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 = 35, normalized size = 0.73 \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 = 0.47, size = 31, normalized size = 0.65
method | result | size |
meijerg | \(-\frac {2 i \left (\frac {3 x}{2}-\ln \left (1+\frac {3 x}{2}\right )\right )}{9}\) | \(16\) |
default | \(-\frac {\left (2+3 x \right ) \left (-3 x +2 \ln \left (2+3 x \right )\right )}{9 \sqrt {-\left (2+3 x \right )^{2}}}\) | \(31\) |
risch | \(\frac {\left (2+3 x \right ) x}{3 \sqrt {-\left (2+3 x \right )^{2}}}-\frac {2 \left (2+3 x \right ) \ln \left (2+3 x \right )}{9 \sqrt {-\left (2+3 x \right )^{2}}}\) | \(45\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.50, size = 21, normalized size = 0.44 \begin {gather*} -\frac {1}{9} \, \sqrt {-9 \, x^{2} - 12 \, x - 4} - \frac {2}{9} i \, \log \left (x + \frac {2}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 2.46, size = 10, normalized size = 0.21 \begin {gather*} -\frac {1}{3} i \, x + \frac {2}{9} i \, \log \left (x + \frac {2}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {- \left (3 x + 2\right )^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 1.37, size = 29, normalized size = 0.60 \begin {gather*} \frac {i \, x}{3 \, \mathrm {sgn}\left (-3 \, x - 2\right )} - \frac {2 i \, \log \left ({\left | 3 \, x + 2 \right |}\right )}{9 \, \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 = 36, normalized size = 0.75 \begin {gather*} -\frac {\sqrt {-9\,x^2-12\,x-4}}{9}+\frac {\ln \left (x+\frac {2}{3}-\frac {\sqrt {-{\left (3\,x+2\right )}^2}\,1{}\mathrm {i}}{3}\right )\,2{}\mathrm {i}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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