Optimal. Leaf size=158 \[ \frac{2}{9} \sqrt [4]{2-3 x^2}+\frac{\log \left (\sqrt{2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )}{9 \sqrt [4]{2}}-\frac{\log \left (\sqrt{2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )}{9 \sqrt [4]{2}}-\frac{1}{9} 2^{3/4} \tan ^{-1}\left (\sqrt [4]{4-6 x^2}+1\right )+\frac{1}{9} 2^{3/4} \tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right ) \]
[Out]
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Rubi [A] time = 0.373465, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417 \[ \frac{2}{9} \sqrt [4]{2-3 x^2}+\frac{\log \left (\sqrt{2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )}{9 \sqrt [4]{2}}-\frac{\log \left (\sqrt{2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )}{9 \sqrt [4]{2}}-\frac{1}{9} 2^{3/4} \tan ^{-1}\left (\sqrt [4]{4-6 x^2}+1\right )+\frac{1}{9} 2^{3/4} \tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right ) \]
Antiderivative was successfully verified.
[In] Int[x^3/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 31.4117, size = 141, normalized size = 0.89 \[ \frac{2 \sqrt [4]{- 3 x^{2} + 2}}{9} + \frac{2^{\frac{3}{4}} \log{\left (- 2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2} + \sqrt{- 3 x^{2} + 2} + \sqrt{2} \right )}}{18} - \frac{2^{\frac{3}{4}} \log{\left (2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2} + \sqrt{- 3 x^{2} + 2} + \sqrt{2} \right )}}{18} - \frac{2^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{2} \sqrt [4]{- 3 x^{2} + 2} - 1 \right )}}{9} - \frac{2^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{2} \sqrt [4]{- 3 x^{2} + 2} + 1 \right )}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3/(-3*x**2+2)**(3/4)/(-3*x**2+4),x)
[Out]
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Mathematica [C] time = 0.0528609, size = 66, normalized size = 0.42 \[ -\frac{2 \left (-4 \left (\frac{2-3 x^2}{4-3 x^2}\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{2}{4-3 x^2}\right )+9 x^2-6\right )}{27 \left (2-3 x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[x^3/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]
[Out]
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Maple [F] time = 0.07, size = 0, normalized size = 0. \[ \int{\frac{{x}^{3}}{-3\,{x}^{2}+4} \left ( -3\,{x}^{2}+2 \right ) ^{-{\frac{3}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/(-3*x^2+2)^(3/4)/(-3*x^2+4),x)
[Out]
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Maxima [A] time = 1.53668, size = 174, normalized size = 1.1 \[ -\frac{1}{9} \cdot 2^{\frac{3}{4}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} + 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{1}{9} \cdot 2^{\frac{3}{4}} \arctan \left (-\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} - 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{1}{18} \cdot 2^{\frac{3}{4}} \log \left (2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) + \frac{1}{18} \cdot 2^{\frac{3}{4}} \log \left (-2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) + \frac{2}{9} \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-x^3/((3*x^2 - 4)*(-3*x^2 + 2)^(3/4)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.241383, size = 261, normalized size = 1.65 \[ \frac{2}{9} \cdot 2^{\frac{3}{4}} \arctan \left (\frac{2^{\frac{3}{4}}}{2^{\frac{3}{4}} + 2 \, \sqrt{2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}} + 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}}\right ) + \frac{2}{9} \cdot 2^{\frac{3}{4}} \arctan \left (-\frac{2^{\frac{3}{4}}}{2^{\frac{3}{4}} - 2 \, \sqrt{-2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}} - 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}}\right ) - \frac{1}{18} \cdot 2^{\frac{3}{4}} \log \left (2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) + \frac{1}{18} \cdot 2^{\frac{3}{4}} \log \left (-2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) + \frac{2}{9} \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-x^3/((3*x^2 - 4)*(-3*x^2 + 2)^(3/4)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \int \frac{x^{3}}{3 x^{2} \left (- 3 x^{2} + 2\right )^{\frac{3}{4}} - 4 \left (- 3 x^{2} + 2\right )^{\frac{3}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3/(-3*x**2+2)**(3/4)/(-3*x**2+4),x)
[Out]
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GIAC/XCAS [A] time = 0.241099, size = 174, normalized size = 1.1 \[ -\frac{1}{9} \cdot 2^{\frac{3}{4}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} + 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{1}{9} \cdot 2^{\frac{3}{4}} \arctan \left (-\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} - 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{1}{18} \cdot 2^{\frac{3}{4}}{\rm ln}\left (2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) + \frac{1}{18} \cdot 2^{\frac{3}{4}}{\rm ln}\left (-2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) + \frac{2}{9} \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-x^3/((3*x^2 - 4)*(-3*x^2 + 2)^(3/4)),x, algorithm="giac")
[Out]