3.1092 \(\int \frac{1}{x^4 \left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx\)

Optimal. Leaf size=165 \[ -\frac{2 \sqrt [4]{3 x^2-1}}{x}+\frac{3}{8} \sqrt{\frac{3}{2}} \tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )-\frac{3}{8} \sqrt{\frac{3}{2}} \tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )-\frac{11 \sqrt{3} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-1}+1\right )^2}} \left (\sqrt{3 x^2-1}+1\right ) F\left (2 \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )|\frac{1}{2}\right )}{8 x}-\frac{\sqrt [4]{3 x^2-1}}{6 x^3} \]

[Out]

-(-1 + 3*x^2)^(1/4)/(6*x^3) - (2*(-1 + 3*x^2)^(1/4))/x + (3*Sqrt[3/2]*ArcTan[(Sq
rt[3/2]*x)/(-1 + 3*x^2)^(1/4)])/8 - (3*Sqrt[3/2]*ArcTanh[(Sqrt[3/2]*x)/(-1 + 3*x
^2)^(1/4)])/8 - (11*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])/(8*x)

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Rubi [A]  time = 0.403795, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{2 \sqrt [4]{3 x^2-1}}{x}+\frac{3}{8} \sqrt{\frac{3}{2}} \tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )-\frac{3}{8} \sqrt{\frac{3}{2}} \tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )-\frac{11 \sqrt{3} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-1}+1\right )^2}} \left (\sqrt{3 x^2-1}+1\right ) F\left (2 \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )|\frac{1}{2}\right )}{8 x}-\frac{\sqrt [4]{3 x^2-1}}{6 x^3} \]

Antiderivative was successfully verified.

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

[Out]

-(-1 + 3*x^2)^(1/4)/(6*x^3) - (2*(-1 + 3*x^2)^(1/4))/x + (3*Sqrt[3/2]*ArcTan[(Sq
rt[3/2]*x)/(-1 + 3*x^2)^(1/4)])/8 - (3*Sqrt[3/2]*ArcTanh[(Sqrt[3/2]*x)/(-1 + 3*x
^2)^(1/4)])/8 - (11*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])/(8*x)

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Rubi in Sympy [A]  time = 22.6108, size = 46, normalized size = 0.28 \[ - \frac{\sqrt [4]{3 x^{2} - 1} \operatorname{appellf_{1}}{\left (- \frac{3}{2},\frac{3}{4},1,- \frac{1}{2},3 x^{2},\frac{3 x^{2}}{2} \right )}}{6 x^{3} \sqrt [4]{- 3 x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.277936, size = 134, normalized size = 0.81 \[ \frac{2 F_1\left (-\frac{3}{2};\frac{3}{4},1;-\frac{1}{2};3 x^2,\frac{3 x^2}{2}\right )}{3 x^3 \left (3 x^2-2\right ) \left (3 x^2-1\right )^{3/4} \left (3 x^2 \left (2 F_1\left (-\frac{1}{2};\frac{3}{4},2;\frac{1}{2};3 x^2,\frac{3 x^2}{2}\right )+3 F_1\left (-\frac{1}{2};\frac{7}{4},1;\frac{1}{2};3 x^2,\frac{3 x^2}{2}\right )\right )-2 F_1\left (-\frac{3}{2};\frac{3}{4},1;-\frac{1}{2};3 x^2,\frac{3 x^2}{2}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.114, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{4} \left ( 3\,{x}^{2}-2 \right ) } \left ( 3\,{x}^{2}-1 \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (3 \, x^{6} - 2 \, x^{4}\right )}{\left (3 \, x^{2} - 1\right )}^{\frac{3}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/((3*x^6 - 2*x^4)*(3*x^2 - 1)^(3/4)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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