3.1089 \(\int \frac{x^2}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx\)

Optimal. Leaf size=61 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{3 \sqrt{6}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{3 \sqrt{6}} \]

[Out]

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])

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Rubi [A]  time = 0.0695003, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{3 \sqrt{6}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{3 \sqrt{6}} \]

Antiderivative was successfully verified.

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

[Out]

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])

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Rubi in Sympy [A]  time = 24.9294, size = 41, normalized size = 0.67 \[ \frac{x^{3} \sqrt [4]{3 x^{2} - 1} \operatorname{appellf_{1}}{\left (\frac{3}{2},\frac{3}{4},1,\frac{5}{2},3 x^{2},\frac{3 x^{2}}{2} \right )}}{6 \sqrt [4]{- 3 x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.072639, size = 134, normalized size = 2.2 \[ \frac{10 x^3 F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right )}{3 \left (3 x^2-2\right ) \left (3 x^2-1\right )^{3/4} \left (3 x^2 \left (2 F_1\left (\frac{5}{2};\frac{3}{4},2;\frac{7}{2};3 x^2,\frac{3 x^2}{2}\right )+3 F_1\left (\frac{5}{2};\frac{7}{4},1;\frac{7}{2};3 x^2,\frac{3 x^2}{2}\right )\right )+10 F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};3 x^2,\frac{3 x^2}{2}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.219538, size = 115, normalized size = 1.89 \[ -\frac{1}{36} \, \sqrt{6}{\left (2 \, \arctan \left (\frac{\sqrt{6}{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}}{3 \, x}\right ) - \log \left (-\frac{3 \, \sqrt{6} x^{2} - 12 \,{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} x + 2 \, \sqrt{6} \sqrt{3 \, x^{2} - 1}}{3 \, x^{2} - 2 \, \sqrt{3 \, x^{2} - 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/36*sqrt(6)*(2*arctan(1/3*sqrt(6)*(3*x^2 - 1)^(1/4)/x) - log(-(3*sqrt(6)*x^2 -
12*(3*x^2 - 1)^(1/4)*x + 2*sqrt(6)*sqrt(3*x^2 - 1))/(3*x^2 - 2*sqrt(3*x^2 - 1))
))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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