Integrand size = 24, antiderivative size = 41 \[ \int \frac {1}{\sqrt {-2-x} \sqrt {-3+x} \sqrt {-1+x}} \, dx=-\frac {2 \sqrt {2+x} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{\sqrt {\frac {2}{3}+\frac {x}{3}}}\right ),\frac {5}{3}\right )}{\sqrt {3} \sqrt {-2-x}} \]
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Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {122, 119} \[ \int \frac {1}{\sqrt {-2-x} \sqrt {-3+x} \sqrt {-1+x}} \, dx=-\frac {2 \sqrt {x+2} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{\sqrt {\frac {x}{3}+\frac {2}{3}}}\right ),\frac {5}{3}\right )}{\sqrt {3} \sqrt {-x-2}} \]
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Rule 119
Rule 122
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {2+x} \int \frac {1}{\sqrt {\frac {2}{3}+\frac {x}{3}} \sqrt {-3+x} \sqrt {-1+x}} \, dx}{\sqrt {3} \sqrt {-2-x}} \\ & = -\frac {2 \sqrt {2+x} F\left (\sin ^{-1}\left (\frac {1}{\sqrt {\frac {2}{3}+\frac {x}{3}}}\right )|\frac {5}{3}\right )}{\sqrt {3} \sqrt {-2-x}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 2.30 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.76 \[ \int \frac {1}{\sqrt {-2-x} \sqrt {-3+x} \sqrt {-1+x}} \, dx=\frac {2 i \sqrt {\frac {-3+x}{-1+x}} \sqrt {\frac {-1+x}{2+x}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {3}}{\sqrt {-2-x}}\right ),\frac {5}{3}\right )}{\sqrt {3} \sqrt {\frac {-3+x}{2+x}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(71\) vs. \(2(32)=64\).
Time = 4.51 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.76
method | result | size |
default | \(-\frac {2 \sqrt {-2-x}\, \sqrt {-3+x}\, \sqrt {-1+x}\, \sqrt {2+x}\, \sqrt {3}\, \sqrt {1-x}\, \sqrt {3-x}\, F\left (\frac {\sqrt {10+5 x}}{5}, \frac {\sqrt {15}}{3}\right )}{3 \left (x^{3}-2 x^{2}-5 x +6\right )}\) | \(72\) |
elliptic | \(\frac {2 \sqrt {-\left (-1+x \right ) \left (-3+x \right ) \left (2+x \right )}\, \sqrt {10+5 x}\, \sqrt {3-3 x}\, \sqrt {15-5 x}\, F\left (\frac {\sqrt {10+5 x}}{5}, \frac {\sqrt {15}}{3}\right )}{15 \sqrt {-2-x}\, \sqrt {-3+x}\, \sqrt {-1+x}\, \sqrt {-x^{3}+2 x^{2}+5 x -6}}\) | \(86\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.20 \[ \int \frac {1}{\sqrt {-2-x} \sqrt {-3+x} \sqrt {-1+x}} \, dx=-2 i \, {\rm weierstrassPInverse}\left (\frac {76}{3}, -\frac {224}{27}, x - \frac {2}{3}\right ) \]
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\[ \int \frac {1}{\sqrt {-2-x} \sqrt {-3+x} \sqrt {-1+x}} \, dx=\int \frac {1}{\sqrt {- x - 2} \sqrt {x - 3} \sqrt {x - 1}}\, dx \]
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\[ \int \frac {1}{\sqrt {-2-x} \sqrt {-3+x} \sqrt {-1+x}} \, dx=\int { \frac {1}{\sqrt {x - 1} \sqrt {x - 3} \sqrt {-x - 2}} \,d x } \]
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\[ \int \frac {1}{\sqrt {-2-x} \sqrt {-3+x} \sqrt {-1+x}} \, dx=\int { \frac {1}{\sqrt {x - 1} \sqrt {x - 3} \sqrt {-x - 2}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {-2-x} \sqrt {-3+x} \sqrt {-1+x}} \, dx=\int \frac {1}{\sqrt {x-1}\,\sqrt {-x-2}\,\sqrt {x-3}} \,d x \]
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