Optimal. Leaf size=143 \[ \frac{\log \left (\sqrt{2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )}{12 \sqrt [4]{2}}-\frac{\log \left (\sqrt{2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )}{12 \sqrt [4]{2}}-\frac{\tan ^{-1}\left (\sqrt [4]{4-6 x^2}+1\right )}{6 \sqrt [4]{2}}+\frac{\tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right )}{6 \sqrt [4]{2}} \]
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Rubi [A] time = 0.265978, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ \frac{\log \left (\sqrt{2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )}{12 \sqrt [4]{2}}-\frac{\log \left (\sqrt{2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )}{12 \sqrt [4]{2}}-\frac{\tan ^{-1}\left (\sqrt [4]{4-6 x^2}+1\right )}{6 \sqrt [4]{2}}+\frac{\tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right )}{6 \sqrt [4]{2}} \]
Antiderivative was successfully verified.
[In] Int[x/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]
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Rubi in Sympy [A] time = 27.3218, size = 128, normalized size = 0.9 \[ \frac{2^{\frac{3}{4}} \log{\left (- 2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2} + \sqrt{- 3 x^{2} + 2} + \sqrt{2} \right )}}{24} - \frac{2^{\frac{3}{4}} \log{\left (2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2} + \sqrt{- 3 x^{2} + 2} + \sqrt{2} \right )}}{24} - \frac{2^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{2} \sqrt [4]{- 3 x^{2} + 2} - 1 \right )}}{12} - \frac{2^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{2} \sqrt [4]{- 3 x^{2} + 2} + 1 \right )}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(-3*x**2+2)**(3/4)/(-3*x**2+4),x)
[Out]
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Mathematica [A] time = 0.0761332, size = 100, normalized size = 0.7 \[ \frac{\log \left (\sqrt{4-6 x^2}-2 \sqrt [4]{4-6 x^2}+2\right )-\log \left (\sqrt{4-6 x^2}+2 \sqrt [4]{4-6 x^2}+2\right )+2 \tan ^{-1}\left (1-\sqrt [4]{4-6 x^2}\right )-2 \tan ^{-1}\left (\sqrt [4]{4-6 x^2}+1\right )}{12 \sqrt [4]{2}} \]
Antiderivative was successfully verified.
[In] Integrate[x/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]
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Maple [F] time = 0.062, size = 0, normalized size = 0. \[ \int{\frac{x}{-3\,{x}^{2}+4} \left ( -3\,{x}^{2}+2 \right ) ^{-{\frac{3}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(-3*x^2+2)^(3/4)/(-3*x^2+4),x)
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Maxima [A] time = 1.50882, size = 159, normalized size = 1.11 \[ -\frac{1}{12} \cdot 2^{\frac{3}{4}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} + 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{1}{12} \cdot 2^{\frac{3}{4}} \arctan \left (-\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} - 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{1}{24} \cdot 2^{\frac{3}{4}} \log \left (2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) + \frac{1}{24} \cdot 2^{\frac{3}{4}} \log \left (-2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-x/((3*x^2 - 4)*(-3*x^2 + 2)^(3/4)),x, algorithm="maxima")
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Fricas [A] time = 0.239979, size = 282, normalized size = 1.97 \[ \frac{1}{96} \cdot 8^{\frac{3}{4}}{\left (4 \, \sqrt{2} \arctan \left (\frac{2}{8^{\frac{1}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 2 \, \sqrt{8^{\frac{1}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} \sqrt{-3 \, x^{2} + 2} + 2} + 2}\right ) + 4 \, \sqrt{2} \arctan \left (\frac{2}{8^{\frac{1}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{-4 \cdot 8^{\frac{1}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 4 \, \sqrt{2} \sqrt{-3 \, x^{2} + 2} + 8} - 2}\right ) - \sqrt{2} \log \left (4 \cdot 8^{\frac{1}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 4 \, \sqrt{2} \sqrt{-3 \, x^{2} + 2} + 8\right ) + \sqrt{2} \log \left (-4 \cdot 8^{\frac{1}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 4 \, \sqrt{2} \sqrt{-3 \, x^{2} + 2} + 8\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-x/((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. \[ - \int \frac{x}{3 x^{2} \left (- 3 x^{2} + 2\right )^{\frac{3}{4}} - 4 \left (- 3 x^{2} + 2\right )^{\frac{3}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(-3*x**2+2)**(3/4)/(-3*x**2+4),x)
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GIAC/XCAS [A] time = 0.237306, size = 159, normalized size = 1.11 \[ -\frac{1}{12} \cdot 2^{\frac{3}{4}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} + 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{1}{12} \cdot 2^{\frac{3}{4}} \arctan \left (-\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} - 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{1}{24} \cdot 2^{\frac{3}{4}}{\rm ln}\left (2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) + \frac{1}{24} \cdot 2^{\frac{3}{4}}{\rm ln}\left (-2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-x/((3*x^2 - 4)*(-3*x^2 + 2)^(3/4)),x, algorithm="giac")
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