Optimal. Leaf size=61 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{3 \sqrt {6}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{3 \sqrt {6}} \]
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Rubi [A] time = 0.02, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {442} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{3 \sqrt {6}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{3 \sqrt {6}} \]
Antiderivative was successfully verified.
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Rule 442
Rubi steps
\begin {align*} \int \frac {x^2}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx &=\frac {\tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )}{3 \sqrt {6}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )}{3 \sqrt {6}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 52, normalized size = 0.85 \[ -\frac {x^3 \left (1-3 x^2\right )^{3/4} F_1\left (\frac {3}{2};\frac {3}{4},1;\frac {5}{2};3 x^2,\frac {3 x^2}{2}\right )}{6 \left (3 x^2-1\right )^{3/4}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.81, size = 104, normalized size = 1.70 \[ -\frac {1}{18} \, \sqrt {6} \arctan \left (\frac {\sqrt {6} {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}}{3 \, x}\right ) + \frac {1}{36} \, \sqrt {6} \log \left (-\frac {9 \, x^{4} - 6 \, \sqrt {6} {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} x^{3} + 12 \, \sqrt {3 \, x^{2} - 1} x^{2} - 4 \, \sqrt {6} {\left (3 \, x^{2} - 1\right )}^{\frac {3}{4}} x + 12 \, x^{2} - 4}{9 \, x^{4} - 12 \, x^{2} + 4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{{\left (3 \, x^{2} - 1\right )}^{\frac {3}{4}} {\left (3 \, x^{2} - 2\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.00, size = 137, normalized size = 2.25 \[ -\frac {\RootOf \left (\textit {\_Z}^{2}-6\right ) \ln \left (\frac {3 \sqrt {3 x^{2}-1}\, x +3 x +\left (3 x^{2}-1\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{2}-6\right )+\left (3 x^{2}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}-6\right )}{3 x^{2}-2}\right )}{18}+\frac {\RootOf \left (\textit {\_Z}^{2}+6\right ) \ln \left (\frac {3 \sqrt {3 x^{2}-1}\, x -3 x +\left (3 x^{2}-1\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{2}+6\right )-\left (3 x^{2}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+6\right )}{3 x^{2}-2}\right )}{18} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{{\left (3 \, x^{2} - 1\right )}^{\frac {3}{4}} {\left (3 \, x^{2} - 2\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^2}{{\left (3\,x^2-1\right )}^{3/4}\,\left (3\,x^2-2\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (3 x^{2} - 2\right ) \left (3 x^{2} - 1\right )^{\frac {3}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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