3.464 \(\int \frac{x^6}{\sqrt{1-x^3}} \, dx\)

Optimal. Leaf size=152 \[ -\frac{16}{55} \sqrt{1-x^3} x-\frac{2}{11} \sqrt{1-x^3} x^4-\frac{32 \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{55 \sqrt [4]{3} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}} \]

[Out]

(-16*x*Sqrt[1 - x^3])/55 - (2*x^4*Sqrt[1 - x^3])/11 - (32*Sqrt[2 + Sqrt[3]]*(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]])/(55*3^(1/4)*Sqrt[(1 - x)/(1 + Sqrt[3] - x)^2
]*Sqrt[1 - x^3])

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Rubi [A]  time = 0.109633, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{16}{55} \sqrt{1-x^3} x-\frac{2}{11} \sqrt{1-x^3} x^4-\frac{32 \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{55 \sqrt [4]{3} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}} \]

Antiderivative was successfully verified.

[In]  Int[x^6/Sqrt[1 - x^3],x]

[Out]

(-16*x*Sqrt[1 - x^3])/55 - (2*x^4*Sqrt[1 - x^3])/11 - (32*Sqrt[2 + Sqrt[3]]*(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]])/(55*3^(1/4)*Sqrt[(1 - x)/(1 + Sqrt[3] - x)^2
]*Sqrt[1 - x^3])

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Rubi in Sympy [A]  time = 6.6533, size = 126, normalized size = 0.83 \[ - \frac{2 x^{4} \sqrt{- x^{3} + 1}}{11} - \frac{16 x \sqrt{- x^{3} + 1}}{55} - \frac{32 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} + x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (- x + 1\right ) F\left (\operatorname{asin}{\left (\frac{- x - \sqrt{3} + 1}{- x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{165 \sqrt{\frac{- x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- x^{3} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6/(-x**3+1)**(1/2),x)

[Out]

-2*x**4*sqrt(-x**3 + 1)/11 - 16*x*sqrt(-x**3 + 1)/55 - 32*3**(3/4)*sqrt((x**2 +
x + 1)/(-x + 1 + sqrt(3))**2)*sqrt(sqrt(3) + 2)*(-x + 1)*elliptic_f(asin((-x - s
qrt(3) + 1)/(-x + 1 + sqrt(3))), -7 - 4*sqrt(3))/(165*sqrt((-x + 1)/(-x + 1 + sq
rt(3))**2)*sqrt(-x**3 + 1))

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Mathematica [C]  time = 0.124205, size = 93, normalized size = 0.61 \[ \frac{2 \left (3 x \left (5 x^6+3 x^3-8\right )+16 i 3^{3/4} \sqrt{(-1)^{5/6} (x-1)} \sqrt{x^2+x+1} F\left (\sin ^{-1}\left (\frac{\sqrt{-i x-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )}{165 \sqrt{1-x^3}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^6/Sqrt[1 - x^3],x]

[Out]

(2*(3*x*(-8 + 3*x^3 + 5*x^6) + (16*I)*3^(3/4)*Sqrt[(-1)^(5/6)*(-1 + x)]*Sqrt[1 +
x + x^2]*EllipticF[ArcSin[Sqrt[-(-1)^(5/6) - I*x]/3^(1/4)], (-1)^(1/3)]))/(165*
Sqrt[1 - x^3])

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Maple [A]  time = 0.031, size = 134, normalized size = 0.9 \[ -{\frac{2\,{x}^{4}}{11}\sqrt{-{x}^{3}+1}}-{\frac{16\,x}{55}\sqrt{-{x}^{3}+1}}-{{\frac{32\,i}{165}}\sqrt{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(-x^3+1)^(1/2),x)

[Out]

-2/11*x^4*(-x^3+1)^(1/2)-16/55*x*(-x^3+1)^(1/2)-32/165*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)*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. \[ \int \frac{x^{6}}{\sqrt{-x^{3} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/sqrt(-x^3 + 1),x, algorithm="maxima")

[Out]

integrate(x^6/sqrt(-x^3 + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{6}}{\sqrt{-x^{3} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/sqrt(-x^3 + 1),x, algorithm="fricas")

[Out]

integral(x^6/sqrt(-x^3 + 1), x)

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Sympy [A]  time = 2.28792, size = 31, normalized size = 0.2 \[ \frac{x^{7} \Gamma \left (\frac{7}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{7}{3} \\ \frac{10}{3} \end{matrix}\middle |{x^{3} e^{2 i \pi }} \right )}}{3 \Gamma \left (\frac{10}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6/(-x**3+1)**(1/2),x)

[Out]

x**7*gamma(7/3)*hyper((1/2, 7/3), (10/3,), x**3*exp_polar(2*I*pi))/(3*gamma(10/3
))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{\sqrt{-x^{3} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/sqrt(-x^3 + 1),x, algorithm="giac")

[Out]

integrate(x^6/sqrt(-x^3 + 1), x)