Optimal. Leaf size=14 \[ 2 F\left (\left .\sin ^{-1}\left (\frac {1}{\sqrt {3-x}}\right )\right |2\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {119}
\begin {gather*} 2 F\left (\left .\text {ArcSin}\left (\frac {1}{\sqrt {3-x}}\right )\right |2\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 119
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-x} \sqrt {2-x} \sqrt {3-x}} \, dx &=2 F\left (\left .\sin ^{-1}\left (\frac {1}{\sqrt {3-x}}\right )\right |2\right )\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.00, size = 67, normalized size = 4.79 \begin {gather*} \frac {2 i \sqrt {\frac {-3+x}{-1+x}} \sqrt {\frac {-2+x}{-1+x}} (-1+x) F\left (\left .i \sinh ^{-1}\left (\frac {1}{\sqrt {1-x}}\right )\right |2\right )}{\sqrt {2-x} \sqrt {3-x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(54\) vs.
\(2(13)=26\).
time = 0.11, size = 55, normalized size = 3.93
method | result | size |
default | \(-\frac {\EllipticF \left (\sqrt {3-x}, \frac {\sqrt {2}}{2}\right ) \sqrt {-2+x}\, \sqrt {-2+2 x}\, \sqrt {2-x}\, \sqrt {2}\, \sqrt {2-2 x}}{2 \left (x^{2}-3 x +2\right )}\) | \(55\) |
elliptic | \(-\frac {\sqrt {-\left (-1+x \right ) \left (-2+x \right ) \left (-3+x \right )}\, \sqrt {-2+2 x}\, \sqrt {-2+x}\, \EllipticF \left (\sqrt {3-x}, \frac {\sqrt {2}}{2}\right )}{\sqrt {1-x}\, \sqrt {2-x}\, \sqrt {-x^{3}+6 x^{2}-11 x +6}}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 12.28, size = 66, normalized size = 4.71 \begin {gather*} \frac {{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 {e^{- 2 i \pi }}{\left (x - 2\right )^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} - \frac {{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 {1}{\left (x - 2\right )^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.07 \begin {gather*} \int \frac {1}{\sqrt {1-x}\,\sqrt {2-x}\,\sqrt {3-x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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