3.1049 \(\int \frac{x^2}{\left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx\)

Optimal. Leaf size=224 \[ \frac{2 \sqrt [4]{3 x^2-1} x}{3 \left (\sqrt{3 x^2-1}+1\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{3 \sqrt{6}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{3 \sqrt{6}}+\frac{\sqrt{\frac{x^2}{\left (\sqrt{3 x^2-1}+1\right )^2}} \left (\sqrt{3 x^2-1}+1\right ) F\left (2 \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )|\frac{1}{2}\right )}{3 \sqrt{3} x}-\frac{2 \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-1}+1\right )^2}} \left (\sqrt{3 x^2-1}+1\right ) E\left (2 \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )|\frac{1}{2}\right )}{3 \sqrt{3} x} \]

[Out]

(2*x*(-1 + 3*x^2)^(1/4))/(3*(1 + Sqrt[-1 + 3*x^2])) - ArcTan[(Sqrt[3/2]*x)/(-1 +
3*x^2)^(1/4)]/(3*Sqrt[6]) - ArcTanh[(Sqrt[3/2]*x)/(-1 + 3*x^2)^(1/4)]/(3*Sqrt[6
]) - (2*Sqrt[x^2/(1 + Sqrt[-1 + 3*x^2])^2]*(1 + Sqrt[-1 + 3*x^2])*EllipticE[2*Ar
cTan[(-1 + 3*x^2)^(1/4)], 1/2])/(3*Sqrt[3]*x) + (Sqrt[x^2/(1 + Sqrt[-1 + 3*x^2])
^2]*(1 + Sqrt[-1 + 3*x^2])*EllipticF[2*ArcTan[(-1 + 3*x^2)^(1/4)], 1/2])/(3*Sqrt
[3]*x)

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Rubi [A]  time = 0.277336, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{2 \sqrt [4]{3 x^2-1} x}{3 \left (\sqrt{3 x^2-1}+1\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{3 \sqrt{6}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{3 \sqrt{6}}+\frac{\sqrt{\frac{x^2}{\left (\sqrt{3 x^2-1}+1\right )^2}} \left (\sqrt{3 x^2-1}+1\right ) F\left (2 \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )|\frac{1}{2}\right )}{3 \sqrt{3} x}-\frac{2 \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-1}+1\right )^2}} \left (\sqrt{3 x^2-1}+1\right ) E\left (2 \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )|\frac{1}{2}\right )}{3 \sqrt{3} x} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*(-1 + 3*x^2)^(1/4))/(3*(1 + Sqrt[-1 + 3*x^2])) - ArcTan[(Sqrt[3/2]*x)/(-1 +
3*x^2)^(1/4)]/(3*Sqrt[6]) - ArcTanh[(Sqrt[3/2]*x)/(-1 + 3*x^2)^(1/4)]/(3*Sqrt[6
]) - (2*Sqrt[x^2/(1 + Sqrt[-1 + 3*x^2])^2]*(1 + Sqrt[-1 + 3*x^2])*EllipticE[2*Ar
cTan[(-1 + 3*x^2)^(1/4)], 1/2])/(3*Sqrt[3]*x) + (Sqrt[x^2/(1 + Sqrt[-1 + 3*x^2])
^2]*(1 + Sqrt[-1 + 3*x^2])*EllipticF[2*ArcTan[(-1 + 3*x^2)^(1/4)], 1/2])/(3*Sqrt
[3]*x)

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Rubi in Sympy [A]  time = 24.7453, size = 41, normalized size = 0.18 \[ \frac{x^{3} \left (3 x^{2} - 1\right )^{\frac{3}{4}} \operatorname{appellf_{1}}{\left (\frac{3}{2},\frac{1}{4},1,\frac{5}{2},3 x^{2},\frac{3 x^{2}}{2} \right )}}{6 \left (- 3 x^{2} + 1\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**3*(3*x**2 - 1)**(3/4)*appellf1(3/2, 1/4, 1, 5/2, 3*x**2, 3*x**2/2)/(6*(-3*x**
2 + 1)**(3/4))

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Mathematica [C]  time = 0.054455, size = 132, normalized size = 0.59 \[ \frac{10 x^3 F_1\left (\frac{3}{2};\frac{1}{4},1;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right )}{3 \left (3 x^2-2\right ) \sqrt [4]{3 x^2-1} \left (3 x^2 \left (2 F_1\left (\frac{5}{2};\frac{1}{4},2;\frac{7}{2};3 x^2,\frac{3 x^2}{2}\right )+F_1\left (\frac{5}{2};\frac{5}{4},1;\frac{7}{2};3 x^2,\frac{3 x^2}{2}\right )\right )+10 F_1\left (\frac{3}{2};\frac{1}{4},1;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(10*x^3*AppellF1[3/2, 1/4, 1, 5/2, 3*x^2, (3*x^2)/2])/(3*(-2 + 3*x^2)*(-1 + 3*x^
2)^(1/4)*(10*AppellF1[3/2, 1/4, 1, 5/2, 3*x^2, (3*x^2)/2] + 3*x^2*(2*AppellF1[5/
2, 1/4, 2, 7/2, 3*x^2, (3*x^2)/2] + AppellF1[5/2, 5/4, 1, 7/2, 3*x^2, (3*x^2)/2]
)))

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Maple [F]  time = 0.086, size = 0, normalized size = 0. \[ \int{\frac{{x}^{2}}{3\,{x}^{2}-2}{\frac{1}{\sqrt [4]{3\,{x}^{2}-1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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