Optimal. Leaf size=57 \[ -\frac {2 \sqrt {1+x} \sqrt {2+x} F\left (\sin ^{-1}\left (\frac {1}{\sqrt {\frac {2}{5}+\frac {x}{5}}}\right )|\frac {1}{5}\right )}{\sqrt {5} \sqrt {-2-x} \sqrt {-1-x}} \]
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Rubi [A]
time = 0.01, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {122, 119}
\begin {gather*} -\frac {2 \sqrt {x+1} \sqrt {x+2} F\left (\text {ArcSin}\left (\frac {1}{\sqrt {\frac {x}{5}+\frac {2}{5}}}\right )|\frac {1}{5}\right )}{\sqrt {5} \sqrt {-x-2} \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 {-2-x} \sqrt {-1-x} \sqrt {-3+x}} \, dx &=\frac {\sqrt {2+x} \int \frac {1}{\sqrt {-1-x} \sqrt {\frac {2}{5}+\frac {x}{5}} \sqrt {-3+x}} \, dx}{\sqrt {5} \sqrt {-2-x}}\\ &=\frac {\left (\sqrt {1+x} \sqrt {2+x}\right ) \int \frac {1}{\sqrt {\frac {2}{5}+\frac {x}{5}} \sqrt {\frac {1}{4}+\frac {x}{4}} \sqrt {-3+x}} \, dx}{2 \sqrt {5} \sqrt {-2-x} \sqrt {-1-x}}\\ &=-\frac {2 \sqrt {1+x} \sqrt {2+x} F\left (\sin ^{-1}\left (\frac {1}{\sqrt {\frac {2}{5}+\frac {x}{5}}}\right )|\frac {1}{5}\right )}{\sqrt {5} \sqrt {-2-x} \sqrt {-1-x}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.51, size = 69, normalized size = 1.21 \begin {gather*} \frac {i \sqrt {1+\frac {4}{-3+x}} \sqrt {1+\frac {5}{-3+x}} (-3+x) F\left (i \sinh ^{-1}\left (\frac {2}{\sqrt {-3+x}}\right )|\frac {5}{4}\right )}{\sqrt {-2-x} \sqrt {-1-x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 51, normalized size = 0.89
method | result | size |
default | \(-\frac {2 \EllipticF \left (\sqrt {2+x}, \frac {\sqrt {5}}{5}\right ) \sqrt {5}\, \sqrt {3-x}\, \sqrt {2+x}\, \sqrt {-3+x}\, \sqrt {-2-x}}{5 \left (x^{2}-x -6\right )}\) | \(51\) |
elliptic | \(\frac {2 \sqrt {\left (-3+x \right ) \left (1+x \right ) \left (2+x \right )}\, \sqrt {2+x}\, \sqrt {15-5 x}\, \EllipticF \left (\sqrt {2+x}, \frac {\sqrt {5}}{5}\right )}{5 \sqrt {-2-x}\, \sqrt {-3+x}\, \sqrt {x^{3}-7 x -6}}\) | \(60\) |
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 [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.22, size = 6, normalized size = 0.11 \begin {gather*} 2 \, {\rm weierstrassPInverse}\left (28, 24, x\right ) \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 - 2} \sqrt {- x - 1} \sqrt {x - 3}}\, 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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