Integrand size = 19, antiderivative size = 26 \[ \int \frac {1}{\sqrt {-3+2 x} \sqrt {2+3 x}} \, dx=\sqrt {\frac {2}{3}} \text {arcsinh}\left (\sqrt {\frac {3}{13}} \sqrt {-3+2 x}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {56, 221} \[ \int \frac {1}{\sqrt {-3+2 x} \sqrt {2+3 x}} \, dx=\sqrt {\frac {2}{3}} \text {arcsinh}\left (\sqrt {\frac {3}{13}} \sqrt {2 x-3}\right ) \]
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Rule 56
Rule 221
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {-3+2 x} \sqrt {2+3 x}} \, dx=\sqrt {\frac {2}{3}} \text {arctanh}\left (\frac {1}{\sqrt {\frac {-9+6 x}{4+6 x}}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(56\) vs. \(2(18)=36\).
Time = 0.66 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.19
method | result | size |
default | \(\frac {\sqrt {\left (-3+2 x \right ) \left (2+3 x \right )}\, \ln \left (\frac {\left (-\frac {5}{2}+6 x \right ) \sqrt {6}}{6}+\sqrt {6 x^{2}-5 x -6}\right ) \sqrt {6}}{6 \sqrt {-3+2 x}\, \sqrt {2+3 x}}\) | \(57\) |
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Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (18) = 36\).
Time = 0.22 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.77 \[ \int \frac {1}{\sqrt {-3+2 x} \sqrt {2+3 x}} \, dx=\frac {1}{12} \, \sqrt {3} \sqrt {2} \log \left (4 \, \sqrt {3} \sqrt {2} {\left (12 \, x - 5\right )} \sqrt {3 \, x + 2} \sqrt {2 \, x - 3} + 288 \, x^{2} - 240 \, x - 119\right ) \]
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Result contains complex when optimal does not.
Time = 0.57 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.15 \[ \int \frac {1}{\sqrt {-3+2 x} \sqrt {2+3 x}} \, dx=\begin {cases} \frac {\sqrt {6} \operatorname {acosh}{\left (\frac {\sqrt {78} \sqrt {x + \frac {2}{3}}}{13} \right )}}{3} & \text {for}\: \left |{x + \frac {2}{3}}\right | > \frac {13}{6} \\- \frac {\sqrt {6} i \operatorname {asin}{\left (\frac {\sqrt {78} \sqrt {x + \frac {2}{3}}}{13} \right )}}{3} & \text {otherwise} \end {cases} \]
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none
Time = 0.30 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\sqrt {-3+2 x} \sqrt {2+3 x}} \, dx=\frac {1}{6} \, \sqrt {6} \log \left (2 \, \sqrt {6} \sqrt {6 \, x^{2} - 5 \, x - 6} + 12 \, x - 5\right ) \]
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none
Time = 0.33 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {1}{\sqrt {-3+2 x} \sqrt {2+3 x}} \, dx=-\frac {1}{3} \, \sqrt {3} \sqrt {2} \log \left ({\left | -\sqrt {2} \sqrt {3 \, x + 2} + \sqrt {6 \, x - 9} \right |}\right ) \]
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Time = 0.12 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65 \[ \int \frac {1}{\sqrt {-3+2 x} \sqrt {2+3 x}} \, dx=\frac {2\,\sqrt {6}\,\mathrm {atanh}\left (\frac {\sqrt {6}\,\left (-\sqrt {2\,x-3}+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,\left (\sqrt {2}-\sqrt {3\,x+2}\right )}\right )}{3} \]
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