Optimal. Leaf size=28 \[ \frac {\sqrt {3 x+2} \log (3 x+2)}{3 \sqrt {-3 x-2}} \]
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Rubi [A] time = 0.00, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {23, 31} \[ \frac {\sqrt {3 x+2} \log (3 x+2)}{3 \sqrt {-3 x-2}} \]
Antiderivative was successfully verified.
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Rule 23
Rule 31
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-2-3 x} \sqrt {2+3 x}} \, dx &=\frac {\sqrt {2+3 x} \int \frac {1}{2+3 x} \, dx}{\sqrt {-2-3 x}}\\ &=\frac {\sqrt {2+3 x} \log (2+3 x)}{3 \sqrt {-2-3 x}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 1.00 \[ \frac {(3 x+2) \log (3 x+2)}{3 \sqrt {-(3 x+2)^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 1, normalized size = 0.04 \[ 0 \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 1.01, size = 11, normalized size = 0.39 \[ -\frac {1}{3} i \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \mathrm {sgn}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 23, normalized size = 0.82 \[ \frac {\sqrt {3 x +2}\, \ln \left (3 x +2\right )}{3 \sqrt {-3 x -2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 2.99, size = 6, normalized size = 0.21 \[ \frac {1}{3} i \, \log \left (x + \frac {2}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 35, normalized size = 1.25 \[ -\frac {4\,\mathrm {atan}\left (\frac {-\sqrt {-3\,x-2}+\sqrt {2}\,1{}\mathrm {i}}{\sqrt {2}-\sqrt {3\,x+2}}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.46, size = 53, normalized size = 1.89 \[ \begin {cases} - \frac {i \log {\left (x + \frac {2}{3} \right )}}{3} & \text {for}\: \left |{x + \frac {2}{3}}\right | < 1 \\\frac {i \log {\left (\frac {1}{x + \frac {2}{3}} \right )}}{3} & \text {for}\: \frac {1}{\left |{x + \frac {2}{3}}\right |} < 1 \\\frac {i {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x + \frac {2}{3}} \right )}}{3} - \frac {i {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x + \frac {2}{3}} \right )}}{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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