3.488 \(\int \frac{x^4}{\sqrt{-1+x^3}} \, dx\)

Optimal. Leaf size=278 \[ -\frac{8 \sqrt{x^3-1}}{7 \left (-x-\sqrt{3}+1\right )}+\frac{2}{7} \sqrt{x^3-1} x^2-\frac{8 \sqrt{2} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{7 \sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}}+\frac{4 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{7 \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}} \]

[Out]

(-8*Sqrt[-1 + x^3])/(7*(1 - Sqrt[3] - x)) + (2*x^2*Sqrt[-1 + x^3])/7 + (4*3^(1/4
)*Sqrt[2 + Sqrt[3]]*(1 - x)*Sqrt[(1 + x + x^2)/(1 - Sqrt[3] - x)^2]*EllipticE[Ar
cSin[(1 + Sqrt[3] - x)/(1 - Sqrt[3] - x)], -7 + 4*Sqrt[3]])/(7*Sqrt[-((1 - x)/(1
- Sqrt[3] - x)^2)]*Sqrt[-1 + x^3]) - (8*Sqrt[2]*(1 - x)*Sqrt[(1 + x + x^2)/(1 -
Sqrt[3] - x)^2]*EllipticF[ArcSin[(1 + Sqrt[3] - x)/(1 - Sqrt[3] - x)], -7 + 4*S
qrt[3]])/(7*3^(1/4)*Sqrt[-((1 - x)/(1 - Sqrt[3] - x)^2)]*Sqrt[-1 + x^3])

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Rubi [A]  time = 0.165347, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{8 \sqrt{x^3-1}}{7 \left (-x-\sqrt{3}+1\right )}+\frac{2}{7} \sqrt{x^3-1} x^2-\frac{8 \sqrt{2} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{7 \sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}}+\frac{4 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{7 \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/Sqrt[-1 + x^3],x]

[Out]

(-8*Sqrt[-1 + x^3])/(7*(1 - Sqrt[3] - x)) + (2*x^2*Sqrt[-1 + x^3])/7 + (4*3^(1/4
)*Sqrt[2 + Sqrt[3]]*(1 - x)*Sqrt[(1 + x + x^2)/(1 - Sqrt[3] - x)^2]*EllipticE[Ar
cSin[(1 + Sqrt[3] - x)/(1 - Sqrt[3] - x)], -7 + 4*Sqrt[3]])/(7*Sqrt[-((1 - x)/(1
- Sqrt[3] - x)^2)]*Sqrt[-1 + x^3]) - (8*Sqrt[2]*(1 - x)*Sqrt[(1 + x + x^2)/(1 -
Sqrt[3] - x)^2]*EllipticF[ArcSin[(1 + Sqrt[3] - x)/(1 - Sqrt[3] - x)], -7 + 4*S
qrt[3]])/(7*3^(1/4)*Sqrt[-((1 - x)/(1 - Sqrt[3] - x)^2)]*Sqrt[-1 + x^3])

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Rubi in Sympy [A]  time = 11.8666, size = 221, normalized size = 0.79 \[ \frac{2 x^{2} \sqrt{x^{3} - 1}}{7} - \frac{8 \sqrt{x^{3} - 1}}{7 \left (- x - \sqrt{3} + 1\right )} + \frac{4 \sqrt [4]{3} \sqrt{\frac{x^{2} + x + 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (- x + 1\right ) E\left (\operatorname{asin}{\left (\frac{- x + 1 + \sqrt{3}}{- x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{7 \sqrt{\frac{x - 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{x^{3} - 1}} - \frac{8 \sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} + x + 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \left (- x + 1\right ) F\left (\operatorname{asin}{\left (\frac{- x + 1 + \sqrt{3}}{- x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{21 \sqrt{\frac{x - 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{x^{3} - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**2*sqrt(x**3 - 1)/7 - 8*sqrt(x**3 - 1)/(7*(-x - sqrt(3) + 1)) + 4*3**(1/4)*s
qrt((x**2 + x + 1)/(-x - sqrt(3) + 1)**2)*sqrt(sqrt(3) + 2)*(-x + 1)*elliptic_e(
asin((-x + 1 + sqrt(3))/(-x - sqrt(3) + 1)), -7 + 4*sqrt(3))/(7*sqrt((x - 1)/(-x
- sqrt(3) + 1)**2)*sqrt(x**3 - 1)) - 8*sqrt(2)*3**(3/4)*sqrt((x**2 + x + 1)/(-x
- sqrt(3) + 1)**2)*(-x + 1)*elliptic_f(asin((-x + 1 + sqrt(3))/(-x - sqrt(3) +
1)), -7 + 4*sqrt(3))/(21*sqrt((x - 1)/(-x - sqrt(3) + 1)**2)*sqrt(x**3 - 1))

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Mathematica [C]  time = 0.288045, size = 135, normalized size = 0.49 \[ \frac{2 \left (3 \left (x^3-1\right ) x^2+4 \sqrt [6]{-1} 3^{3/4} \sqrt{(-1)^{5/6} (x-1)} \sqrt{x^2+x+1} \left (\sqrt [3]{-1} F\left (\sin ^{-1}\left (\frac{\sqrt{-i x-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )-i \sqrt{3} E\left (\sin ^{-1}\left (\frac{\sqrt{-i x-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )\right )}{21 \sqrt{x^3-1}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^4/Sqrt[-1 + x^3],x]

[Out]

(2*(3*x^2*(-1 + x^3) + 4*(-1)^(1/6)*3^(3/4)*Sqrt[(-1)^(5/6)*(-1 + x)]*Sqrt[1 + x
+ x^2]*((-I)*Sqrt[3]*EllipticE[ArcSin[Sqrt[-(-1)^(5/6) - I*x]/3^(1/4)], (-1)^(1
/3)] + (-1)^(1/3)*EllipticF[ArcSin[Sqrt[-(-1)^(5/6) - I*x]/3^(1/4)], (-1)^(1/3)]
)))/(21*Sqrt[-1 + x^3])

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Maple [A]  time = 0.024, size = 186, normalized size = 0.7 \[{\frac{2\,{x}^{2}}{7}\sqrt{{x}^{3}-1}}+{\frac{-4\,i\sqrt{3}-12}{7}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}}\sqrt{{\frac{1}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}} \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }}\sqrt{{\frac{1}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}} \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) }} \left ( \left ({\frac{3}{2}}-{\frac{i}{2}}\sqrt{3} \right ){\it EllipticE} \left ( \sqrt{{\frac{-1+x}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}},\sqrt{{\frac{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}} \right ) + \left ( -{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ){\it EllipticF} \left ( \sqrt{{\frac{-1+x}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}},\sqrt{{\frac{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}} \right ) \right ){\frac{1}{\sqrt{{x}^{3}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(x^3-1)^(1/2),x)

[Out]

2/7*x^2*(x^3-1)^(1/2)+8/7*(-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)*((3/2-1/2*I*3^(1/2))*EllipticE(((-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))+(-1/2+1
/2*I*3^(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. \[ \int \frac{x^{4}}{\sqrt{x^{3} - 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^4/sqrt(x^3 - 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{4}}{\sqrt{x^{3} - 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^4/sqrt(x^3 - 1), x)

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Sympy [A]  time = 2.09823, size = 27, normalized size = 0.1 \[ - \frac{i x^{5} \Gamma \left (\frac{5}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{5}{3} \\ \frac{8}{3} \end{matrix}\middle |{x^{3}} \right )}}{3 \Gamma \left (\frac{8}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-I*x**5*gamma(5/3)*hyper((1/2, 5/3), (8/3,), x**3)/(3*gamma(8/3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\sqrt{x^{3} - 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^4/sqrt(x^3 - 1), x)