Optimal. Leaf size=57 \[ -\frac{2 \sqrt{x+1} \sqrt{x+3} \text{EllipticF}\left (\sin ^{-1}\left (\frac{1}{\sqrt{\frac{x}{5}+\frac{3}{5}}}\right ),\frac{2}{5}\right )}{\sqrt{5} \sqrt{-x-3} \sqrt{-x-1}} \]
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Rubi [A] time = 0.0190624, 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} \[ -\frac{2 \sqrt{x+1} \sqrt{x+3} F\left (\sin ^{-1}\left (\frac{1}{\sqrt{\frac{x}{5}+\frac{3}{5}}}\right )|\frac{2}{5}\right )}{\sqrt{5} \sqrt{-x-3} \sqrt{-x-1}} \]
Antiderivative was successfully verified.
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Rule 121
Rule 118
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-3-x} \sqrt{-1-x} \sqrt{-2+x}} \, dx &=\frac{\sqrt{3+x} \int \frac{1}{\sqrt{-1-x} \sqrt{\frac{3}{5}+\frac{x}{5}} \sqrt{-2+x}} \, dx}{\sqrt{5} \sqrt{-3-x}}\\ &=\frac{\left (\sqrt{1+x} \sqrt{3+x}\right ) \int \frac{1}{\sqrt{\frac{3}{5}+\frac{x}{5}} \sqrt{\frac{1}{3}+\frac{x}{3}} \sqrt{-2+x}} \, dx}{\sqrt{15} \sqrt{-3-x} \sqrt{-1-x}}\\ &=-\frac{2 \sqrt{1+x} \sqrt{3+x} F\left (\sin ^{-1}\left (\frac{1}{\sqrt{\frac{3}{5}+\frac{x}{5}}}\right )|\frac{2}{5}\right )}{\sqrt{5} \sqrt{-3-x} \sqrt{-1-x}}\\ \end{align*}
Mathematica [C] time = 0.0493458, size = 75, normalized size = 1.32 \[ \frac{2 i \sqrt{\frac{3}{x-2}+1} \sqrt{\frac{5}{x-2}+1} (x-2) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{3}}{\sqrt{x-2}}\right ),\frac{5}{3}\right )}{\sqrt{-3 (x-2)-15} \sqrt{-x-1}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.032, size = 54, normalized size = 1. \begin{align*}{\frac{2\,\sqrt{3}}{3\,{x}^{2}+3\,x-18}{\it EllipticF} \left ({\frac{1}{2}\sqrt{-2-2\,x}},{\frac{i}{3}}\sqrt{6} \right ) \sqrt{3+x}\sqrt{2-x}\sqrt{-2+x}\sqrt{-3-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 - 2} \sqrt{-x - 1} \sqrt{-x - 3}}\,{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 - 2} \sqrt{-x - 1} \sqrt{-x - 3}}{x^{3} + 2 \, x^{2} - 5 \, 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 - 3} \sqrt{- x - 1} \sqrt{x - 2}}\, 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 - 2} \sqrt{-x - 1} \sqrt{-x - 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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