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

Optimal. Leaf size=47 \[ \frac{2}{9} x \sqrt [4]{3 x^2+2}-\frac{4\ 2^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{9 \sqrt{3}} \]

[Out]

(2*x*(2 + 3*x^2)^(1/4))/9 - (4*2^(3/4)*EllipticF[ArcTan[Sqrt[3/2]*x]/2, 2])/(9*S
qrt[3])

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Rubi [A]  time = 0.0334056, antiderivative size = 47, 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 \[ \frac{2}{9} x \sqrt [4]{3 x^2+2}-\frac{4\ 2^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{9 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(2 + 3*x^2)^(3/4),x]

[Out]

(2*x*(2 + 3*x^2)^(1/4))/9 - (4*2^(3/4)*EllipticF[ArcTan[Sqrt[3/2]*x]/2, 2])/(9*S
qrt[3])

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Rubi in Sympy [A]  time = 3.58923, size = 41, normalized size = 0.87 \[ \frac{2 x \sqrt [4]{3 x^{2} + 2}}{9} - \frac{4 \cdot 2^{\frac{3}{4}} \sqrt{3} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{6} x}{2} \right )}}{2}\middle | 2\right )}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(3*x**2+2)**(3/4),x)

[Out]

2*x*(3*x**2 + 2)**(1/4)/9 - 4*2**(3/4)*sqrt(3)*elliptic_f(atan(sqrt(6)*x/2)/2, 2
)/27

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Mathematica [C]  time = 0.0207826, size = 41, normalized size = 0.87 \[ \frac{2}{9} x \left (\sqrt [4]{3 x^2+2}-\sqrt [4]{2} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{3 x^2}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(2 + 3*x^2)^(3/4),x]

[Out]

(2*x*((2 + 3*x^2)^(1/4) - 2^(1/4)*Hypergeometric2F1[1/2, 3/4, 3/2, (-3*x^2)/2]))
/9

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Maple [C]  time = 0.029, size = 31, normalized size = 0.7 \[{\frac{2\,x}{9}\sqrt [4]{3\,{x}^{2}+2}}-{\frac{2\,\sqrt [4]{2}x}{9}{\mbox{$_2$F$_1$}({\frac{1}{2}},{\frac{3}{4}};\,{\frac{3}{2}};\,-{\frac{3\,{x}^{2}}{2}})}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(3*x^2+2)^(3/4),x)

[Out]

2/9*x*(3*x^2+2)^(1/4)-2/9*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. \[ \int \frac{x^{2}}{{\left (3 \, x^{2} + 2\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(3*x^2 + 2)^(3/4),x, algorithm="maxima")

[Out]

integrate(x^2/(3*x^2 + 2)^(3/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{2}}{{\left (3 \, x^{2} + 2\right )}^{\frac{3}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(3*x^2 + 2)^(3/4),x, algorithm="fricas")

[Out]

integral(x^2/(3*x^2 + 2)^(3/4), x)

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Sympy [A]  time = 2.12111, size = 27, normalized size = 0.57 \[ \frac{\sqrt [4]{2} x^{3}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{3 x^{2} e^{i \pi }}{2}} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(3*x**2+2)**(3/4),x)

[Out]

2**(1/4)*x**3*hyper((3/4, 3/2), (5/2,), 3*x**2*exp_polar(I*pi)/2)/6

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (3 \, x^{2} + 2\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(3*x^2 + 2)^(3/4),x, algorithm="giac")

[Out]

integrate(x^2/(3*x^2 + 2)^(3/4), x)