Optimal. Leaf size=129 \[ \frac {\tanh ^{-1}\left (\frac {2\ 2^{3/4}-2 \sqrt [4]{2} \sqrt {3 x^2+2}}{2 \sqrt {3} x \sqrt [4]{3 x^2+2}}\right )}{3 \sqrt [4]{2} \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {2 \sqrt [4]{2} \sqrt {3 x^2+2}+2\ 2^{3/4}}{2 \sqrt {3} x \sqrt [4]{3 x^2+2}}\right )}{3 \sqrt [4]{2} \sqrt {3}} \]
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Rubi [A] time = 0.02, antiderivative size = 129, 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 = {441} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {2\ 2^{3/4}-2 \sqrt [4]{2} \sqrt {3 x^2+2}}{2 \sqrt {3} x \sqrt [4]{3 x^2+2}}\right )}{3 \sqrt [4]{2} \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {2 \sqrt [4]{2} \sqrt {3 x^2+2}+2\ 2^{3/4}}{2 \sqrt {3} x \sqrt [4]{3 x^2+2}}\right )}{3 \sqrt [4]{2} \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 441
Rubi steps
\begin {align*} \int \frac {x^2}{\left (2+3 x^2\right )^{3/4} \left (4+3 x^2\right )} \, dx &=-\frac {\tan ^{-1}\left (\frac {2\ 2^{3/4}+2 \sqrt [4]{2} \sqrt {2+3 x^2}}{2 \sqrt {3} x \sqrt [4]{2+3 x^2}}\right )}{3 \sqrt [4]{2} \sqrt {3}}+\frac {\tanh ^{-1}\left (\frac {2\ 2^{3/4}-2 \sqrt [4]{2} \sqrt {2+3 x^2}}{2 \sqrt {3} x \sqrt [4]{2+3 x^2}}\right )}{3 \sqrt [4]{2} \sqrt {3}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 37, normalized size = 0.29 \begin {gather*} \frac {x^3 F_1\left (\frac {3}{2};\frac {3}{4},1;\frac {5}{2};-\frac {3 x^2}{2},-\frac {3 x^2}{4}\right )}{12\ 2^{3/4}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 2.18, size = 136, normalized size = 1.05 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\frac {\sqrt [4]{2} \sqrt {3 x^2+2}}{\sqrt {3}}-\frac {\sqrt {3} x^2}{2 \sqrt [4]{2}}}{x \sqrt [4]{3 x^2+2}}\right )}{6 \sqrt [4]{2} \sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {2 \sqrt [4]{2} \sqrt {3} x \sqrt [4]{3 x^2+2}}{3 x^2+2 \sqrt {2} \sqrt {3 x^2+2}}\right )}{6 \sqrt [4]{2} \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.08, size = 282, normalized size = 2.19 \begin {gather*} \frac {1}{216} \cdot 72^{\frac {3}{4}} \sqrt {2} \arctan \left (\frac {72^{\frac {1}{4}} \sqrt {6} \sqrt {2} x \sqrt {\frac {72^{\frac {3}{4}} \sqrt {2} {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}} x + 18 \, \sqrt {2} x^{2} + 24 \, \sqrt {3 \, x^{2} + 2}}{x^{2}}} - 12 \cdot 72^{\frac {1}{4}} \sqrt {2} {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}} - 36 \, x}{36 \, x}\right ) + \frac {1}{216} \cdot 72^{\frac {3}{4}} \sqrt {2} \arctan \left (\frac {72^{\frac {1}{4}} \sqrt {6} \sqrt {2} x \sqrt {-\frac {72^{\frac {3}{4}} \sqrt {2} {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}} x - 18 \, \sqrt {2} x^{2} - 24 \, \sqrt {3 \, x^{2} + 2}}{x^{2}}} - 12 \cdot 72^{\frac {1}{4}} \sqrt {2} {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}} + 36 \, x}{36 \, x}\right ) - \frac {1}{864} \cdot 72^{\frac {3}{4}} \sqrt {2} \log \left (\frac {96 \, {\left (72^{\frac {3}{4}} \sqrt {2} {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}} x + 18 \, \sqrt {2} x^{2} + 24 \, \sqrt {3 \, x^{2} + 2}\right )}}{x^{2}}\right ) + \frac {1}{864} \cdot 72^{\frac {3}{4}} \sqrt {2} \log \left (-\frac {96 \, {\left (72^{\frac {3}{4}} \sqrt {2} {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}} x - 18 \, \sqrt {2} x^{2} - 24 \, \sqrt {3 \, x^{2} + 2}\right )}}{x^{2}}\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} + 4\right )} {\left (3 \, x^{2} + 2\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 = 7.44, size = 186, normalized size = 1.44 \begin {gather*} -\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+18\right )^{2}\right ) \ln \left (\frac {3 x \RootOf \left (\textit {\_Z}^{4}+18\right )^{2}+\left (3 x^{2}+2\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+18\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+18\right )^{2}+9 \sqrt {3 x^{2}+2}\, x -6 \left (3 x^{2}+2\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+18\right )^{2}\right )}{3 x^{2}+4}\right )}{18}-\frac {\RootOf \left (\textit {\_Z}^{4}+18\right ) \ln \left (-\frac {3 x \RootOf \left (\textit {\_Z}^{4}+18\right )^{2}+\left (3 x^{2}+2\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{4}+18\right )^{3}-9 \sqrt {3 x^{2}+2}\, x +6 \left (3 x^{2}+2\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+18\right )}{3 x^{2}+4}\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} + 4\right )} {\left (3 \, x^{2} + 2\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 = -0.01 \begin {gather*} \int \frac {x^2}{{\left (3\,x^2+2\right )}^{3/4}\,\left (3\,x^2+4\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}}{\left (3 x^{2} + 2\right )^{\frac {3}{4}} \left (3 x^{2} + 4\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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