Optimal. Leaf size=28 \[ \frac {x}{6 \sqrt {6} \sqrt {1-2 x} \sqrt {2 x+1}} \]
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Rubi [A] time = 0.00, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {39} \[ \frac {x}{6 \sqrt {6} \sqrt {1-2 x} \sqrt {2 x+1}} \]
Antiderivative was successfully verified.
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Rule 39
Rubi steps
\begin {align*} \int \frac {1}{(3-6 x)^{3/2} (2+4 x)^{3/2}} \, dx &=\frac {x}{6 \sqrt {6} \sqrt {1-2 x} \sqrt {1+2 x}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 16, normalized size = 0.57 \[ \frac {x}{6 \sqrt {6-24 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 26, normalized size = 0.93 \[ -\frac {\sqrt {4 \, x + 2} x \sqrt {-6 \, x + 3}}{36 \, {\left (4 \, x^{2} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.06, size = 71, normalized size = 2.54 \[ -\frac {\sqrt {6} {\left (\sqrt {-4 \, x + 2} - 2\right )}}{288 \, \sqrt {4 \, x + 2}} - \frac {\sqrt {6} \sqrt {4 \, x + 2} \sqrt {-4 \, x + 2}}{288 \, {\left (2 \, x - 1\right )}} + \frac {\sqrt {6} \sqrt {4 \, x + 2}}{288 \, {\left (\sqrt {-4 \, x + 2} - 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 28, normalized size = 1.00 \[ -\frac {\left (2 x -1\right ) \left (2 x +1\right ) x}{\left (-6 x +3\right )^{\frac {3}{2}} \left (4 x +2\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 12, normalized size = 0.43 \[ \frac {x}{6 \, \sqrt {-24 \, x^{2} + 6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.46, size = 24, normalized size = 0.86 \[ -\frac {x\,\sqrt {3-6\,x}}{\sqrt {4\,x+2}\,\left (36\,x-18\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 85.28, size = 156, normalized size = 5.57 \[ \begin {cases} - \frac {2 \sqrt {6} i \sqrt {x - \frac {1}{2}} \left (x + \frac {1}{2}\right )}{144 \left (x + \frac {1}{2}\right )^{\frac {3}{2}} - 144 \sqrt {x + \frac {1}{2}}} + \frac {\sqrt {6} i \sqrt {x - \frac {1}{2}}}{144 \left (x + \frac {1}{2}\right )^{\frac {3}{2}} - 144 \sqrt {x + \frac {1}{2}}} & \text {for}\: \left |{x + \frac {1}{2}}\right | > 1 \\- \frac {2 \sqrt {6} \sqrt {\frac {1}{2} - x} \left (x + \frac {1}{2}\right )}{144 \left (x + \frac {1}{2}\right )^{\frac {3}{2}} - 144 \sqrt {x + \frac {1}{2}}} + \frac {\sqrt {6} \sqrt {\frac {1}{2} - x}}{144 \left (x + \frac {1}{2}\right )^{\frac {3}{2}} - 144 \sqrt {x + \frac {1}{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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