3.1074 \(\int \frac{x^2}{\left (-2-3 x^2\right ) \left (-1-3 x^2\right )^{3/4}} \, dx\)

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 = 0.0685388, 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 = 24.9799, size = 46, normalized size = 0.75 \[ \frac{x^{3} \sqrt [4]{- 3 x^{2} - 1} \operatorname{appellf_{1}}{\left (\frac{3}{2},\frac{3}{4},1,\frac{5}{2},- 3 x^{2},- \frac{3 x^{2}}{2} \right )}}{6 \sqrt [4]{3 x^{2} + 1}} \]

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 = 0.229369, size = 134, normalized size = 2.2 \[ \frac{10 x^3 F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-3 x^2,-\frac{3 x^2}{2}\right )}{3 \left (-3 x^2-1\right )^{3/4} \left (3 x^2+2\right ) \left (3 x^2 \left (2 F_1\left (\frac{5}{2};\frac{3}{4},2;\frac{7}{2};-3 x^2,-\frac{3 x^2}{2}\right )+3 F_1\left (\frac{5}{2};\frac{7}{4},1;\frac{7}{2};-3 x^2,-\frac{3 x^2}{2}\right )\right )-10 F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-3 x^2,-\frac{3 x^2}{2}\right )\right )} \]

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*(-1 - 3*x^2)^(3/4)*(2
+ 3*x^2)*(-10*AppellF1[3/2, 3/4, 1, 5/2, -3*x^2, (-3*x^2)/2] + 3*x^2*(2*AppellF
1[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 = 0.056, 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. \[ -\int \frac{x^{2}}{{\left (3 \, x^{2} + 2\right )}{\left (-3 \, x^{2} - 1\right )}^{\frac{3}{4}}}\,{d x} \]

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, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.224488, size = 158, normalized size = 2.59 \[ -\frac{1}{36} \, \sqrt{6}{\left (\log \left (\frac{\sqrt{6}{\left (3 \, x + \sqrt{6}{\left (-3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right )}}{6 \, x}\right ) - \log \left (-\frac{\sqrt{6}{\left (3 \, x - \sqrt{6}{\left (-3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right )}}{6 \, x}\right ) + i \, \log \left (\frac{\sqrt{6}{\left (3 i \, x + \sqrt{6}{\left (-3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right )}}{6 \, x}\right ) - i \, \log \left (\frac{\sqrt{6}{\left (-3 i \, x + \sqrt{6}{\left (-3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right )}}{6 \, x}\right )\right )} \]

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, algorithm="fricas")

[Out]

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

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

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

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

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, algorithm="giac")

[Out]

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