3.1038 \(\int \frac{x^2}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx\)

Optimal. Leaf size=148 \[ \frac{\sqrt [4]{2} \tan ^{-1}\left (\frac{2^{3/4}-\sqrt [4]{2} \sqrt{2-3 x^2}}{\sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{3 \sqrt{3}}+\frac{\sqrt [4]{2} \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt{2-3 x^2}+2^{3/4}}{\sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{3 \sqrt{3}}-\frac{2 \sqrt [4]{2} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{3 \sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.163983, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{\sqrt [4]{2} \tan ^{-1}\left (\frac{2^{3/4}-\sqrt [4]{2} \sqrt{2-3 x^2}}{\sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{3 \sqrt{3}}+\frac{\sqrt [4]{2} \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt{2-3 x^2}+2^{3/4}}{\sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{3 \sqrt{3}}-\frac{2 \sqrt [4]{2} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/((2 - 3*x^2)^(1/4)*(4 - 3*x^2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 9.36029, size = 27, normalized size = 0.18 \[ \frac{2^{\frac{3}{4}} x^{3} \operatorname{appellf_{1}}{\left (\frac{3}{2},\frac{1}{4},1,\frac{5}{2},\frac{3 x^{2}}{2},\frac{3 x^{2}}{4} \right )}}{24} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2**(3/4)*x**3*appellf1(3/2, 1/4, 1, 5/2, 3*x**2/2, 3*x**2/4)/24

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Mathematica [C]  time = 0.05643, size = 140, normalized size = 0.95 \[ -\frac{20 x^3 F_1\left (\frac{3}{2};\frac{1}{4},1;\frac{5}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right )}{3 \sqrt [4]{2-3 x^2} \left (3 x^2-4\right ) \left (3 x^2 \left (2 F_1\left (\frac{5}{2};\frac{1}{4},2;\frac{7}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right )+F_1\left (\frac{5}{2};\frac{5}{4},1;\frac{7}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right )\right )+20 F_1\left (\frac{3}{2};\frac{1}{4},1;\frac{5}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-20*x^3*AppellF1[3/2, 1/4, 1, 5/2, (3*x^2)/2, (3*x^2)/4])/(3*(2 - 3*x^2)^(1/4)*
(-4 + 3*x^2)*(20*AppellF1[3/2, 1/4, 1, 5/2, (3*x^2)/2, (3*x^2)/4] + 3*x^2*(2*App
ellF1[5/2, 1/4, 2, 7/2, (3*x^2)/2, (3*x^2)/4] + AppellF1[5/2, 5/4, 1, 7/2, (3*x^
2)/2, (3*x^2)/4])))

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \[ \int{\frac{{x}^{2}}{-3\,{x}^{2}+4}{\frac{1}{\sqrt [4]{-3\,{x}^{2}+2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{x^{2}}{3 x^{2} \sqrt [4]{- 3 x^{2} + 2} - 4 \sqrt [4]{- 3 x^{2} + 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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