Optimal. Leaf size=28 \[ \frac {\sqrt {2+3 x} \log (2+3 x)}{3 \sqrt {-2-3 x}} \]
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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}
\begin {gather*} \frac {\sqrt {3 x+2} \log (3 x+2)}{3 \sqrt {-3 x-2}} \end {gather*}
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 = 0.00, size = 28, normalized size = 1.00 \begin {gather*} \frac {(2+3 x) \log (2+3 x)}{3 \sqrt {-(2+3 x)^2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 2.39, size = 94, normalized size = 3.36 \begin {gather*} \text {Piecewise}\left [\left \{\left \{0,\text {Abs}\left [\frac {2}{3}+x\right ]<1\text {\&\&}\frac {3}{\text {Abs}\left [2+3 x\right ]}<1\right \},\left \{\left (-\frac {I}{3}\right ) \text {Log}\left [\frac {2}{3}+x\right ],\text {Abs}\left [\frac {2}{3}+x\right ]<1\right \},\left \{\frac {I \text {Log}\left [\frac {3}{2+3 x}\right ]}{3},\frac {3}{\text {Abs}\left [2+3 x\right ]}<1\right \}\right \},-\frac {I \text {meijerg}\left [\left \{\left \{1,1\right \},\left \{\right \}\right \},\left \{\left \{\right \},\left \{0,0\right \}\right \},\frac {2}{3}+x\right ]}{3}+\frac {I \text {meijerg}\left [\left \{\left \{\right \},\left \{1,1\right \}\right \},\left \{\left \{0,0\right \},\left \{\right \}\right \},\frac {2}{3}+x\right ]}{3}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.16, size = 23, normalized size = 0.82
method | result | size |
meijerg | \(-\frac {i \ln \left (1+\frac {3 x}{2}\right )}{3}\) | \(10\) |
default | \(\frac {\ln \left (2+3 x \right ) \sqrt {2+3 x}}{3 \sqrt {-2-3 x}}\) | \(23\) |
risch | \(-\frac {i \sqrt {\frac {-2-3 x}{2+3 x}}\, \sqrt {2+3 x}\, \ln \left (2+3 x \right )}{3 \sqrt {-2-3 x}}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.34, size = 6, normalized size = 0.21 \begin {gather*} \frac {1}{3} 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 [A]
time = 0.30, size = 1, normalized size = 0.04 \begin {gather*} 0 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.59, size = 66, normalized size = 2.36 \begin {gather*} \begin {cases} 0 & \text {for}\: \frac {1}{\left |{x + \frac {2}{3}}\right |} < 1 \wedge \left |{x + \frac {2}{3}}\right | < 1 \\- \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} \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 = 0.00, size = 14, normalized size = 0.50 \begin {gather*} -\frac {1}{3} \mathrm {sign}\left (x\right ) \mathrm {i} \ln \left |-3 x-2\right | \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.22, size = 35, normalized size = 1.25 \begin {gather*} -\frac {4\,\mathrm {atan}\left (\frac {-\sqrt {-3\,x-2}+\sqrt {2}\,1{}\mathrm {i}}{\sqrt {2}-\sqrt {3\,x+2}}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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