3.506 \(\int \frac{x^4}{\sqrt{-1-x^3}} \, dx\)

Optimal. Leaf size=264 \[ \frac{8 \sqrt{-x^3-1}}{7 \left (x-\sqrt{3}+1\right )}-\frac{2}{7} \sqrt{-x^3-1} x^2+\frac{8 \sqrt{2} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{7 \sqrt [4]{3} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}}-\frac{4 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{7 \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}} \]

[Out]

(-2*x^2*Sqrt[-1 - x^3])/7 + (8*Sqrt[-1 - x^3])/(7*(1 - Sqrt[3] + x)) - (4*3^(1/4
)*Sqrt[2 + Sqrt[3]]*(1 + x)*Sqrt[(1 - x + x^2)/(1 - Sqrt[3] + x)^2]*EllipticE[Ar
cSin[(1 + Sqrt[3] + x)/(1 - Sqrt[3] + x)], -7 + 4*Sqrt[3]])/(7*Sqrt[-((1 + x)/(1
- Sqrt[3] + x)^2)]*Sqrt[-1 - x^3]) + (8*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*S
qrt[3]])/(7*3^(1/4)*Sqrt[-((1 + x)/(1 - Sqrt[3] + x)^2)]*Sqrt[-1 - x^3])

_______________________________________________________________________________________

Rubi [A]  time = 0.155917, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{8 \sqrt{-x^3-1}}{7 \left (x-\sqrt{3}+1\right )}-\frac{2}{7} \sqrt{-x^3-1} x^2+\frac{8 \sqrt{2} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{7 \sqrt [4]{3} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}}-\frac{4 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{7 \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/Sqrt[-1 - x^3],x]

[Out]

(-2*x^2*Sqrt[-1 - x^3])/7 + (8*Sqrt[-1 - x^3])/(7*(1 - Sqrt[3] + x)) - (4*3^(1/4
)*Sqrt[2 + Sqrt[3]]*(1 + x)*Sqrt[(1 - x + x^2)/(1 - Sqrt[3] + x)^2]*EllipticE[Ar
cSin[(1 + Sqrt[3] + x)/(1 - Sqrt[3] + x)], -7 + 4*Sqrt[3]])/(7*Sqrt[-((1 + x)/(1
- Sqrt[3] + x)^2)]*Sqrt[-1 - x^3]) + (8*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*S
qrt[3]])/(7*3^(1/4)*Sqrt[-((1 + x)/(1 - Sqrt[3] + x)^2)]*Sqrt[-1 - x^3])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 11.7093, size = 231, normalized size = 0.88 \[ - \frac{2 x^{2} \sqrt{- x^{3} - 1}}{7} + \frac{8 \sqrt{- x^{3} - 1}}{7 \left (x - \sqrt{3} + 1\right )} - \frac{4 \sqrt [4]{3} \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (x + 1\right ) E\left (\operatorname{asin}{\left (\frac{x + 1 + \sqrt{3}}{x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{7 \sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{- x^{3} - 1}} + \frac{8 \sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \left (x + 1\right ) F\left (\operatorname{asin}{\left (\frac{x + 1 + \sqrt{3}}{x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{21 \sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{- x^{3} - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(-x**3-1)**(1/2),x)

[Out]

-2*x**2*sqrt(-x**3 - 1)/7 + 8*sqrt(-x**3 - 1)/(7*(x - sqrt(3) + 1)) - 4*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))/(7*sqrt((-x - 1)/(x -
sqrt(3) + 1)**2)*sqrt(-x**3 - 1)) + 8*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))/(21*sqrt((-x - 1)/(x - sqrt(3) + 1)**2)*sqrt(-x**3 - 1))

_______________________________________________________________________________________

Mathematica [C]  time = 0.437166, size = 157, normalized size = 0.59 \[ \frac{2 \left (3 x^2 \left (x^3+1\right )-4 (-1)^{5/6} 3^{3/4} \sqrt{-(-1)^{5/6}+i x} \sqrt{-\sqrt [3]{-1} x^2-(-1)^{2/3} x+1} \left (\sqrt [3]{-1} F\left (\sin ^{-1}\left (\frac{\sqrt{-\sqrt [6]{-1} \left (x+(-1)^{2/3}\right )}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )-i \sqrt{3} E\left (\sin ^{-1}\left (\frac{\sqrt{-\sqrt [6]{-1} \left (x+(-1)^{2/3}\right )}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )\right )}{21 \sqrt{-x^3-1}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^4/Sqrt[-1 - x^3],x]

[Out]

(2*(3*x^2*(1 + x^3) - 4*(-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[ArcSin[Sqrt[-((-1)^(
1/6)*((-1)^(2/3) + x))]/3^(1/4)], (-1)^(1/3)])))/(21*Sqrt[-1 - x^3])

_______________________________________________________________________________________

Maple [A]  time = 0.031, size = 175, normalized size = 0.7 \[ -{\frac{2\,{x}^{2}}{7}\sqrt{-{x}^{3}-1}}+{{\frac{8\,i}{21}}\sqrt{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}} \left ( \left ({\frac{3}{2}}+{\frac{i}{2}}\sqrt{3} \right ){\it EllipticE} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ) -{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ) \right ){\frac{1}{\sqrt{-{x}^{3}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(-x^3-1)^(1/2),x)

[Out]

-2/7*x^2*(-x^3-1)^(1/2)+8/21*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)))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\sqrt{-x^{3} - 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/sqrt(-x^3 - 1),x, algorithm="maxima")

[Out]

integrate(x^4/sqrt(-x^3 - 1), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{4}}{\sqrt{-x^{3} - 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/sqrt(-x^3 - 1),x, algorithm="fricas")

[Out]

integral(x^4/sqrt(-x^3 - 1), x)

_______________________________________________________________________________________

Sympy [A]  time = 1.94288, size = 32, normalized size = 0.12 \[ - \frac{i x^{5} \Gamma \left (\frac{5}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{5}{3} \\ \frac{8}{3} \end{matrix}\middle |{x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac{8}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(-x**3-1)**(1/2),x)

[Out]

-I*x**5*gamma(5/3)*hyper((1/2, 5/3), (8/3,), x**3*exp_polar(I*pi))/(3*gamma(8/3)
)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\sqrt{-x^{3} - 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/sqrt(-x^3 - 1),x, algorithm="giac")

[Out]

integrate(x^4/sqrt(-x^3 - 1), x)