Optimal. Leaf size=49 \[ -\frac{\sqrt [4]{3 x^2+2}}{2 x}-\frac{\sqrt{3} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{2 \sqrt [4]{2}} \]
[Out]
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Rubi [A] time = 0.0328475, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{\sqrt [4]{3 x^2+2}}{2 x}-\frac{\sqrt{3} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{2 \sqrt [4]{2}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^2*(2 + 3*x^2)^(3/4)),x]
[Out]
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Rubi in Sympy [A] time = 3.37698, size = 39, normalized size = 0.8 \[ - \frac{2^{\frac{3}{4}} \sqrt{3} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{6} x}{2} \right )}}{2}\middle | 2\right )}{4} - \frac{\sqrt [4]{3 x^{2} + 2}}{2 x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**2/(3*x**2+2)**(3/4),x)
[Out]
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Mathematica [C] time = 0.0255852, size = 46, normalized size = 0.94 \[ -\frac{3 x \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{3 x^2}{2}\right )}{4\ 2^{3/4}}-\frac{\sqrt [4]{3 x^2+2}}{2 x} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^2*(2 + 3*x^2)^(3/4)),x]
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Maple [C] time = 0.033, size = 33, normalized size = 0.7 \[ -{\frac{1}{2\,x}\sqrt [4]{3\,{x}^{2}+2}}-{\frac{3\,\sqrt [4]{2}x}{8}{\mbox{$_2$F$_1$}({\frac{1}{2}},{\frac{3}{4}};\,{\frac{3}{2}};\,-{\frac{3\,{x}^{2}}{2}})}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^2/(3*x^2+2)^(3/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (3 \, x^{2} + 2\right )}^{\frac{3}{4}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((3*x^2 + 2)^(3/4)*x^2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (3 \, x^{2} + 2\right )}^{\frac{3}{4}} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((3*x^2 + 2)^(3/4)*x^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.75271, size = 29, normalized size = 0.59 \[ - \frac{\sqrt [4]{2}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{1}{2} \end{matrix}\middle |{\frac{3 x^{2} e^{i \pi }}{2}} \right )}}{2 x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**2/(3*x**2+2)**(3/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (3 \, x^{2} + 2\right )}^{\frac{3}{4}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((3*x^2 + 2)^(3/4)*x^2),x, algorithm="giac")
[Out]