3.1037 \(\int \frac{x^4}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx\)

Optimal. Leaf size=164 \[ \frac{2}{45} \left (2-3 x^2\right )^{3/4} x+\frac{4 \sqrt [4]{2} \tan ^{-1}\left (\frac{2^{3/4}-\sqrt [4]{2} \sqrt{2-3 x^2}}{\sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{9 \sqrt{3}}+\frac{4 \sqrt [4]{2} \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt{2-3 x^2}+2^{3/4}}{\sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{9 \sqrt{3}}-\frac{16 \sqrt [4]{2} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{15 \sqrt{3}} \]

[Out]

(2*x*(2 - 3*x^2)^(3/4))/45 + (4*2^(1/4)*ArcTan[(2^(3/4) - 2^(1/4)*Sqrt[2 - 3*x^2
])/(Sqrt[3]*x*(2 - 3*x^2)^(1/4))])/(9*Sqrt[3]) + (4*2^(1/4)*ArcTanh[(2^(3/4) + 2
^(1/4)*Sqrt[2 - 3*x^2])/(Sqrt[3]*x*(2 - 3*x^2)^(1/4))])/(9*Sqrt[3]) - (16*2^(1/4
)*EllipticE[ArcSin[Sqrt[3/2]*x]/2, 2])/(15*Sqrt[3])

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Rubi [A]  time = 0.20539, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{2}{45} \left (2-3 x^2\right )^{3/4} x+\frac{4 \sqrt [4]{2} \tan ^{-1}\left (\frac{2^{3/4}-\sqrt [4]{2} \sqrt{2-3 x^2}}{\sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{9 \sqrt{3}}+\frac{4 \sqrt [4]{2} \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt{2-3 x^2}+2^{3/4}}{\sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{9 \sqrt{3}}-\frac{16 \sqrt [4]{2} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{15 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*(2 - 3*x^2)^(3/4))/45 + (4*2^(1/4)*ArcTan[(2^(3/4) - 2^(1/4)*Sqrt[2 - 3*x^2
])/(Sqrt[3]*x*(2 - 3*x^2)^(1/4))])/(9*Sqrt[3]) + (4*2^(1/4)*ArcTanh[(2^(3/4) + 2
^(1/4)*Sqrt[2 - 3*x^2])/(Sqrt[3]*x*(2 - 3*x^2)^(1/4))])/(9*Sqrt[3]) - (16*2^(1/4
)*EllipticE[ArcSin[Sqrt[3/2]*x]/2, 2])/(15*Sqrt[3])

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Rubi in Sympy [A]  time = 8.29578, size = 27, normalized size = 0.16 \[ \frac{2^{\frac{3}{4}} x^{5} \operatorname{appellf_{1}}{\left (\frac{5}{2},\frac{1}{4},1,\frac{7}{2},\frac{3 x^{2}}{2},\frac{3 x^{2}}{4} \right )}}{40} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2**(3/4)*x**5*appellf1(5/2, 1/4, 1, 7/2, 3*x**2/2, 3*x**2/4)/40

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Mathematica [C]  time = 0.351942, size = 273, normalized size = 1.66 \[ \frac{2 x \left (-\frac{240 x^2 F_1\left (\frac{3}{2};\frac{1}{4},1;\frac{5}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right )}{\left (3 x^2-4\right ) \left (3 x^2 \left (2 F_1\left (\frac{5}{2};\frac{1}{4},2;\frac{7}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right )+F_1\left (\frac{5}{2};\frac{5}{4},1;\frac{7}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right )\right )+20 F_1\left (\frac{3}{2};\frac{1}{4},1;\frac{5}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right )\right )}+\frac{32 F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right )}{\left (3 x^2-4\right ) \left (x^2 \left (2 F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right )+F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right )\right )+4 F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right )\right )}-3 x^2+2\right )}{45 \sqrt [4]{2-3 x^2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*x*(2 - 3*x^2 + (32*AppellF1[1/2, 1/4, 1, 3/2, (3*x^2)/2, (3*x^2)/4])/((-4 + 3
*x^2)*(4*AppellF1[1/2, 1/4, 1, 3/2, (3*x^2)/2, (3*x^2)/4] + x^2*(2*AppellF1[3/2,
1/4, 2, 5/2, (3*x^2)/2, (3*x^2)/4] + AppellF1[3/2, 5/4, 1, 5/2, (3*x^2)/2, (3*x
^2)/4]))) - (240*x^2*AppellF1[3/2, 1/4, 1, 5/2, (3*x^2)/2, (3*x^2)/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*AppellF1[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])))))/(45*(2 - 3*x^2)^(1/4))

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Maple [F]  time = 0.098, size = 0, normalized size = 0. \[ \int{\frac{{x}^{4}}{-3\,{x}^{2}+4}{\frac{1}{\sqrt [4]{-3\,{x}^{2}+2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-integrate(x^4/((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^4/((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. \[ - \int \frac{x^{4}}{3 x^{2} \sqrt [4]{- 3 x^{2} + 2} - 4 \sqrt [4]{- 3 x^{2} + 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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