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} \begin {gather*} \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}} \end {gather*}
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 = 0.10, size = 52, normalized size = 0.85 \begin {gather*} -\frac {x^3 \left (3 x^2+1\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}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 2.11, size = 79, normalized size = 1.30 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x \left (-3 x^2-1\right )^{3/4}}{3 x^2+1}\right )}{3 \sqrt {6}}-\frac {\tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x \left (-3 x^2-1\right )^{3/4}}{3 x^2+1}\right )}{3 \sqrt {6}} \end {gather*}
Antiderivative was successfully verified.
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fricas [C] time = 0.93, size = 115, normalized size = 1.89 \begin {gather*} -\frac {1}{36} \, \sqrt {6} \log \left (\frac {\sqrt {6} x + 2 \, {\left (-3 \, x^{2} - 1\right )}^{\frac {1}{4}}}{2 \, x}\right ) + \frac {1}{36} \, \sqrt {6} \log \left (-\frac {\sqrt {6} x - 2 \, {\left (-3 \, x^{2} - 1\right )}^{\frac {1}{4}}}{2 \, x}\right ) - \frac {1}{36} i \, \sqrt {6} \log \left (\frac {i \, \sqrt {6} x + 2 \, {\left (-3 \, x^{2} - 1\right )}^{\frac {1}{4}}}{2 \, x}\right ) + \frac {1}{36} i \, \sqrt {6} \log \left (\frac {-i \, \sqrt {6} x + 2 \, {\left (-3 \, x^{2} - 1\right )}^{\frac {1}{4}}}{2 \, x}\right ) \end {gather*}
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 \begin {gather*} \int -\frac {x^{2}}{{\left (3 \, x^{2} + 2\right )} {\left (-3 \, x^{2} - 1\right )}^{\frac {3}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.32, size = 138, normalized size = 2.26 \begin {gather*} \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} \end {gather*}
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 \begin {gather*} -\int \frac {x^{2}}{{\left (3 \, x^{2} + 2\right )} {\left (-3 \, x^{2} - 1\right )}^{\frac {3}{4}}}\,{d x} \end {gather*}
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 \begin {gather*} -\int \frac {x^2}{{\left (-3\,x^2-1\right )}^{3/4}\,\left (3\,x^2+2\right )} \,d x \end {gather*}
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 \begin {gather*} - \int \frac {x^{2}}{3 x^{2} \left (- 3 x^{2} - 1\right )^{\frac {3}{4}} + 2 \left (- 3 x^{2} - 1\right )^{\frac {3}{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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