3.487 \(\int \frac{x^7}{\sqrt{-1+x^3}} \, dx\)

Optimal. Leaf size=294 \[ -\frac{80 \sqrt{x^3-1}}{91 \left (-x-\sqrt{3}+1\right )}+\frac{2}{13} \sqrt{x^3-1} x^5+\frac{20}{91} \sqrt{x^3-1} x^2-\frac{80 \sqrt{2} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{91 \sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}}+\frac{40 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{91 \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}} \]

[Out]

(-80*Sqrt[-1 + x^3])/(91*(1 - Sqrt[3] - x)) + (20*x^2*Sqrt[-1 + x^3])/91 + (2*x^
5*Sqrt[-1 + x^3])/13 + (40*3^(1/4)*Sqrt[2 + Sqrt[3]]*(1 - x)*Sqrt[(1 + x + x^2)/
(1 - Sqrt[3] - x)^2]*EllipticE[ArcSin[(1 + Sqrt[3] - x)/(1 - Sqrt[3] - x)], -7 +
4*Sqrt[3]])/(91*Sqrt[-((1 - x)/(1 - Sqrt[3] - x)^2)]*Sqrt[-1 + x^3]) - (80*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*Sqrt[3]])/(91*3^(1/4)*Sqrt[-((1 - x)/(1 - Sqrt
[3] - x)^2)]*Sqrt[-1 + x^3])

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Rubi [A]  time = 0.205058, antiderivative size = 294, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{80 \sqrt{x^3-1}}{91 \left (-x-\sqrt{3}+1\right )}+\frac{2}{13} \sqrt{x^3-1} x^5+\frac{20}{91} \sqrt{x^3-1} x^2-\frac{80 \sqrt{2} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{91 \sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}}+\frac{40 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{91 \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}} \]

Antiderivative was successfully verified.

[In]  Int[x^7/Sqrt[-1 + x^3],x]

[Out]

(-80*Sqrt[-1 + x^3])/(91*(1 - Sqrt[3] - x)) + (20*x^2*Sqrt[-1 + x^3])/91 + (2*x^
5*Sqrt[-1 + x^3])/13 + (40*3^(1/4)*Sqrt[2 + Sqrt[3]]*(1 - x)*Sqrt[(1 + x + x^2)/
(1 - Sqrt[3] - x)^2]*EllipticE[ArcSin[(1 + Sqrt[3] - x)/(1 - Sqrt[3] - x)], -7 +
4*Sqrt[3]])/(91*Sqrt[-((1 - x)/(1 - Sqrt[3] - x)^2)]*Sqrt[-1 + x^3]) - (80*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*Sqrt[3]])/(91*3^(1/4)*Sqrt[-((1 - x)/(1 - Sqrt
[3] - x)^2)]*Sqrt[-1 + x^3])

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Rubi in Sympy [A]  time = 14.3131, size = 236, normalized size = 0.8 \[ \frac{2 x^{5} \sqrt{x^{3} - 1}}{13} + \frac{20 x^{2} \sqrt{x^{3} - 1}}{91} - \frac{80 \sqrt{x^{3} - 1}}{91 \left (- x - \sqrt{3} + 1\right )} + \frac{40 \sqrt [4]{3} \sqrt{\frac{x^{2} + x + 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (- x + 1\right ) E\left (\operatorname{asin}{\left (\frac{- x + 1 + \sqrt{3}}{- x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{91 \sqrt{\frac{x - 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{x^{3} - 1}} - \frac{80 \sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} + x + 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \left (- x + 1\right ) F\left (\operatorname{asin}{\left (\frac{- x + 1 + \sqrt{3}}{- x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{273 \sqrt{\frac{x - 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{x^{3} - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(x**3-1)**(1/2),x)

[Out]

2*x**5*sqrt(x**3 - 1)/13 + 20*x**2*sqrt(x**3 - 1)/91 - 80*sqrt(x**3 - 1)/(91*(-x
- sqrt(3) + 1)) + 40*3**(1/4)*sqrt((x**2 + x + 1)/(-x - sqrt(3) + 1)**2)*sqrt(s
qrt(3) + 2)*(-x + 1)*elliptic_e(asin((-x + 1 + sqrt(3))/(-x - sqrt(3) + 1)), -7
+ 4*sqrt(3))/(91*sqrt((x - 1)/(-x - sqrt(3) + 1)**2)*sqrt(x**3 - 1)) - 80*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))/(273*sqrt((x - 1)/(-x - s
qrt(3) + 1)**2)*sqrt(x**3 - 1))

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Mathematica [C]  time = 0.223445, size = 142, normalized size = 0.48 \[ \frac{2 \left (3 \left (x^3-1\right ) \left (7 x^3+10\right ) x^2+40 \sqrt [6]{-1} 3^{3/4} \sqrt{(-1)^{5/6} (x-1)} \sqrt{x^2+x+1} \left (\sqrt [3]{-1} F\left (\sin ^{-1}\left (\frac{\sqrt{-i x-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )-i \sqrt{3} E\left (\sin ^{-1}\left (\frac{\sqrt{-i x-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )\right )}{273 \sqrt{x^3-1}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^7/Sqrt[-1 + x^3],x]

[Out]

(2*(3*x^2*(-1 + x^3)*(10 + 7*x^3) + 40*(-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)])))/(273*Sqrt[-1 + x^3])

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Maple [A]  time = 0.026, size = 198, normalized size = 0.7 \[{\frac{2\,{x}^{5}}{13}\sqrt{{x}^{3}-1}}+{\frac{20\,{x}^{2}}{91}\sqrt{{x}^{3}-1}}+{\frac{-120-40\,i\sqrt{3}}{91}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}}\sqrt{{\frac{1}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}} \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }}\sqrt{{\frac{1}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}} \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) }} \left ( \left ({\frac{3}{2}}-{\frac{i}{2}}\sqrt{3} \right ){\it EllipticE} \left ( \sqrt{{\frac{-1+x}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}},\sqrt{{\frac{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}} \right ) + \left ( -{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ){\it EllipticF} \left ( \sqrt{{\frac{-1+x}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}},\sqrt{{\frac{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}} \right ) \right ){\frac{1}{\sqrt{{x}^{3}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(x^3-1)^(1/2),x)

[Out]

2/13*x^5*(x^3-1)^(1/2)+20/91*x^2*(x^3-1)^(1/2)+80/91*(-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. \[ \int \frac{x^{7}}{\sqrt{x^{3} - 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/sqrt(x^3 - 1),x, algorithm="maxima")

[Out]

integrate(x^7/sqrt(x^3 - 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{7}}{\sqrt{x^{3} - 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/sqrt(x^3 - 1),x, algorithm="fricas")

[Out]

integral(x^7/sqrt(x^3 - 1), x)

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Sympy [A]  time = 2.56774, size = 27, normalized size = 0.09 \[ - \frac{i x^{8} \Gamma \left (\frac{8}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{8}{3} \\ \frac{11}{3} \end{matrix}\middle |{x^{3}} \right )}}{3 \Gamma \left (\frac{11}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(x**3-1)**(1/2),x)

[Out]

-I*x**8*gamma(8/3)*hyper((1/2, 8/3), (11/3,), x**3)/(3*gamma(11/3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{7}}{\sqrt{x^{3} - 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/sqrt(x^3 - 1),x, algorithm="giac")

[Out]

integrate(x^7/sqrt(x^3 - 1), x)