3.1044 \(\int \frac{x^3}{\left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx\)

Optimal. Leaf size=48 \[ \frac{2}{27} \left (3 x^2-1\right )^{3/4}+\frac{2}{9} \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )-\frac{2}{9} \tanh ^{-1}\left (\sqrt [4]{3 x^2-1}\right ) \]

[Out]

(2*(-1 + 3*x^2)^(3/4))/27 + (2*ArcTan[(-1 + 3*x^2)^(1/4)])/9 - (2*ArcTanh[(-1 +
3*x^2)^(1/4)])/9

_______________________________________________________________________________________

Rubi [A]  time = 0.118385, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{2}{27} \left (3 x^2-1\right )^{3/4}+\frac{2}{9} \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )-\frac{2}{9} \tanh ^{-1}\left (\sqrt [4]{3 x^2-1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*(-1 + 3*x^2)^(3/4))/27 + (2*ArcTan[(-1 + 3*x^2)^(1/4)])/9 - (2*ArcTanh[(-1 +
3*x^2)^(1/4)])/9

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 13.9646, size = 42, normalized size = 0.88 \[ \frac{2 \left (3 x^{2} - 1\right )^{\frac{3}{4}}}{27} + \frac{2 \operatorname{atan}{\left (\sqrt [4]{3 x^{2} - 1} \right )}}{9} - \frac{2 \operatorname{atanh}{\left (\sqrt [4]{3 x^{2} - 1} \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(3*x**2 - 1)**(3/4)/27 + 2*atan((3*x**2 - 1)**(1/4))/9 - 2*atanh((3*x**2 - 1)*
*(1/4))/9

_______________________________________________________________________________________

Mathematica [C]  time = 0.0285521, size = 34, normalized size = 0.71 \[ \frac{2}{27} \left (3 x^2-1\right )^{3/4} \left (1-2 \, _2F_1\left (\frac{3}{4},1;\frac{7}{4};3 x^2-1\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*(-1 + 3*x^2)^(3/4)*(1 - 2*Hypergeometric2F1[3/4, 1, 7/4, -1 + 3*x^2]))/27

_______________________________________________________________________________________

Maple [F]  time = 0.095, size = 0, normalized size = 0. \[ \int{\frac{{x}^{3}}{3\,{x}^{2}-2}{\frac{1}{\sqrt [4]{3\,{x}^{2}-1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.49916, size = 70, normalized size = 1.46 \[ \frac{2}{27} \,{\left (3 \, x^{2} - 1\right )}^{\frac{3}{4}} + \frac{2}{9} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right ) - \frac{1}{9} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + 1\right ) + \frac{1}{9} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/27*(3*x^2 - 1)^(3/4) + 2/9*arctan((3*x^2 - 1)^(1/4)) - 1/9*log((3*x^2 - 1)^(1/
4) + 1) + 1/9*log((3*x^2 - 1)^(1/4) - 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.235034, size = 70, normalized size = 1.46 \[ \frac{2}{27} \,{\left (3 \, x^{2} - 1\right )}^{\frac{3}{4}} + \frac{2}{9} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right ) - \frac{1}{9} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + 1\right ) + \frac{1}{9} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/27*(3*x^2 - 1)^(3/4) + 2/9*arctan((3*x^2 - 1)^(1/4)) - 1/9*log((3*x^2 - 1)^(1/
4) + 1) + 1/9*log((3*x^2 - 1)^(1/4) - 1)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.234374, size = 72, normalized size = 1.5 \[ \frac{2}{27} \,{\left (3 \, x^{2} - 1\right )}^{\frac{3}{4}} + \frac{2}{9} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}}\right ) - \frac{1}{9} \,{\rm ln}\left ({\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} + 1\right ) + \frac{1}{9} \,{\rm ln}\left ({\left |{\left (3 \, x^{2} - 1\right )}^{\frac{1}{4}} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/27*(3*x^2 - 1)^(3/4) + 2/9*arctan((3*x^2 - 1)^(1/4)) - 1/9*ln((3*x^2 - 1)^(1/4
) + 1) + 1/9*ln(abs((3*x^2 - 1)^(1/4) - 1))