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

Optimal. Leaf size=149 \[ \frac{16}{55} \sqrt{-x^3-1} x-\frac{2}{11} \sqrt{-x^3-1} x^4+\frac{32 \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{55 \sqrt [4]{3} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}} \]

[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 = 0.110919, antiderivative size = 149, 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 \[ \frac{16}{55} \sqrt{-x^3-1} x-\frac{2}{11} \sqrt{-x^3-1} x^4+\frac{32 \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{55 \sqrt [4]{3} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}} \]

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 = 6.65249, size = 129, normalized size = 0.87 \[ - \frac{2 x^{4} \sqrt{- x^{3} - 1}}{11} + \frac{16 x \sqrt{- x^{3} - 1}}{55} + \frac{32 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (x + 1\right ) F\left (\operatorname{asin}{\left (\frac{x + 1 + \sqrt{3}}{x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{165 \sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{- x^{3} - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**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 - sqrt(3) + 1)**2)*sqrt(-sqrt(3) + 2)*(x + 1)*elliptic_f(asin((x + 1 +
sqrt(3))/(x - sqrt(3) + 1)), -7 + 4*sqrt(3))/(165*sqrt((-x - 1)/(x - sqrt(3) +
1)**2)*sqrt(-x**3 - 1))

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Mathematica [C]  time = 0.216143, size = 115, normalized size = 0.77 \[ \frac{2 \left (3 x \left (5 x^6-3 x^3-8\right )+16 (-1)^{5/6} 3^{3/4} \sqrt{-(-1)^{5/6}+i x} \sqrt{-\sqrt [3]{-1} x^2-(-1)^{2/3} x+1} F\left (\sin ^{-1}\left (\frac{\sqrt{-\sqrt [6]{-1} \left (x+(-1)^{2/3}\right )}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )}{165 \sqrt{-x^3-1}} \]

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

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Maple [A]  time = 0.033, 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 = 2.27244, size = 32, normalized size = 0.21 \[ - \frac{i x^{7} \Gamma \left (\frac{7}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{7}{3} \\ \frac{10}{3} \end{matrix}\middle |{x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac{10}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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