Optimal. Leaf size=26 \[ \sqrt {\frac {2}{3}} \sinh ^{-1}\left (\sqrt {\frac {3}{13}} \sqrt {-3+2 x}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 26, 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 = {56, 221}
\begin {gather*} \sqrt {\frac {2}{3}} \sinh ^{-1}\left (\sqrt {\frac {3}{13}} \sqrt {2 x-3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 56
Rule 221
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-3+2 x} \sqrt {2+3 x}} \, dx &=\sqrt {2} \text {Subst}\left (\int \frac {1}{\sqrt {13+3 x^2}} \, dx,x,\sqrt {-3+2 x}\right )\\ &=\sqrt {\frac {2}{3}} \sinh ^{-1}\left (\sqrt {\frac {3}{13}} \sqrt {-3+2 x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 26, normalized size = 1.00 \begin {gather*} \sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {1}{\sqrt {\frac {-9+6 x}{4+6 x}}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(56\) vs.
\(2(18)=36\).
time = 0.17, size = 57, normalized size = 2.19
method | result | size |
default | \(\frac {\sqrt {\left (2 x -3\right ) \left (2+3 x \right )}\, \ln \left (\frac {\left (-\frac {5}{2}+6 x \right ) \sqrt {6}}{6}+\sqrt {6 x^{2}-5 x -6}\right ) \sqrt {6}}{6 \sqrt {2 x -3}\, \sqrt {2+3 x}}\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 28, normalized size = 1.08 \begin {gather*} \frac {1}{6} \, \sqrt {6} \log \left (2 \, \sqrt {6} \sqrt {6 \, x^{2} - 5 \, x - 6} + 12 \, x - 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 46 vs.
\(2 (18) = 36\).
time = 1.24, size = 46, normalized size = 1.77 \begin {gather*} \frac {1}{12} \, \sqrt {3} \sqrt {2} \log \left (4 \, \sqrt {3} \sqrt {2} {\left (12 \, x - 5\right )} \sqrt {3 \, x + 2} \sqrt {2 \, x - 3} + 288 \, x^{2} - 240 \, x - 119\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.50, size = 56, normalized size = 2.15 \begin {gather*} \begin {cases} \frac {\sqrt {6} \operatorname {acosh}{\left (\frac {\sqrt {78} \sqrt {x + \frac {2}{3}}}{13} \right )}}{3} & \text {for}\: \left |{x + \frac {2}{3}}\right | > \frac {13}{6} \\- \frac {\sqrt {6} i \operatorname {asin}{\left (\frac {\sqrt {78} \sqrt {x + \frac {2}{3}}}{13} \right )}}{3} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.28, size = 30, normalized size = 1.15 \begin {gather*} -\frac {1}{3} \, \sqrt {3} \sqrt {2} \log \left ({\left | -\sqrt {2} \sqrt {3 \, x + 2} + \sqrt {6 \, x - 9} \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.12, size = 43, normalized size = 1.65 \begin {gather*} \frac {2\,\sqrt {6}\,\mathrm {atanh}\left (\frac {\sqrt {6}\,\left (-\sqrt {2\,x-3}+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,\left (\sqrt {2}-\sqrt {3\,x+2}\right )}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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