Optimal. Leaf size=221 \[ -\frac{3 \sqrt [4]{3 x^2-2} x}{2 \left (\sqrt{3 x^2-2}+\sqrt{2}\right )}+\frac{\left (3 x^2-2\right )^{3/4}}{2 x}-\frac{\sqrt{3} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{2\ 2^{3/4} x}+\frac{\sqrt{3} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{2^{3/4} x} \]
[Out]
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Rubi [A] time = 0.242435, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{3 \sqrt [4]{3 x^2-2} x}{2 \left (\sqrt{3 x^2-2}+\sqrt{2}\right )}+\frac{\left (3 x^2-2\right )^{3/4}}{2 x}-\frac{\sqrt{3} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{2\ 2^{3/4} x}+\frac{\sqrt{3} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{2^{3/4} x} \]
Antiderivative was successfully verified.
[In] Int[1/(x^2*(-2 + 3*x^2)^(1/4)),x]
[Out]
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Rubi in Sympy [A] time = 3.93418, size = 54, normalized size = 0.24 \[ - \frac{\sqrt{6} \sqrt [4]{- \frac{3 x^{2}}{2} + 1} E\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{6} x}{2} \right )}}{2}\middle | 2\right )}{2 \sqrt [4]{3 x^{2} - 2}} + \frac{\left (3 x^{2} - 2\right )^{\frac{3}{4}}}{2 x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**2/(3*x**2-2)**(1/4),x)
[Out]
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Mathematica [C] time = 0.0263848, size = 63, normalized size = 0.29 \[ \frac{-3\ 2^{3/4} \sqrt [4]{2-3 x^2} x^2 \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{3 x^2}{2}\right )+12 x^2-8}{8 x \sqrt [4]{3 x^2-2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^2*(-2 + 3*x^2)^(1/4)),x]
[Out]
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Maple [C] time = 0.057, size = 55, normalized size = 0.3 \[{\frac{1}{2\,x} \left ( 3\,{x}^{2}-2 \right ) ^{{\frac{3}{4}}}}-{\frac{3\,{2}^{3/4}x}{8}\sqrt [4]{-{\it signum} \left ( -1+{\frac{3\,{x}^{2}}{2}} \right ) }{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{3}{2}};\,{\frac{3\,{x}^{2}}{2}})}{\frac{1}{\sqrt [4]{{\it signum} \left ( -1+{\frac{3\,{x}^{2}}{2}} \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^2/(3*x^2-2)^(1/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (3 \, x^{2} - 2\right )}^{\frac{1}{4}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((3*x^2 - 2)^(1/4)*x^2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (3 \, x^{2} - 2\right )}^{\frac{1}{4}} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((3*x^2 - 2)^(1/4)*x^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.49537, size = 31, normalized size = 0.14 \[ \frac{2^{\frac{3}{4}} e^{\frac{3 i \pi }{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{1}{2} \end{matrix}\middle |{\frac{3 x^{2}}{2}} \right )}}{2 x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**2/(3*x**2-2)**(1/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (3 \, x^{2} - 2\right )}^{\frac{1}{4}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((3*x^2 - 2)^(1/4)*x^2),x, algorithm="giac")
[Out]