Optimal. Leaf size=48 \[ \frac {1}{4} \sqrt {4 x^2+12 x+9}-\frac {3 (2 x+3) \log (2 x+3)}{4 \sqrt {4 x^2+12 x+9}} \]
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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 = {640, 608, 31} \begin {gather*} \frac {1}{4} \sqrt {4 x^2+12 x+9}-\frac {3 (2 x+3) \log (2 x+3)}{4 \sqrt {4 x^2+12 x+9}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 608
Rule 640
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {9+12 x+4 x^2}} \, dx &=\frac {1}{4} \sqrt {9+12 x+4 x^2}-\frac {3}{2} \int \frac {1}{\sqrt {9+12 x+4 x^2}} \, dx\\ &=\frac {1}{4} \sqrt {9+12 x+4 x^2}-\frac {(3 (6+4 x)) \int \frac {1}{6+4 x} \, dx}{2 \sqrt {9+12 x+4 x^2}}\\ &=\frac {1}{4} \sqrt {9+12 x+4 x^2}-\frac {3 (3+2 x) \log (3+2 x)}{4 \sqrt {9+12 x+4 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 33, normalized size = 0.69 \begin {gather*} \frac {(2 x+3) (2 x-3 \log (2 x+3)+3)}{4 \sqrt {(2 x+3)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.10, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {9+12 x+4 x^2}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.40, size = 12, normalized size = 0.25 \begin {gather*} \frac {1}{2} \, x - \frac {3}{4} \, \log \left (2 \, x + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 36, normalized size = 0.75 \begin {gather*} \frac {1}{4} \, \sqrt {4 \, x^{2} + 12 \, x + 9} + \frac {3}{4} \, \log \left ({\left | -2 \, x + \sqrt {4 \, x^{2} + 12 \, x + 9} - 3 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 29, normalized size = 0.60 \begin {gather*} -\frac {\left (2 x +3\right ) \left (-2 x +3 \ln \left (2 x +3\right )\right )}{4 \sqrt {\left (2 x +3\right )^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.81, size = 21, normalized size = 0.44 \begin {gather*} \frac {1}{4} \, \sqrt {4 \, x^{2} + 12 \, x + 9} - \frac {3}{4} \, \log \left (x + \frac {3}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 32, normalized size = 0.67 \begin {gather*} \frac {\sqrt {4\,x^2+12\,x+9}}{4}-\frac {3\,\ln \left (x+\frac {\sqrt {{\left (2\,x+3\right )}^2}}{2}+\frac {3}{2}\right )}{4} \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 (2 x + 3\right )^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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