3.865 \(\int \frac{x^2}{\sqrt [4]{2+3 x^2}} \, dx\)

Optimal. Leaf size=63 \[ \frac{2}{15} \left (3 x^2+2\right )^{3/4} x-\frac{8 x}{15 \sqrt [4]{3 x^2+2}}+\frac{8 \sqrt [4]{2} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{15 \sqrt{3}} \]

[Out]

(-8*x)/(15*(2 + 3*x^2)^(1/4)) + (2*x*(2 + 3*x^2)^(3/4))/15 + (8*2^(1/4)*Elliptic
E[ArcTan[Sqrt[3/2]*x]/2, 2])/(15*Sqrt[3])

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Rubi [A]  time = 0.0454174, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{2}{15} \left (3 x^2+2\right )^{3/4} x-\frac{8 x}{15 \sqrt [4]{3 x^2+2}}+\frac{8 \sqrt [4]{2} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{15 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(2 + 3*x^2)^(1/4),x]

[Out]

(-8*x)/(15*(2 + 3*x^2)^(1/4)) + (2*x*(2 + 3*x^2)^(3/4))/15 + (8*2^(1/4)*Elliptic
E[ArcTan[Sqrt[3/2]*x]/2, 2])/(15*Sqrt[3])

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{2 x \left (3 x^{2} + 2\right )^{\frac{3}{4}}}{15} - \frac{4 \int \frac{1}{\sqrt [4]{3 x^{2} + 2}}\, dx}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(3*x**2+2)**(1/4),x)

[Out]

2*x*(3*x**2 + 2)**(3/4)/15 - 4*Integral((3*x**2 + 2)**(-1/4), x)/15

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Mathematica [C]  time = 0.0245888, size = 41, normalized size = 0.65 \[ \frac{2}{15} x \left (\left (3 x^2+2\right )^{3/4}-2^{3/4} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{3 x^2}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(2 + 3*x^2)^(1/4),x]

[Out]

(2*x*((2 + 3*x^2)^(3/4) - 2^(3/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (-3*x^2)/2]))
/15

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Maple [C]  time = 0.028, size = 31, normalized size = 0.5 \[{\frac{2\,x}{15} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{4}}}}-{\frac{2\,{2}^{3/4}x}{15}{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{3}{2}};\,-{\frac{3\,{x}^{2}}{2}})}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(3*x^2+2)^(1/4),x)

[Out]

2/15*x*(3*x^2+2)^(3/4)-2/15*2^(3/4)*x*hypergeom([1/4,1/2],[3/2],-3/2*x^2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (3 \, x^{2} + 2\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(3*x^2 + 2)^(1/4),x, algorithm="maxima")

[Out]

integrate(x^2/(3*x^2 + 2)^(1/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{2}}{{\left (3 \, x^{2} + 2\right )}^{\frac{1}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(3*x^2 + 2)^(1/4),x, algorithm="fricas")

[Out]

integral(x^2/(3*x^2 + 2)^(1/4), x)

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Sympy [A]  time = 2.1087, size = 27, normalized size = 0.43 \[ \frac{2^{\frac{3}{4}} x^{3}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{3 x^{2} e^{i \pi }}{2}} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(3*x**2+2)**(1/4),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(3*x^2 + 2)^(1/4),x, algorithm="giac")

[Out]

integrate(x^2/(3*x^2 + 2)^(1/4), x)