Optimal. Leaf size=158 \[ \frac {2}{9} \sqrt [4]{2-3 x^2}+\frac {\log \left (\sqrt {2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2}\right )}{9 \sqrt [4]{2}}-\frac {\log \left (\sqrt {2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2}\right )}{9 \sqrt [4]{2}}-\frac {1}{9} 2^{3/4} \tan ^{-1}\left (\sqrt [4]{4-6 x^2}+1\right )+\frac {1}{9} 2^{3/4} \tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right ) \]
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Rubi [A] time = 0.15, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {443, 261, 444, 63, 211, 1165, 628, 1162, 617, 204} \[ \frac {2}{9} \sqrt [4]{2-3 x^2}+\frac {\log \left (\sqrt {2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2}\right )}{9 \sqrt [4]{2}}-\frac {\log \left (\sqrt {2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2}\right )}{9 \sqrt [4]{2}}-\frac {1}{9} 2^{3/4} \tan ^{-1}\left (\sqrt [4]{4-6 x^2}+1\right )+\frac {1}{9} 2^{3/4} \tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 204
Rule 211
Rule 261
Rule 443
Rule 444
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {x^3}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx &=\int \left (-\frac {x}{3 \left (2-3 x^2\right )^{3/4}}+\frac {4 x}{3 \left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )}\right ) \, dx\\ &=-\left (\frac {1}{3} \int \frac {x}{\left (2-3 x^2\right )^{3/4}} \, dx\right )+\frac {4}{3} \int \frac {x}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx\\ &=\frac {2}{9} \sqrt [4]{2-3 x^2}+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{(2-3 x)^{3/4} (4-3 x)} \, dx,x,x^2\right )\\ &=\frac {2}{9} \sqrt [4]{2-3 x^2}-\frac {8}{9} \operatorname {Subst}\left (\int \frac {1}{2+x^4} \, dx,x,\sqrt [4]{2-3 x^2}\right )\\ &=\frac {2}{9} \sqrt [4]{2-3 x^2}-\frac {1}{9} \left (2 \sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}-x^2}{2+x^4} \, dx,x,\sqrt [4]{2-3 x^2}\right )-\frac {1}{9} \left (2 \sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}+x^2}{2+x^4} \, dx,x,\sqrt [4]{2-3 x^2}\right )\\ &=\frac {2}{9} \sqrt [4]{2-3 x^2}+\frac {\operatorname {Subst}\left (\int \frac {2^{3/4}+2 x}{-\sqrt {2}-2^{3/4} x-x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )}{9 \sqrt [4]{2}}+\frac {\operatorname {Subst}\left (\int \frac {2^{3/4}-2 x}{-\sqrt {2}+2^{3/4} x-x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )}{9 \sqrt [4]{2}}-\frac {1}{9} \sqrt {2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {2}-2^{3/4} x+x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )-\frac {1}{9} \sqrt {2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {2}+2^{3/4} x+x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )\\ &=\frac {2}{9} \sqrt [4]{2-3 x^2}+\frac {\log \left (\sqrt {2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2-3 x^2}\right )}{9 \sqrt [4]{2}}-\frac {\log \left (\sqrt {2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2-3 x^2}\right )}{9 \sqrt [4]{2}}-\frac {1}{9} 2^{3/4} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt [4]{4-6 x^2}\right )+\frac {1}{9} 2^{3/4} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt [4]{4-6 x^2}\right )\\ &=\frac {2}{9} \sqrt [4]{2-3 x^2}-\frac {1}{9} 2^{3/4} \tan ^{-1}\left (1+\sqrt [4]{4-6 x^2}\right )+\frac {1}{9} 2^{3/4} \tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right )+\frac {\log \left (\sqrt {2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2-3 x^2}\right )}{9 \sqrt [4]{2}}-\frac {\log \left (\sqrt {2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2-3 x^2}\right )}{9 \sqrt [4]{2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 146, normalized size = 0.92 \[ \frac {1}{18} \left (4 \sqrt [4]{2-3 x^2}+2^{3/4} \log \left (\sqrt {2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2}\right )-2^{3/4} \log \left (\sqrt {2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt {2}\right )+2\ 2^{3/4} \tan ^{-1}\left (1-\sqrt [4]{4-6 x^2}\right )-2\ 2^{3/4} \tan ^{-1}\left (\sqrt [4]{4-6 x^2}+1\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 186, normalized size = 1.18 \[ \frac {2}{9} \cdot 2^{\frac {3}{4}} \arctan \left (2^{\frac {1}{4}} \sqrt {2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}} - 2^{\frac {1}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} - 1\right ) + \frac {2}{9} \cdot 2^{\frac {3}{4}} \arctan \left (2^{\frac {1}{4}} \sqrt {-2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}} - 2^{\frac {1}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + 1\right ) - \frac {1}{18} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) + \frac {1}{18} \cdot 2^{\frac {3}{4}} \log \left (-2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) + \frac {2}{9} \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.46, size = 129, normalized size = 0.82 \[ -\frac {1}{9} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{9} \cdot 2^{\frac {3}{4}} \arctan \left (-\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} - 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{18} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) + \frac {1}{18} \cdot 2^{\frac {3}{4}} \log \left (-2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) + \frac {2}{9} \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.39, size = 528, normalized size = 3.34 \[ -\frac {2 \left (3 x^{2}-2\right )}{9 \left (-3 x^{2}+2\right )^{\frac {3}{4}}}-\frac {\left (-\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) \ln \left (\frac {-27 x^{6}-18 \left (-27 x^{6}+54 x^{4}-36 x^{2}+8\right )^{\frac {1}{4}} x^{4} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right )+36 x^{4}+6 \sqrt {-27 x^{6}+54 x^{4}-36 x^{2}+8}\, x^{2} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2}+24 \left (-27 x^{6}+54 x^{4}-36 x^{2}+8\right )^{\frac {1}{4}} x^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right )+2 \left (-27 x^{6}+54 x^{4}-36 x^{2}+8\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+2\right )^{2}-12 x^{2}-4 \sqrt {-27 x^{6}+54 x^{4}-36 x^{2}+8}\, \RootOf \left (\textit {\_Z}^{4}+2\right )^{2}-8 \left (-27 x^{6}+54 x^{4}-36 x^{2}+8\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right )}{\left (3 x^{2}-4\right ) \left (3 x^{2}-2\right )^{2}}\right )}{9}+\frac {\RootOf \left (\textit {\_Z}^{4}+2\right ) \ln \left (\frac {-27 x^{6}+18 \left (-27 x^{6}+54 x^{4}-36 x^{2}+8\right )^{\frac {1}{4}} x^{4} \RootOf \left (\textit {\_Z}^{4}+2\right )+36 x^{4}-6 \sqrt {-27 x^{6}+54 x^{4}-36 x^{2}+8}\, x^{2} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2}-24 \left (-27 x^{6}+54 x^{4}-36 x^{2}+8\right )^{\frac {1}{4}} x^{2} \RootOf \left (\textit {\_Z}^{4}+2\right )+2 \left (-27 x^{6}+54 x^{4}-36 x^{2}+8\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{4}+2\right )^{3}-12 x^{2}+4 \sqrt {-27 x^{6}+54 x^{4}-36 x^{2}+8}\, \RootOf \left (\textit {\_Z}^{4}+2\right )^{2}+8 \left (-27 x^{6}+54 x^{4}-36 x^{2}+8\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+2\right )}{\left (3 x^{2}-4\right ) \left (3 x^{2}-2\right )^{2}}\right )}{9}\right ) \left (-\left (3 x^{2}-2\right )^{3}\right )^{\frac {1}{4}}}{\left (-3 x^{2}+2\right )^{\frac {3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.01, size = 129, normalized size = 0.82 \[ -\frac {1}{9} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{9} \cdot 2^{\frac {3}{4}} \arctan \left (-\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} - 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{18} \cdot 2^{\frac {3}{4}} \log \left (2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) + \frac {1}{18} \cdot 2^{\frac {3}{4}} \log \left (-2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) + \frac {2}{9} \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 60, normalized size = 0.38 \[ \frac {2\,{\left (2-3\,x^2\right )}^{1/4}}{9}+2^{3/4}\,\mathrm {atan}\left (2^{1/4}\,{\left (2-3\,x^2\right )}^{1/4}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{9}-\frac {1}{9}{}\mathrm {i}\right )+2^{3/4}\,\mathrm {atan}\left (2^{1/4}\,{\left (2-3\,x^2\right )}^{1/4}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{9}+\frac {1}{9}{}\mathrm {i}\right ) \]
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 \[ - \int \frac {x^{3}}{3 x^{2} \left (2 - 3 x^{2}\right )^{\frac {3}{4}} - 4 \left (2 - 3 x^{2}\right )^{\frac {3}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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