Optimal. Leaf size=39 \[ -\frac{1}{6} \sqrt{2-3 x^2} \sqrt{3 x^2-1}-\frac{1}{12} \sin ^{-1}\left (3-6 x^2\right ) \]
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Rubi [A] time = 0.0310672, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {444, 50, 53, 619, 216} \[ -\frac{1}{6} \sqrt{2-3 x^2} \sqrt{3 x^2-1}-\frac{1}{12} \sin ^{-1}\left (3-6 x^2\right ) \]
Antiderivative was successfully verified.
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Rule 444
Rule 50
Rule 53
Rule 619
Rule 216
Rubi steps
\begin{align*} \int \frac{x \sqrt{-1+3 x^2}}{\sqrt{2-3 x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{-1+3 x}}{\sqrt{2-3 x}} \, dx,x,x^2\right )\\ &=-\frac{1}{6} \sqrt{2-3 x^2} \sqrt{-1+3 x^2}+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{\sqrt{2-3 x} \sqrt{-1+3 x}} \, dx,x,x^2\right )\\ &=-\frac{1}{6} \sqrt{2-3 x^2} \sqrt{-1+3 x^2}+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-2+9 x-9 x^2}} \, dx,x,x^2\right )\\ &=-\frac{1}{6} \sqrt{2-3 x^2} \sqrt{-1+3 x^2}-\frac{1}{36} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{9}}} \, dx,x,9 \left (1-2 x^2\right )\right )\\ &=-\frac{1}{6} \sqrt{2-3 x^2} \sqrt{-1+3 x^2}-\frac{1}{12} \sin ^{-1}\left (3-6 x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0123984, size = 37, normalized size = 0.95 \[ \frac{1}{6} \left (-\sqrt{-9 x^4+9 x^2-2}-\sin ^{-1}\left (\sqrt{2-3 x^2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 60, normalized size = 1.5 \begin{align*}{\frac{1}{12}\sqrt{-3\,{x}^{2}+2}\sqrt{3\,{x}^{2}-1} \left ( -2\,\sqrt{-9\,{x}^{4}+9\,{x}^{2}-2}+\arcsin \left ( 6\,{x}^{2}-3 \right ) \right ){\frac{1}{\sqrt{-9\,{x}^{4}+9\,{x}^{2}-2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47783, size = 36, normalized size = 0.92 \begin{align*} -\frac{1}{6} \, \sqrt{-9 \, x^{4} + 9 \, x^{2} - 2} + \frac{1}{12} \, \arcsin \left (6 \, x^{2} - 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.80314, size = 166, normalized size = 4.26 \begin{align*} -\frac{1}{6} \, \sqrt{3 \, x^{2} - 1} \sqrt{-3 \, x^{2} + 2} - \frac{1}{12} \, \arctan \left (\frac{3 \, \sqrt{3 \, x^{2} - 1}{\left (2 \, x^{2} - 1\right )} \sqrt{-3 \, x^{2} + 2}}{2 \,{\left (9 \, x^{4} - 9 \, x^{2} + 2\right )}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.1469, size = 66, normalized size = 1.69 \begin{align*} \frac{\begin{cases} - \frac{\sqrt{2 - 3 x^{2}} \sqrt{3 x^{2} - 1}}{2} + \frac{\operatorname{asin}{\left (\sqrt{3 x^{2} - 1} \right )}}{2} & \text{for}\: \left (x \geq \frac{\sqrt{3}}{3} \wedge x < \frac{\sqrt{6}}{3}\right ) \vee \left (x \leq - \frac{\sqrt{3}}{3} \wedge x > - \frac{\sqrt{6}}{3}\right ) \end{cases}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16384, size = 45, normalized size = 1.15 \begin{align*} -\frac{1}{6} \, \sqrt{3 \, x^{2} - 1} \sqrt{-3 \, x^{2} + 2} + \frac{1}{6} \, \arcsin \left (\sqrt{3 \, x^{2} - 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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