Optimal. Leaf size=41 \[ -\frac {2 \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {1}{\sqrt {\frac {1}{3}+\frac {x}{3}}}\right )|\frac {4}{3}\right )}{\sqrt {3} \sqrt {-1-x}} \]
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Rubi [A]
time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {122, 119}
\begin {gather*} -\frac {2 \sqrt {x+1} F\left (\text {ArcSin}\left (\frac {1}{\sqrt {\frac {x}{3}+\frac {1}{3}}}\right )|\frac {4}{3}\right )}{\sqrt {3} \sqrt {-x-1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 119
Rule 122
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-1-x} \sqrt {-3+x} \sqrt {-2+x}} \, dx &=\frac {\sqrt {1+x} \int \frac {1}{\sqrt {\frac {1}{3}+\frac {x}{3}} \sqrt {-3+x} \sqrt {-2+x}} \, dx}{\sqrt {3} \sqrt {-1-x}}\\ &=-\frac {2 \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {1}{\sqrt {\frac {1}{3}+\frac {x}{3}}}\right )|\frac {4}{3}\right )}{\sqrt {3} \sqrt {-1-x}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.65, size = 72, normalized size = 1.76 \begin {gather*} \frac {2 i \sqrt {\frac {-3+x}{-2+x}} \sqrt {\frac {-2+x}{1+x}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {3}}{\sqrt {-1-x}}\right )|\frac {4}{3}\right )}{\sqrt {3} \sqrt {\frac {-3+x}{1+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. \(67\) vs.
\(2(32)=64\).
time = 0.11, size = 68, normalized size = 1.66
method | result | size |
default | \(-\frac {2 \sqrt {-1-x}\, \sqrt {-3+x}\, \sqrt {-2+x}\, \sqrt {1+x}\, \sqrt {3}\, \sqrt {2-x}\, \sqrt {3-x}\, \EllipticF \left (\frac {\sqrt {1+x}}{2}, \frac {2 \sqrt {3}}{3}\right )}{3 \left (x^{3}-4 x^{2}+x +6\right )}\) | \(68\) |
elliptic | \(\frac {2 \sqrt {-\left (-2+x \right ) \left (-3+x \right ) \left (1+x \right )}\, \sqrt {1+x}\, \sqrt {6-3 x}\, \sqrt {3-x}\, \EllipticF \left (\frac {\sqrt {1+x}}{2}, \frac {2 \sqrt {3}}{3}\right )}{3 \sqrt {-1-x}\, \sqrt {-3+x}\, \sqrt {-2+x}\, \sqrt {-x^{3}+4 x^{2}-x -6}}\) | \(82\) |
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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {- x - 1} \sqrt {x - 3} \sqrt {x - 2}}\, dx \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 = -0.02 \begin {gather*} \int \frac {1}{\sqrt {-x-1}\,\sqrt {x-2}\,\sqrt {x-3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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