Optimal. Leaf size=41 \[ -\frac {2 \sqrt {x+2} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {1}{\sqrt {\frac {x}{3}+\frac {2}{3}}}\right ),\frac {5}{3}\right )}{\sqrt {3} \sqrt {-x-2}} \]
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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 = {121, 118} \[ -\frac {2 \sqrt {x+2} F\left (\sin ^{-1}\left (\frac {1}{\sqrt {\frac {x}{3}+\frac {2}{3}}}\right )|\frac {5}{3}\right )}{\sqrt {3} \sqrt {-x-2}} \]
Antiderivative was successfully verified.
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Rule 118
Rule 121
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-2-x} \sqrt {-3+x} \sqrt {-1+x}} \, dx &=\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*}
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Mathematica [C] time = 0.13, size = 72, normalized size = 1.76 \[ \frac {2 i \sqrt {\frac {x-3}{x-1}} \sqrt {\frac {x-1}{x+2}} \operatorname {EllipticF}\left (i \sinh ^{-1}\left (\frac {\sqrt {3}}{\sqrt {-x-2}}\right ),\frac {5}{3}\right )}{\sqrt {3} \sqrt {\frac {x-3}{x+2}}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {x - 1} \sqrt {x - 3} \sqrt {-x - 2}}{x^{3} - 2 \, x^{2} - 5 \, x + 6}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {x - 1} \sqrt {x - 3} \sqrt {-x - 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 72, normalized size = 1.76 \[ -\frac {2 \sqrt {-x -2}\, \sqrt {x -3}\, \sqrt {x -1}\, \sqrt {x +2}\, \sqrt {3}\, \sqrt {-x +1}\, \sqrt {-x +3}\, \EllipticF \left (\frac {\sqrt {5 x +10}}{5}, \frac {\sqrt {15}}{3}\right )}{3 \left (x^{3}-2 x^{2}-5 x +6\right )} \]
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 \[ \int \frac {1}{\sqrt {x - 1} \sqrt {x - 3} \sqrt {-x - 2}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {- x - 2} \sqrt {x - 3} \sqrt {x - 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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