3.881 \(\int \frac{1}{x^2 \left (2+3 x^2\right )^{3/4}} \, dx\)

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]

-(2 + 3*x^2)^(1/4)/(2*x) - (Sqrt[3]*EllipticF[ArcTan[Sqrt[3/2]*x]/2, 2])/(2*2^(1
/4))

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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]

-(2 + 3*x^2)^(1/4)/(2*x) - (Sqrt[3]*EllipticF[ArcTan[Sqrt[3/2]*x]/2, 2])/(2*2^(1
/4))

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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]

-2**(3/4)*sqrt(3)*elliptic_f(atan(sqrt(6)*x/2)/2, 2)/4 - (3*x**2 + 2)**(1/4)/(2*
x)

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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]

[Out]

-(2 + 3*x^2)^(1/4)/(2*x) - (3*x*Hypergeometric2F1[1/2, 3/4, 3/2, (-3*x^2)/2])/(4
*2^(3/4))

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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]

-1/2*(3*x^2+2)^(1/4)/x-3/8*2^(1/4)*x*hypergeom([1/2,3/4],[3/2],-3/2*x^2)

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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]

integrate(1/((3*x^2 + 2)^(3/4)*x^2), x)

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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]

integral(1/((3*x^2 + 2)^(3/4)*x^2), x)

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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]

-2**(1/4)*hyper((-1/2, 3/4), (1/2,), 3*x**2*exp_polar(I*pi)/2)/(2*x)

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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]

integrate(1/((3*x^2 + 2)^(3/4)*x^2), x)