Optimal. Leaf size=57 \[ -\frac{2 \sqrt{x+1} \sqrt{x+2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{1}{\sqrt{\frac{x}{5}+\frac{2}{5}}}\right ),\frac{1}{5}\right )}{\sqrt{5} \sqrt{-x-2} \sqrt{-x-1}} \]
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Rubi [A] time = 0.0189782, antiderivative size = 57, normalized size of antiderivative = 1., 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} \[ -\frac{2 \sqrt{x+1} \sqrt{x+2} F\left (\sin ^{-1}\left (\frac{1}{\sqrt{\frac{x}{5}+\frac{2}{5}}}\right )|\frac{1}{5}\right )}{\sqrt{5} \sqrt{-x-2} \sqrt{-x-1}} \]
Antiderivative was successfully verified.
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Rule 121
Rule 118
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*}
Mathematica [C] time = 0.0498898, size = 69, normalized size = 1.21 \[ \frac{i \sqrt{\frac{4}{x-3}+1} \sqrt{\frac{5}{x-3}+1} (x-3) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{2}{\sqrt{x-3}}\right ),\frac{5}{4}\right )}{\sqrt{-x-2} \sqrt{-x-1}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.03, size = 46, normalized size = 0.8 \begin{align*}{\frac{1}{{x}^{2}-x-6}{\it EllipticF} \left ( \sqrt{-1-x},{\frac{i}{2}} \right ) \sqrt{2+x}\sqrt{3-x}\sqrt{-3+x}\sqrt{-2-x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x - 3} \sqrt{-x - 1} \sqrt{-x - 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{x - 3} \sqrt{-x - 1} \sqrt{-x - 2}}{x^{3} - 7 \, x - 6}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- x - 2} \sqrt{- x - 1} \sqrt{x - 3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x - 3} \sqrt{-x - 1} \sqrt{-x - 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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