Integrand size = 26, antiderivative size = 27 \[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-x} \sqrt {2+3 x}} \, dx=-\sqrt {\frac {2}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{2}} \sqrt {-x}\right ),-1\right ) \]
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Time = 0.00 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {116} \[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-x} \sqrt {2+3 x}} \, dx=-\sqrt {\frac {2}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{2}} \sqrt {-x}\right ),-1\right ) \]
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Rule 116
Rubi steps \begin{align*} \text {integral}& = -\sqrt {\frac {2}{3}} F\left (\left .\sin ^{-1}\left (\sqrt {\frac {3}{2}} \sqrt {-x}\right )\right |-1\right ) \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-x} \sqrt {2+3 x}} \, dx=\frac {x \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {9 x^2}{4}\right )}{\sqrt {-x}} \]
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Time = 1.74 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78
method | result | size |
default | \(\frac {F\left (\frac {\sqrt {4+6 x}}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {3}}{3}\) | \(21\) |
elliptic | \(\frac {\sqrt {x \left (9 x^{2}-4\right )}\, \sqrt {4+6 x}\, \sqrt {-6 x}\, F\left (\frac {\sqrt {4+6 x}}{2}, \frac {\sqrt {2}}{2}\right )}{6 \sqrt {-x}\, \sqrt {2+3 x}\, \sqrt {9 x^{3}-4 x}}\) | \(64\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.22 \[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-x} \sqrt {2+3 x}} \, dx=\frac {2}{3} \, {\rm weierstrassPInverse}\left (\frac {16}{9}, 0, x\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (22) = 44\).
Time = 11.47 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.04 \[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-x} \sqrt {2+3 x}} \, dx=\frac {\sqrt {6} i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {1}{2}, 1, 1 & \frac {3}{4}, \frac {3}{4}, \frac {5}{4} \\\frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4} & 0 \end {matrix} \middle | {\frac {4 e^{- 2 i \pi }}{9 x^{2}}} \right )}}{24 \pi ^{\frac {3}{2}}} - \frac {\sqrt {6} i {G_{6, 6}^{3, 5}\left (\begin {matrix} - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4} & 1 \\0, \frac {1}{2}, 0 & - \frac {1}{4}, \frac {1}{4}, \frac {1}{4} \end {matrix} \middle | {\frac {4}{9 x^{2}}} \right )}}{24 \pi ^{\frac {3}{2}}} \]
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\[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-x} \sqrt {2+3 x}} \, dx=\int { \frac {1}{\sqrt {-x} \sqrt {3 \, x + 2} \sqrt {-3 \, x + 2}} \,d x } \]
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\[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-x} \sqrt {2+3 x}} \, dx=\int { \frac {1}{\sqrt {-x} \sqrt {3 \, x + 2} \sqrt {-3 \, x + 2}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-x} \sqrt {2+3 x}} \, dx=\int \frac {1}{\sqrt {-x}\,\sqrt {2-3\,x}\,\sqrt {3\,x+2}} \,d x \]
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