Optimal. Leaf size=70 \[ -\frac{\text{EllipticF}\left (\cos ^{-1}\left (\sqrt{\frac{3}{2}} x\right ),2\right )}{9 \sqrt{3}}-\frac{1}{9} \sqrt{2-3 x^2} \sqrt{3 x^2-1} x-\frac{E\left (\left .\cos ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{3 \sqrt{3}} \]
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Rubi [A] time = 0.0455551, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {478, 524, 425, 420} \[ -\frac{1}{9} \sqrt{2-3 x^2} \sqrt{3 x^2-1} x-\frac{F\left (\left .\cos ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{9 \sqrt{3}}-\frac{E\left (\left .\cos ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 478
Rule 524
Rule 425
Rule 420
Rubi steps
\begin{align*} \int \frac{x^2 \sqrt{-1+3 x^2}}{\sqrt{2-3 x^2}} \, dx &=-\frac{1}{9} x \sqrt{2-3 x^2} \sqrt{-1+3 x^2}+\frac{1}{9} \int \frac{-2+9 x^2}{\sqrt{2-3 x^2} \sqrt{-1+3 x^2}} \, dx\\ &=-\frac{1}{9} x \sqrt{2-3 x^2} \sqrt{-1+3 x^2}+\frac{1}{9} \int \frac{1}{\sqrt{2-3 x^2} \sqrt{-1+3 x^2}} \, dx+\frac{1}{3} \int \frac{\sqrt{-1+3 x^2}}{\sqrt{2-3 x^2}} \, dx\\ &=-\frac{1}{9} x \sqrt{2-3 x^2} \sqrt{-1+3 x^2}-\frac{E\left (\left .\cos ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{3 \sqrt{3}}-\frac{F\left (\left .\cos ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{9 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0710557, size = 86, normalized size = 1.23 \[ \frac{\sqrt{3-9 x^2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right ),2\right )+3 x \sqrt{2-3 x^2} \left (1-3 x^2\right )-3 \sqrt{3-9 x^2} E\left (\left .\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{27 \sqrt{3 x^2-1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 129, normalized size = 1.8 \begin{align*} -{\frac{\sqrt{2}}{972\,{x}^{4}-972\,{x}^{2}+216}\sqrt{3\,{x}^{2}-1}\sqrt{-6\,{x}^{2}+4} \left ( 54\,{x}^{5}+\sqrt{2}\sqrt{3}\sqrt{-6\,{x}^{2}+4}\sqrt{-3\,{x}^{2}+1}{\it EllipticF} \left ({\frac{x\sqrt{2}\sqrt{3}}{2}},\sqrt{2} \right ) -3\,\sqrt{2}\sqrt{3}\sqrt{-6\,{x}^{2}+4}\sqrt{-3\,{x}^{2}+1}{\it EllipticE} \left ( 1/2\,x\sqrt{2}\sqrt{3},\sqrt{2} \right ) -54\,{x}^{3}+12\,x \right ) } \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{\sqrt{3 \, x^{2} - 1} x^{2}}{\sqrt{-3 \, x^{2} + 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{3 \, x^{2} - 1} \sqrt{-3 \, x^{2} + 2} x^{2}}{3 \, x^{2} - 2}, 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{x^{2} \sqrt{3 x^{2} - 1}}{\sqrt{2 - 3 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{\sqrt{3 \, x^{2} - 1} x^{2}}{\sqrt{-3 \, x^{2} + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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