Optimal. Leaf size=57 \[ -\frac {\sqrt {x+3} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {2}{\sqrt {x+3}}\right ),\frac {5}{4}\right )}{\sqrt {-x-3}}-\frac {i K\left (-\frac {1}{4}\right ) \sqrt {x+3}}{\sqrt {-x-3}} \]
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Rubi [A] time = 0.01, antiderivative size = 36, normalized size of antiderivative = 0.63, 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 = {121, 118} \[ -\frac {\sqrt {x+3} F\left (\sin ^{-1}\left (\frac {1}{\sqrt {\frac {x}{4}+\frac {3}{4}}}\right )|\frac {5}{4}\right )}{\sqrt {-x-3}} \]
Warning: Unable to verify antiderivative.
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Rule 118
Rule 121
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-3-x} \sqrt {-2+x} \sqrt {-1+x}} \, dx &=\frac {\sqrt {3+x} \int \frac {1}{\sqrt {\frac {3}{4}+\frac {x}{4}} \sqrt {-2+x} \sqrt {-1+x}} \, dx}{2 \sqrt {-3-x}}\\ &=-\frac {\sqrt {3+x} F\left (\sin ^{-1}\left (\frac {1}{\sqrt {\frac {3}{4}+\frac {x}{4}}}\right )|\frac {5}{4}\right )}{\sqrt {-3-x}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 63, normalized size = 1.11 \[ \frac {i \sqrt {\frac {x-2}{x-1}} \sqrt {\frac {x-1}{x+3}} \operatorname {EllipticF}\left (i \sinh ^{-1}\left (\frac {2}{\sqrt {-x-3}}\right ),\frac {5}{4}\right )}{\sqrt {\frac {x-2}{x+3}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {x - 1} \sqrt {x - 2} \sqrt {-x - 3}}{x^{3} - 7 \, x + 6}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {x - 1} \sqrt {x - 2} \sqrt {-x - 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 65, normalized size = 1.14 \[ \frac {\sqrt {-x -3}\, \sqrt {x -2}\, \sqrt {x -1}\, \sqrt {x +3}\, \sqrt {-x +1}\, \sqrt {-x +2}\, \EllipticF \left (\frac {\sqrt {5 x +15}}{5}, \frac {\sqrt {5}}{2}\right )}{-x^{3}+7 x -6} \]
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 \[ \int \frac {1}{\sqrt {x - 1} \sqrt {x - 2} \sqrt {-x - 3}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {x-1}\,\sqrt {x-2}\,\sqrt {-x-3}} \,d x \]
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 \[ \int \frac {1}{\sqrt {- x - 3} \sqrt {x - 2} \sqrt {x - 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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