2}} \, dx\\ &=\frac {1}{9} \sqrt {4+12 x+9 x^2}-\frac {(2 (6+9 x)) \int \frac {1}{6+9 x} \, dx}{3 \sqrt {4+12 x+9 x^2}}\\ &=\frac {1}{9} \sqrt {4+12 x+9 x^2}-\frac {2 (2+3 x) \log (2+3 x)}{9 \sqrt {4+12 x+9 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 33, normalized size = 0.69 \begin {gather*} \frac {(2+3 x) (2+3 x-2 \log (2+3 x))}{9 \sqrt {(2+3 x)^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 29, normalized size = 0.60
method | result | size |
default | \(-\frac {\left (2+3 x \right ) \left (-3 x +2 \ln \left (2+3 x \right )\right )}{9 \sqrt {\left (2+3 x \right )^{2}}}\) | \(29\) |
risch | \(\frac {\sqrt {\left (2+3 x \right )^{2}}\, x}{6+9 x}-\frac {2 \sqrt {\left (2+3 x \right )^{2}}\, \ln \left (2+3 x \right )}{9 \left (2+3 x \right )}\) | \(45\) |
meijerg | \(\frac {\frac {2 x}{3}-\frac {4 \ln \left (1+\frac {3 x}{2}\right )}{9}}{\sqrt {\left (2+3 x \right )^{2}}}+\frac {2 x \left (\frac {3 x}{2}-\ln \left (1+\frac {3 x}{2}\right )\right )}{3 \sqrt {\left (2+3 x \right )^{2}}}\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 21, normalized size = 0.44 \begin {gather*} \frac {1}{9} \, \sqrt {9 \, x^{2} + 12 \, x + 4} - \frac {2}{9} \, \log \left (x + \frac {2}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.51, size = 12, normalized size = 0.25 \begin {gather*} \frac {1}{3} \, x - \frac {2}{9} \, \log \left (3 \, x + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 12, normalized size = 0.25 \begin {gather*} \frac {x}{3} - \frac {2 \log {\left (3 x + 2 \right )}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.79, size = 25, normalized size = 0.52 \begin {gather*} \frac {1}{3} \, x \mathrm {sgn}\left (3 \, x + 2\right ) - \frac {2}{9} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \mathrm {sgn}\left (3 \, x + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.27, size = 32, normalized size = 0.67 \begin {gather*} \frac {\sqrt {9\,x^2+12\,x+4}}{9}-\frac {2\,\ln \left (x+\frac {\sqrt {{\left (3\,x+2\right )}^2}}{3}+\frac {2}{3}\right )}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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