Integrand size = 19, antiderivative size = 43 \[ \int \sqrt {3-6 x} \sqrt {2+4 x} \, dx=\sqrt {\frac {3}{2}} \sqrt {1-2 x} x \sqrt {1+2 x}+\frac {1}{2} \sqrt {\frac {3}{2}} \arcsin (2 x) \]
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Time = 0.00 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {38, 41, 222} \[ \int \sqrt {3-6 x} \sqrt {2+4 x} \, dx=\frac {1}{2} \sqrt {\frac {3}{2}} \arcsin (2 x)+\sqrt {\frac {3}{2}} \sqrt {1-2 x} \sqrt {2 x+1} x \]
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Rule 38
Rule 41
Rule 222
Rubi steps \begin{align*} \text {integral}& = \sqrt {\frac {3}{2}} \sqrt {1-2 x} x \sqrt {1+2 x}+3 \int \frac {1}{\sqrt {3-6 x} \sqrt {2+4 x}} \, dx \\ & = \sqrt {\frac {3}{2}} \sqrt {1-2 x} x \sqrt {1+2 x}+3 \int \frac {1}{\sqrt {6-24 x^2}} \, dx \\ & = \sqrt {\frac {3}{2}} \sqrt {1-2 x} x \sqrt {1+2 x}+\frac {1}{2} \sqrt {\frac {3}{2}} \sin ^{-1}(2 x) \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.98 \[ \int \sqrt {3-6 x} \sqrt {2+4 x} \, dx=\sqrt {\frac {3}{2}} \left (x \sqrt {1-4 x^2}+\arctan \left (\frac {\sqrt {1-4 x^2}}{1-2 x}\right )\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(69\) vs. \(2(30)=60\).
Time = 0.36 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.63
method | result | size |
default | \(\frac {\left (2+4 x \right )^{\frac {3}{2}} \sqrt {3-6 x}}{8}-\frac {\sqrt {3-6 x}\, \sqrt {2+4 x}}{4}+\frac {\sqrt {\left (2+4 x \right ) \left (3-6 x \right )}\, \arcsin \left (2 x \right ) \sqrt {6}}{4 \sqrt {2+4 x}\, \sqrt {3-6 x}}\) | \(70\) |
risch | \(-\frac {x \left (-1+2 x \right ) \left (1+2 x \right ) \sqrt {\left (2+4 x \right ) \left (3-6 x \right )}\, \sqrt {6}}{2 \sqrt {-\left (-1+2 x \right ) \left (1+2 x \right )}\, \sqrt {3-6 x}\, \sqrt {2+4 x}}+\frac {\sqrt {\left (2+4 x \right ) \left (3-6 x \right )}\, \arcsin \left (2 x \right ) \sqrt {6}}{4 \sqrt {2+4 x}\, \sqrt {3-6 x}}\) | \(95\) |
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Time = 0.22 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.21 \[ \int \sqrt {3-6 x} \sqrt {2+4 x} \, dx=\frac {1}{2} \, \sqrt {4 \, x + 2} x \sqrt {-6 \, x + 3} - \frac {1}{4} \, \sqrt {3} \sqrt {2} \arctan \left (\frac {\sqrt {3} \sqrt {2} \sqrt {4 \, x + 2} \sqrt {-6 \, x + 3}}{12 \, x}\right ) \]
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Result contains complex when optimal does not.
Time = 4.78 (sec) , antiderivative size = 187, normalized size of antiderivative = 4.35 \[ \int \sqrt {3-6 x} \sqrt {2+4 x} \, dx=\begin {cases} - \frac {\sqrt {6} i \operatorname {acosh}{\left (\sqrt {x + \frac {1}{2}} \right )}}{2} + \frac {\sqrt {6} i \left (x + \frac {1}{2}\right )^{\frac {5}{2}}}{\sqrt {x - \frac {1}{2}}} - \frac {3 \sqrt {6} i \left (x + \frac {1}{2}\right )^{\frac {3}{2}}}{2 \sqrt {x - \frac {1}{2}}} + \frac {\sqrt {6} i \sqrt {x + \frac {1}{2}}}{2 \sqrt {x - \frac {1}{2}}} & \text {for}\: \left |{x + \frac {1}{2}}\right | > 1 \\\frac {\sqrt {6} \operatorname {asin}{\left (\sqrt {x + \frac {1}{2}} \right )}}{2} - \frac {\sqrt {6} \left (x + \frac {1}{2}\right )^{\frac {5}{2}}}{\sqrt {\frac {1}{2} - x}} + \frac {3 \sqrt {6} \left (x + \frac {1}{2}\right )^{\frac {3}{2}}}{2 \sqrt {\frac {1}{2} - x}} - \frac {\sqrt {6} \sqrt {x + \frac {1}{2}}}{2 \sqrt {\frac {1}{2} - x}} & \text {otherwise} \end {cases} \]
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Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.51 \[ \int \sqrt {3-6 x} \sqrt {2+4 x} \, dx=\frac {1}{2} \, \sqrt {-24 \, x^{2} + 6} x + \frac {1}{4} \, \sqrt {6} \arcsin \left (2 \, x\right ) \]
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Time = 0.30 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.28 \[ \int \sqrt {3-6 x} \sqrt {2+4 x} \, dx=\frac {1}{2} \, \sqrt {3} \sqrt {2} {\left (\sqrt {2 \, x + 1} {\left (x - 1\right )} \sqrt {-2 \, x + 1} + \sqrt {2 \, x + 1} \sqrt {-2 \, x + 1} + \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {2 \, x + 1}\right )\right )} \]
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Time = 0.29 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.02 \[ \int \sqrt {3-6 x} \sqrt {2+4 x} \, dx=\frac {x\,\sqrt {4\,x+2}\,\sqrt {3-6\,x}}{2}-\frac {\sqrt {6}\,\ln \left (x-\frac {\sqrt {1-2\,x}\,\sqrt {2\,x+1}\,1{}\mathrm {i}}{2}\right )\,1{}\mathrm {i}}{4} \]
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