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

Optimal. Leaf size=282 \[ -\frac{80 \sqrt{-x^3-1}}{91 \left (x-\sqrt{3}+1\right )}-\frac{2}{13} \sqrt{-x^3-1} x^5+\frac{20}{91} \sqrt{-x^3-1} x^2-\frac{80 \sqrt{2} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{91 \sqrt [4]{3} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}}+\frac{40 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{91 \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}} \]

[Out]

(20*x^2*Sqrt[-1 - x^3])/91 - (2*x^5*Sqrt[-1 - x^3])/13 - (80*Sqrt[-1 - x^3])/(91
*(1 - Sqrt[3] + x)) + (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 = 0.195482, antiderivative size = 282, 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 \[ -\frac{80 \sqrt{-x^3-1}}{91 \left (x-\sqrt{3}+1\right )}-\frac{2}{13} \sqrt{-x^3-1} x^5+\frac{20}{91} \sqrt{-x^3-1} x^2-\frac{80 \sqrt{2} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{91 \sqrt [4]{3} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}}+\frac{40 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{91 \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}} \]

Antiderivative was successfully verified.

[In]  Int[x^7/Sqrt[-1 - x^3],x]

[Out]

(20*x^2*Sqrt[-1 - x^3])/91 - (2*x^5*Sqrt[-1 - x^3])/13 - (80*Sqrt[-1 - x^3])/(91
*(1 - Sqrt[3] + x)) + (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 = 14.5118, size = 248, normalized size = 0.88 \[ - \frac{2 x^{5} \sqrt{- x^{3} - 1}}{13} + \frac{20 x^{2} \sqrt{- x^{3} - 1}}{91} - \frac{80 \sqrt{- x^{3} - 1}}{91 \left (x - \sqrt{3} + 1\right )} + \frac{40 \sqrt [4]{3} \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (x + 1\right ) E\left (\operatorname{asin}{\left (\frac{x + 1 + \sqrt{3}}{x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{91 \sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{- x^{3} - 1}} - \frac{80 \sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \left (x + 1\right ) F\left (\operatorname{asin}{\left (\frac{x + 1 + \sqrt{3}}{x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{273 \sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{- x^{3} - 1}} \]

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
(sqrt(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 - sqrt(
3) + 1)**2)*sqrt(-x**3 - 1))

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Mathematica [C]  time = 0.507829, size = 164, normalized size = 0.58 \[ \frac{2 \left (3 \left (x^3+1\right ) \left (7 x^3-10\right ) x^2+40 (-1)^{5/6} 3^{3/4} \sqrt{-(-1)^{5/6}+i x} \sqrt{-\sqrt [3]{-1} x^2-(-1)^{2/3} x+1} \left (\sqrt [3]{-1} F\left (\sin ^{-1}\left (\frac{\sqrt{-\sqrt [6]{-1} \left (x+(-1)^{2/3}\right )}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )-i \sqrt{3} E\left (\sin ^{-1}\left (\frac{\sqrt{-\sqrt [6]{-1} \left (x+(-1)^{2/3}\right )}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )\right )}{273 \sqrt{-x^3-1}} \]

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)^(5/6)*3^(3/4)*Sqrt[-(-1)^(5/6) + I*x
]*Sqrt[1 - (-1)^(2/3)*x - (-1)^(1/3)*x^2]*((-I)*Sqrt[3]*EllipticE[ArcSin[Sqrt[-(
(-1)^(1/6)*((-1)^(2/3) + x))]/3^(1/4)], (-1)^(1/3)] + (-1)^(1/3)*EllipticF[ArcSi
n[Sqrt[-((-1)^(1/6)*((-1)^(2/3) + x))]/3^(1/4)], (-1)^(1/3)])))/(273*Sqrt[-1 - x
^3])

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Maple [A]  time = 0.032, size = 189, normalized size = 0.7 \[ -{\frac{2\,{x}^{5}}{13}\sqrt{-{x}^{3}-1}}+{\frac{20\,{x}^{2}}{91}\sqrt{-{x}^{3}-1}}-{{\frac{80\,i}{273}}\sqrt{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}} \left ( \left ({\frac{3}{2}}+{\frac{i}{2}}\sqrt{3} \right ){\it EllipticE} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ) -{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ) \right ){\frac{1}{\sqrt{-{x}^{3}-1}}}} \]

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))-E
llipticF(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 = 2.40444, size = 32, normalized size = 0.11 \[ - \frac{i x^{8} \Gamma \left (\frac{8}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{8}{3} \\ \frac{11}{3} \end{matrix}\middle |{x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac{11}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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