3.1051 \(\int \frac{1}{x^2 \left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx\)

Optimal. Leaf size=246 \[ \frac{3 \sqrt [4]{3 x^2-1} x}{2 \left (\sqrt{3 x^2-1}+1\right )}-\frac{\left (3 x^2-1\right )^{3/4}}{2 x}-\frac{1}{4} \sqrt{\frac{3}{2}} \tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )-\frac{1}{4} \sqrt{\frac{3}{2}} \tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )+\frac{\sqrt{3} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-1}+1\right )^2}} \left (\sqrt{3 x^2-1}+1\right ) F\left (2 \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )|\frac{1}{2}\right )}{4 x}-\frac{\sqrt{3} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-1}+1\right )^2}} \left (\sqrt{3 x^2-1}+1\right ) E\left (2 \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )|\frac{1}{2}\right )}{2 x} \]

[Out]

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

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Rubi [A]  time = 0.319627, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \frac{3 \sqrt [4]{3 x^2-1} x}{2 \left (\sqrt{3 x^2-1}+1\right )}-\frac{\left (3 x^2-1\right )^{3/4}}{2 x}-\frac{1}{4} \sqrt{\frac{3}{2}} \tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )-\frac{1}{4} \sqrt{\frac{3}{2}} \tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )+\frac{\sqrt{3} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-1}+1\right )^2}} \left (\sqrt{3 x^2-1}+1\right ) F\left (2 \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )|\frac{1}{2}\right )}{4 x}-\frac{\sqrt{3} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-1}+1\right )^2}} \left (\sqrt{3 x^2-1}+1\right ) E\left (2 \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )|\frac{1}{2}\right )}{2 x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 23.7882, size = 42, normalized size = 0.17 \[ - \frac{\left (3 x^{2} - 1\right )^{\frac{3}{4}} \operatorname{appellf_{1}}{\left (- \frac{1}{2},\frac{1}{4},1,\frac{1}{2},3 x^{2},\frac{3 x^{2}}{2} \right )}}{2 x \left (- 3 x^{2} + 1\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.179085, size = 144, normalized size = 0.59 \[ \frac{\frac{15 x^4 F_1\left (\frac{3}{2};\frac{1}{4},1;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right )}{\left (3 x^2-2\right ) \left (3 x^2 \left (2 F_1\left (\frac{5}{2};\frac{1}{4},2;\frac{7}{2};3 x^2,\frac{3 x^2}{2}\right )+F_1\left (\frac{5}{2};\frac{5}{4},1;\frac{7}{2};3 x^2,\frac{3 x^2}{2}\right )\right )+10 F_1\left (\frac{3}{2};\frac{1}{4},1;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right )\right )}-3 x^2+1}{2 x \sqrt [4]{3 x^2-1}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(1 - 3*x^2 + (15*x^4*AppellF1[3/2, 1/4, 1, 5/2, 3*x^2, (3*x^2)/2])/((-2 + 3*x^2)
*(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]))))/(2*
x*(-1 + 3*x^2)^(1/4))

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Maple [F]  time = 0.104, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2} \left ( 3\,{x}^{2}-2 \right ) }{\frac{1}{\sqrt [4]{3\,{x}^{2}-1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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