3.1091 \(\int \frac{1}{x^2 \left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx\)

Optimal. Leaf size=149 \[ -\frac{\sqrt [4]{3 x^2-1}}{2 x}+\frac{1}{4} \sqrt{\frac{3}{2}} \tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )-\frac{1}{4} \sqrt{\frac{3}{2}} \tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )-\frac{\sqrt{3} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-1}+1\right )^2}} \left (\sqrt{3 x^2-1}+1\right ) F\left (2 \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )|\frac{1}{2}\right )}{2 x} \]

[Out]

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

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Rubi [A]  time = 0.307079, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{\sqrt [4]{3 x^2-1}}{2 x}+\frac{1}{4} \sqrt{\frac{3}{2}} \tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )-\frac{1}{4} \sqrt{\frac{3}{2}} \tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )-\frac{\sqrt{3} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-1}+1\right )^2}} \left (\sqrt{3 x^2-1}+1\right ) F\left (2 \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )|\frac{1}{2}\right )}{2 x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.1584, size = 42, normalized size = 0.28 \[ - \frac{\sqrt [4]{3 x^{2} - 1} \operatorname{appellf_{1}}{\left (- \frac{1}{2},\frac{3}{4},1,\frac{1}{2},3 x^{2},\frac{3 x^{2}}{2} \right )}}{2 x \sqrt [4]{- 3 x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.278853, size = 132, normalized size = 0.89 \[ -\frac{2 F_1\left (-\frac{1}{2};\frac{3}{4},1;\frac{1}{2};3 x^2,\frac{3 x^2}{2}\right )}{x \left (3 x^2-2\right ) \left (3 x^2-1\right )^{3/4} \left (3 x^2 \left (2 F_1\left (\frac{1}{2};\frac{3}{4},2;\frac{3}{2};3 x^2,\frac{3 x^2}{2}\right )+3 F_1\left (\frac{1}{2};\frac{7}{4},1;\frac{3}{2};3 x^2,\frac{3 x^2}{2}\right )\right )+2 F_1\left (-\frac{1}{2};\frac{3}{4},1;\frac{1}{2};3 x^2,\frac{3 x^2}{2}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.107, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2} \left ( 3\,{x}^{2}-2 \right ) } \left ( 3\,{x}^{2}-1 \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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