Optimal. Leaf size=48 \[ \frac{1}{3} \sqrt{-x^4-2 x^2+3} x+\frac{4 F\left (\sin ^{-1}(x)|-\frac{1}{3}\right )}{\sqrt{3}}-\frac{2 E\left (\sin ^{-1}(x)|-\frac{1}{3}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0428554, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {1988, 1091, 1180, 524, 424, 419} \[ \frac{1}{3} \sqrt{-x^4-2 x^2+3} x+\frac{4 F\left (\sin ^{-1}(x)|-\frac{1}{3}\right )}{\sqrt{3}}-\frac{2 E\left (\sin ^{-1}(x)|-\frac{1}{3}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1988
Rule 1091
Rule 1180
Rule 524
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \sqrt{\left (1-x^2\right ) \left (3+x^2\right )} \, dx &=\int \sqrt{3-2 x^2-x^4} \, dx\\ &=\frac{1}{3} x \sqrt{3-2 x^2-x^4}+\frac{1}{3} \int \frac{6-2 x^2}{\sqrt{3-2 x^2-x^4}} \, dx\\ &=\frac{1}{3} x \sqrt{3-2 x^2-x^4}+\frac{2}{3} \int \frac{6-2 x^2}{\sqrt{2-2 x^2} \sqrt{6+2 x^2}} \, dx\\ &=\frac{1}{3} x \sqrt{3-2 x^2-x^4}-\frac{2}{3} \int \frac{\sqrt{6+2 x^2}}{\sqrt{2-2 x^2}} \, dx+8 \int \frac{1}{\sqrt{2-2 x^2} \sqrt{6+2 x^2}} \, dx\\ &=\frac{1}{3} x \sqrt{3-2 x^2-x^4}-\frac{2 E\left (\sin ^{-1}(x)|-\frac{1}{3}\right )}{\sqrt{3}}+\frac{4 F\left (\sin ^{-1}(x)|-\frac{1}{3}\right )}{\sqrt{3}}\\ \end{align*}
Mathematica [C] time = 0.0571395, size = 59, normalized size = 1.23 \[ \frac{1}{3} \left (\sqrt{-x^4-2 x^2+3} x-4 i F\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{3}}\right )\right |-3\right )-2 i E\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{3}}\right )\right |-3\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.017, size = 114, normalized size = 2.4 \begin{align*}{\frac{x}{3}\sqrt{-{x}^{4}-2\,{x}^{2}+3}}+{\frac{2\,{\it EllipticF} \left ( x,i/3\sqrt{3} \right ) }{3}\sqrt{-{x}^{2}+1}\sqrt{3\,{x}^{2}+9}{\frac{1}{\sqrt{-{x}^{4}-2\,{x}^{2}+3}}}}+{\frac{2\,{\it EllipticF} \left ( x,i/3\sqrt{3} \right ) -2\,{\it EllipticE} \left ( x,i/3\sqrt{3} \right ) }{3}\sqrt{-{x}^{2}+1}\sqrt{3\,{x}^{2}+9}{\frac{1}{\sqrt{-{x}^{4}-2\,{x}^{2}+3}}}} \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 \sqrt{-{\left (x^{2} + 3\right )}{\left (x^{2} - 1\right )}}\,{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 (\sqrt{-x^{4} - 2 \, x^{2} + 3}, 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 \sqrt{\left (1 - x^{2}\right ) \left (x^{2} + 3\right )}\, 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 \sqrt{-{\left (x^{2} + 3\right )}{\left (x^{2} - 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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