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

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

[Out]

(80*Sqrt[1 - x^3])/(91*(1 + Sqrt[3] - x)) - (20*x^2*Sqrt[1 - x^3])/91 - (2*x^5*S
qrt[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*S
qrt[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 = 0.214793, antiderivative size = 294, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{80 \sqrt{1-x^3}}{91 \left (-x+\sqrt{3}+1\right )}-\frac{2}{13} \sqrt{1-x^3} x^5-\frac{20}{91} \sqrt{1-x^3} x^2+\frac{80 \sqrt{2} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{91 \sqrt [4]{3} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}}-\frac{40 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{91 \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}} \]

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*S
qrt[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*S
qrt[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 = 17.0492, size = 240, normalized size = 0.82 \[ - \frac{2 x^{5} \sqrt{- x^{3} + 1}}{13} - \frac{20 x^{2} \sqrt{- x^{3} + 1}}{91} + \frac{80 \sqrt{- x^{3} + 1}}{91 \left (- x + 1 + \sqrt{3}\right )} - \frac{40 \sqrt [4]{3} \sqrt{\frac{x^{2} + x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (- x + 1\right ) E\left (\operatorname{asin}{\left (\frac{- x - \sqrt{3} + 1}{- x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{91 \sqrt{\frac{- x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- x^{3} + 1}} + \frac{80 \sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} + x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \left (- x + 1\right ) F\left (\operatorname{asin}{\left (\frac{- x - \sqrt{3} + 1}{- x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{273 \sqrt{\frac{- x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- x^{3} + 1}} \]

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 + 1 + sqrt(3))) - 40*3**(1/4)*sqrt((x**2 + x + 1)/(-x + 1 + sqrt(3))**2)*sq
rt(-sqrt(3) + 2)*(-x + 1)*elliptic_e(asin((-x - sqrt(3) + 1)/(-x + 1 + sqrt(3)))
, -7 - 4*sqrt(3))/(91*sqrt((-x + 1)/(-x + 1 + sqrt(3))**2)*sqrt(-x**3 + 1)) + 80
*sqrt(2)*3**(3/4)*sqrt((x**2 + x + 1)/(-x + 1 + sqrt(3))**2)*(-x + 1)*elliptic_f
(asin((-x - sqrt(3) + 1)/(-x + 1 + sqrt(3))), -7 - 4*sqrt(3))/(273*sqrt((-x + 1)
/(-x + 1 + sqrt(3))**2)*sqrt(-x**3 + 1))

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Mathematica [C]  time = 0.26666, size = 144, normalized size = 0.49 \[ \frac{2 \left (3 \left (x^3-1\right ) \left (7 x^3+10\right ) x^2+40 \sqrt [6]{-1} 3^{3/4} \sqrt{(-1)^{5/6} (x-1)} \sqrt{x^2+x+1} \left (\sqrt [3]{-1} F\left (\sin ^{-1}\left (\frac{\sqrt{-i x-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )-i \sqrt{3} E\left (\sin ^{-1}\left (\frac{\sqrt{-i x-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )\right )}{273 \sqrt{1-x^3}} \]

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 = 0.031, size = 187, normalized size = 0.6 \[ -{\frac{2\,{x}^{5}}{13}\sqrt{-{x}^{3}+1}}-{\frac{20\,{x}^{2}}{91}\sqrt{-{x}^{3}+1}}-{{\frac{80\,i}{273}}\sqrt{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}} \left ( \left ( -{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3} \right ){\it EllipticE} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ) +{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ) \right ){\frac{1}{\sqrt{-{x}^{3}+1}}}} \]

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)-80/273*I*3^(1/2)*(I*(x+1/2-1/2
*I*3^(1/2))*3^(1/2))^(1/2)*((-1+x)/(-3/2+1/2*I*3^(1/2)))^(1/2)*(-I*(x+1/2+1/2*I*
3^(1/2))*3^(1/2))^(1/2)/(-x^3+1)^(1/2)*((-3/2+1/2*I*3^(1/2))*EllipticE(1/3*3^(1/
2)*(I*(x+1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2),(I*3^(1/2)/(-3/2+1/2*I*3^(1/2)))^(1/2
))+EllipticF(1/3*3^(1/2)*(I*(x+1/2-1/2*I*3^(1/2))*3^(1/2))^(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 = 2.53454, size = 31, normalized size = 0.11 \[ \frac{x^{8} \Gamma \left (\frac{8}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{8}{3} \\ \frac{11}{3} \end{matrix}\middle |{x^{3} e^{2 i \pi }} \right )}}{3 \Gamma \left (\frac{11}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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