3.1071 \(\int \frac{1}{x^2 \left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx\)

Optimal. Leaf size=166 \[ -\frac{\sqrt [4]{2-3 x^2}}{8 x}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{2^{3/4}-\sqrt [4]{2} \sqrt{2-3 x^2}}{\sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{16 \sqrt [4]{2}}-\frac{\sqrt{3} \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt{2-3 x^2}+2^{3/4}}{\sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{16 \sqrt [4]{2}}+\frac{\sqrt{3} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{4 \sqrt [4]{2}} \]

[Out]

-(2 - 3*x^2)^(1/4)/(8*x) + (Sqrt[3]*ArcTan[(2^(3/4) - 2^(1/4)*Sqrt[2 - 3*x^2])/(
Sqrt[3]*x*(2 - 3*x^2)^(1/4))])/(16*2^(1/4)) - (Sqrt[3]*ArcTanh[(2^(3/4) + 2^(1/4
)*Sqrt[2 - 3*x^2])/(Sqrt[3]*x*(2 - 3*x^2)^(1/4))])/(16*2^(1/4)) + (Sqrt[3]*Ellip
ticF[ArcSin[Sqrt[3/2]*x]/2, 2])/(4*2^(1/4))

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Rubi [A]  time = 0.266509, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{\sqrt [4]{2-3 x^2}}{8 x}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{2^{3/4}-\sqrt [4]{2} \sqrt{2-3 x^2}}{\sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{16 \sqrt [4]{2}}-\frac{\sqrt{3} \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt{2-3 x^2}+2^{3/4}}{\sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{16 \sqrt [4]{2}}+\frac{\sqrt{3} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{4 \sqrt [4]{2}} \]

Antiderivative was successfully verified.

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

[Out]

-(2 - 3*x^2)^(1/4)/(8*x) + (Sqrt[3]*ArcTan[(2^(3/4) - 2^(1/4)*Sqrt[2 - 3*x^2])/(
Sqrt[3]*x*(2 - 3*x^2)^(1/4))])/(16*2^(1/4)) - (Sqrt[3]*ArcTanh[(2^(3/4) + 2^(1/4
)*Sqrt[2 - 3*x^2])/(Sqrt[3]*x*(2 - 3*x^2)^(1/4))])/(16*2^(1/4)) + (Sqrt[3]*Ellip
ticF[ArcSin[Sqrt[3/2]*x]/2, 2])/(4*2^(1/4))

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Rubi in Sympy [A]  time = 9.00535, size = 29, normalized size = 0.17 \[ - \frac{\sqrt [4]{2} \operatorname{appellf_{1}}{\left (- \frac{1}{2},\frac{3}{4},1,\frac{1}{2},\frac{3 x^{2}}{2},\frac{3 x^{2}}{4} \right )}}{8 x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2**(1/4)*appellf1(-1/2, 3/4, 1, 1/2, 3*x**2/2, 3*x**2/4)/(8*x)

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Mathematica [C]  time = 0.252745, size = 140, normalized size = 0.84 \[ \frac{4 F_1\left (-\frac{1}{2};\frac{3}{4},1;\frac{1}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right )}{x \left (2-3 x^2\right )^{3/4} \left (3 x^2-4\right ) \left (3 x^2 \left (2 F_1\left (\frac{1}{2};\frac{3}{4},2;\frac{3}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right )+3 F_1\left (\frac{1}{2};\frac{7}{4},1;\frac{3}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right )\right )+4 F_1\left (-\frac{1}{2};\frac{3}{4},1;\frac{1}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.083, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2} \left ( -3\,{x}^{2}+4 \right ) } \left ( -3\,{x}^{2}+2 \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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