Optimal. Leaf size=92 \[ -\frac {\sqrt {x+2} \sqrt {x+3} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {2}{\sqrt {x+3}}\right ),\frac {1}{4}\right )}{\sqrt {-x-3} \sqrt {-x-2}}-\frac {2 i K(4) \sqrt {x+2}}{\sqrt {-x-2}}+\frac {K\left (\frac {3}{4}\right ) \sqrt {x+3}}{\sqrt {-x-3}} \]
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Rubi [A] time = 0.02, antiderivative size = 52, normalized size of antiderivative = 0.57, 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 = {121, 118} \[ -\frac {\sqrt {x+2} \sqrt {x+3} F\left (\sin ^{-1}\left (\frac {1}{\sqrt {\frac {x}{4}+\frac {3}{4}}}\right )|\frac {1}{4}\right )}{\sqrt {-x-3} \sqrt {-x-2}} \]
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 {-2-x} \sqrt {\frac {3}{4}+\frac {x}{4}} \sqrt {-1+x}} \, dx}{2 \sqrt {-3-x}}\\ &=\frac {\left (\sqrt {2+x} \sqrt {3+x}\right ) \int \frac {1}{\sqrt {\frac {3}{4}+\frac {x}{4}} \sqrt {\frac {2}{3}+\frac {x}{3}} \sqrt {-1+x}} \, dx}{2 \sqrt {3} \sqrt {-3-x} \sqrt {-2-x}}\\ &=-\frac {\sqrt {2+x} \sqrt {3+x} F\left (\sin ^{-1}\left (\frac {1}{\sqrt {\frac {3}{4}+\frac {x}{4}}}\right )|\frac {1}{4}\right )}{\sqrt {-3-x} \sqrt {-2-x}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 75, normalized size = 0.82 \[ \frac {2 i \sqrt {\frac {3}{x-1}+1} \sqrt {\frac {4}{x-1}+1} (x-1) \operatorname {EllipticF}\left (i \sinh ^{-1}\left (\frac {\sqrt {3}}{\sqrt {x-1}}\right ),\frac {4}{3}\right )}{\sqrt {-3 (x-1)-12} \sqrt {-x-2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x - 1} \sqrt {-x - 2} \sqrt {-x - 3}}{x^{3} + 4 \, x^{2} + x - 6}, x\right ) \]
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 = 54, normalized size = 0.59 \[ \frac {2 \sqrt {x +3}\, \sqrt {3}\, \sqrt {-x +1}\, \sqrt {x -1}\, \sqrt {-x -3}\, \EllipticF \left (\sqrt {-x -2}, \frac {i \sqrt {3}}{3}\right )}{3 \left (x^{2}+2 x -3\right )} \]
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 \[ \int \frac {1}{\sqrt {x - 1} \sqrt {-x - 2} \sqrt {-x - 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {x-1}\,\sqrt {-x-2}\,\sqrt {-x-3}} \,d x \]
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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