3.483 \(\int \frac{x^3}{\sqrt{-1+x^3}} \, dx\)

Optimal. Leaf size=137 \[ \frac{2}{5} x \sqrt{x^3-1}-\frac{4 \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{5 \sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}} \]

[Out]

(2*x*Sqrt[-1 + x^3])/5 - (4*Sqrt[2 - Sqrt[3]]*(1 - x)*Sqrt[(1 + x + x^2)/(1 - Sq
rt[3] - x)^2]*EllipticF[ArcSin[(1 + Sqrt[3] - x)/(1 - Sqrt[3] - x)], -7 + 4*Sqrt
[3]])/(5*3^(1/4)*Sqrt[-((1 - x)/(1 - Sqrt[3] - x)^2)]*Sqrt[-1 + x^3])

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Rubi [A]  time = 0.0716452, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{2}{5} x \sqrt{x^3-1}-\frac{4 \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{5 \sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}} \]

Antiderivative was successfully verified.

[In]  Int[x^3/Sqrt[-1 + x^3],x]

[Out]

(2*x*Sqrt[-1 + x^3])/5 - (4*Sqrt[2 - Sqrt[3]]*(1 - x)*Sqrt[(1 + x + x^2)/(1 - Sq
rt[3] - x)^2]*EllipticF[ArcSin[(1 + Sqrt[3] - x)/(1 - Sqrt[3] - x)], -7 + 4*Sqrt
[3]])/(5*3^(1/4)*Sqrt[-((1 - x)/(1 - Sqrt[3] - x)^2)]*Sqrt[-1 + x^3])

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Rubi in Sympy [A]  time = 4.21705, size = 107, normalized size = 0.78 \[ \frac{2 x \sqrt{x^{3} - 1}}{5} - \frac{4 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} + x + 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (- x + 1\right ) F\left (\operatorname{asin}{\left (\frac{- x + 1 + \sqrt{3}}{- x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{15 \sqrt{\frac{x - 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{x^{3} - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x*sqrt(x**3 - 1)/5 - 4*3**(3/4)*sqrt((x**2 + x + 1)/(-x - sqrt(3) + 1)**2)*sqr
t(-sqrt(3) + 2)*(-x + 1)*elliptic_f(asin((-x + 1 + sqrt(3))/(-x - sqrt(3) + 1)),
-7 + 4*sqrt(3))/(15*sqrt((x - 1)/(-x - sqrt(3) + 1)**2)*sqrt(x**3 - 1))

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Mathematica [C]  time = 0.103438, size = 84, normalized size = 0.61 \[ \frac{2 \left (3 x \left (x^3-1\right )+2 i 3^{3/4} \sqrt{(-1)^{5/6} (x-1)} \sqrt{x^2+x+1} F\left (\sin ^{-1}\left (\frac{\sqrt{-i x-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )}{15 \sqrt{x^3-1}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^3/Sqrt[-1 + x^3],x]

[Out]

(2*(3*x*(-1 + x^3) + (2*I)*3^(3/4)*Sqrt[(-1)^(5/6)*(-1 + x)]*Sqrt[1 + x + x^2]*E
llipticF[ArcSin[Sqrt[-(-1)^(5/6) - I*x]/3^(1/4)], (-1)^(1/3)]))/(15*Sqrt[-1 + x^
3])

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Maple [A]  time = 0.024, size = 127, normalized size = 0.9 \[{\frac{2\,x}{5}\sqrt{{x}^{3}-1}}+{\frac{-6-2\,i\sqrt{3}}{5}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}}\sqrt{{\frac{1}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}} \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }}\sqrt{{\frac{1}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}} \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) }}{\it EllipticF} \left ( \sqrt{{\frac{-1+x}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}},\sqrt{{\frac{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{{x}^{3}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(x^3-1)^(1/2),x)

[Out]

2/5*x*(x^3-1)^(1/2)+4/5*(-3/2-1/2*I*3^(1/2))*((-1+x)/(-3/2-1/2*I*3^(1/2)))^(1/2)
*((x+1/2-1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2)*((x+1/2+1/2*I*3^(1/2))/(3/2+1
/2*I*3^(1/2)))^(1/2)/(x^3-1)^(1/2)*EllipticF(((-1+x)/(-3/2-1/2*I*3^(1/2)))^(1/2)
,((3/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\sqrt{x^{3} - 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^3/sqrt(x^3 - 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^3/sqrt(x^3 - 1), x)

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Sympy [A]  time = 1.91851, size = 27, normalized size = 0.2 \[ - \frac{i x^{4} \Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{x^{3}} \right )}}{3 \Gamma \left (\frac{7}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-I*x**4*gamma(4/3)*hyper((1/2, 4/3), (7/3,), x**3)/(3*gamma(7/3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\sqrt{x^{3} - 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^3/sqrt(x^3 - 1), x)