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

Optimal. Leaf size=276 \[ \frac{8 \sqrt{1-x^3}}{7 \left (-x+\sqrt{3}+1\right )}-\frac{2}{7} \sqrt{1-x^3} x^2+\frac{8 \sqrt{2} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{7 \sqrt [4]{3} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}}-\frac{4 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{7 \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}} \]

[Out]

(8*Sqrt[1 - x^3])/(7*(1 + Sqrt[3] - x)) - (2*x^2*Sqrt[1 - x^3])/7 - (4*3^(1/4)*S
qrt[2 - Sqrt[3]]*(1 - x)*Sqrt[(1 + x + x^2)/(1 + Sqrt[3] - x)^2]*EllipticE[ArcSi
n[(1 - Sqrt[3] - x)/(1 + Sqrt[3] - x)], -7 - 4*Sqrt[3]])/(7*Sqrt[(1 - x)/(1 + Sq
rt[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*Sqrt[3]]
)/(7*3^(1/4)*Sqrt[(1 - x)/(1 + Sqrt[3] - x)^2]*Sqrt[1 - x^3])

_______________________________________________________________________________________

Rubi [A]  time = 0.174844, antiderivative size = 276, 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 \[ \frac{8 \sqrt{1-x^3}}{7 \left (-x+\sqrt{3}+1\right )}-\frac{2}{7} \sqrt{1-x^3} x^2+\frac{8 \sqrt{2} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{7 \sqrt [4]{3} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}}-\frac{4 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{7 \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/Sqrt[1 - x^3],x]

[Out]

(8*Sqrt[1 - x^3])/(7*(1 + Sqrt[3] - x)) - (2*x^2*Sqrt[1 - x^3])/7 - (4*3^(1/4)*S
qrt[2 - Sqrt[3]]*(1 - x)*Sqrt[(1 + x + x^2)/(1 + Sqrt[3] - x)^2]*EllipticE[ArcSi
n[(1 - Sqrt[3] - x)/(1 + Sqrt[3] - x)], -7 - 4*Sqrt[3]])/(7*Sqrt[(1 - x)/(1 + Sq
rt[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*Sqrt[3]]
)/(7*3^(1/4)*Sqrt[(1 - x)/(1 + Sqrt[3] - x)^2]*Sqrt[1 - x^3])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 14.0515, size = 224, normalized size = 0.81 \[ - \frac{2 x^{2} \sqrt{- x^{3} + 1}}{7} + \frac{8 \sqrt{- x^{3} + 1}}{7 \left (- x + 1 + \sqrt{3}\right )} - \frac{4 \sqrt [4]{3} \sqrt{\frac{x^{2} + x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (- x + 1\right ) E\left (\operatorname{asin}{\left (\frac{- x - \sqrt{3} + 1}{- x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{7 \sqrt{\frac{- x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- x^{3} + 1}} + \frac{8 \sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} + x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \left (- x + 1\right ) F\left (\operatorname{asin}{\left (\frac{- x - \sqrt{3} + 1}{- x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{21 \sqrt{\frac{- x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- x^{3} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*x**2*sqrt(-x**3 + 1)/7 + 8*sqrt(-x**3 + 1)/(7*(-x + 1 + sqrt(3))) - 4*3**(1/4
)*sqrt((x**2 + x + 1)/(-x + 1 + sqrt(3))**2)*sqrt(-sqrt(3) + 2)*(-x + 1)*ellipti
c_e(asin((-x - sqrt(3) + 1)/(-x + 1 + sqrt(3))), -7 - 4*sqrt(3))/(7*sqrt((-x + 1
)/(-x + 1 + sqrt(3))**2)*sqrt(-x**3 + 1)) + 8*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))/(21*sqrt((-x + 1)/(-x + 1 + sqrt(3))**2)*sqrt(-x**3 +
1))

_______________________________________________________________________________________

Mathematica [C]  time = 0.296948, size = 137, normalized size = 0.5 \[ \frac{2 \left (3 \left (x^3-1\right ) x^2+4 \sqrt [6]{-1} 3^{3/4} \sqrt{(-1)^{5/6} (x-1)} \sqrt{x^2+x+1} \left (\sqrt [3]{-1} F\left (\sin ^{-1}\left (\frac{\sqrt{-i x-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )-i \sqrt{3} E\left (\sin ^{-1}\left (\frac{\sqrt{-i x-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )\right )}{21 \sqrt{1-x^3}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^4/Sqrt[1 - x^3],x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.031, size = 173, normalized size = 0.6 \[ -{\frac{2\,{x}^{2}}{7}\sqrt{-{x}^{3}+1}}-{{\frac{8\,i}{21}}\sqrt{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}} \left ( \left ( -{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3} \right ){\it EllipticE} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ) +{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ) \right ){\frac{1}{\sqrt{-{x}^{3}+1}}}} \]

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 = 2.05211, size = 31, normalized size = 0.11 \[ \frac{x^{5} \Gamma \left (\frac{5}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{5}{3} \\ \frac{8}{3} \end{matrix}\middle |{x^{3} e^{2 i \pi }} \right )}}{3 \Gamma \left (\frac{8}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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