3.452 \(\int \frac{x^4}{\sqrt{1+x^3}} \, dx\)

Optimal. Leaf size=246 \[ -\frac{8 \sqrt{x^3+1}}{7 \left (x+\sqrt{3}+1\right )}+\frac{2}{7} \sqrt{x^3+1} x^2-\frac{8 \sqrt{2} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{7 \sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}}+\frac{4 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{7 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}} \]

[Out]

(2*x^2*Sqrt[1 + x^3])/7 - (8*Sqrt[1 + x^3])/(7*(1 + Sqrt[3] + x)) + (4*3^(1/4)*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])

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Rubi [A]  time = 0.141592, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{8 \sqrt{x^3+1}}{7 \left (x+\sqrt{3}+1\right )}+\frac{2}{7} \sqrt{x^3+1} x^2-\frac{8 \sqrt{2} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{7 \sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}}+\frac{4 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{7 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}} \]

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)*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])

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Rubi in Sympy [A]  time = 11.2357, size = 224, normalized size = 0.91 \[ \frac{2 x^{2} \sqrt{x^{3} + 1}}{7} - \frac{8 \sqrt{x^{3} + 1}}{7 \left (x + 1 + \sqrt{3}\right )} + \frac{4 \sqrt [4]{3} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (x + 1\right ) E\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{7 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{x^{3} + 1}} - \frac{8 \sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (x + 1\right ) F\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{21 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{x^{3} + 1}} \]

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)*sq
rt((x**2 - x + 1)/(x + 1 + sqrt(3))**2)*sqrt(-sqrt(3) + 2)*(x + 1)*elliptic_e(as
in((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))

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Mathematica [A]  time = 0.58398, size = 138, normalized size = 0.56 \[ \frac{2 \left (4\ 3^{3/4} \sqrt{-\sqrt [6]{-1} \left (x+(-1)^{2/3}\right )} \sqrt{(-1)^{2/3} x^2+\sqrt [3]{-1} x+1} \left ((-1)^{5/6} F\left (\sin ^{-1}\left (\frac{\sqrt{-(-1)^{5/6} (x+1)}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )+\sqrt{3} E\left (\sin ^{-1}\left (\frac{\sqrt{-(-1)^{5/6} (x+1)}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )+3 \left (x^3+1\right ) x^2\right )}{21 \sqrt{x^3+1}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^4/Sqrt[1 + x^3],x]

[Out]

(2*(3*x^2*(1 + x^3) + 4*3^(3/4)*Sqrt[-((-1)^(1/6)*((-1)^(2/3) + x))]*Sqrt[1 + (-
1)^(1/3)*x + (-1)^(2/3)*x^2]*(Sqrt[3]*EllipticE[ArcSin[Sqrt[-((-1)^(5/6)*(1 + x)
)]/3^(1/4)], (-1)^(1/3)] + (-1)^(5/6)*EllipticF[ArcSin[Sqrt[-((-1)^(5/6)*(1 + x)
)]/3^(1/4)], (-1)^(1/3)])))/(21*Sqrt[1 + x^3])

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Maple [A]  time = 0.024, size = 186, normalized size = 0.8 \[{\frac{2\,{x}^{2}}{7}\sqrt{{x}^{3}+1}}-{\frac{-4\,i\sqrt{3}+12}{7}\sqrt{{\frac{1+x}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}}\sqrt{{\frac{1}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}} \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }}\sqrt{{\frac{1}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}} \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) }} \left ( \left ( -{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3} \right ){\it EllipticE} \left ( \sqrt{{\frac{1+x}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}},\sqrt{{\frac{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}} \right ) + \left ({\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ){\it EllipticF} \left ( \sqrt{{\frac{1+x}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}},\sqrt{{\frac{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}} \right ) \right ){\frac{1}{\sqrt{{x}^{3}+1}}}} \]

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)-8/7*(3/2-1/2*I*3^(1/2))*((1+x)/(3/2-1/2*I*3^(1/2)))^(1/2)*
((x-1/2-1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2+1/2*I*3^(1/2))/(-3/2+
1/2*I*3^(1/2)))^(1/2)/(x^3+1)^(1/2)*((-3/2-1/2*I*3^(1/2))*EllipticE(((1+x)/(3/2-
1/2*I*3^(1/2)))^(1/2),((-3/2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2))+(1/2+1/
2*I*3^(1/2))*EllipticF(((1+x)/(3/2-1/2*I*3^(1/2)))^(1/2),((-3/2+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. \[ \int \frac{x^{4}}{\sqrt{x^{3} + 1}}\,{d x} \]

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)

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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)

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Sympy [A]  time = 1.95516, size = 29, normalized size = 0.12 \[ \frac{x^{5} \Gamma \left (\frac{5}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{5}{3} \\ \frac{8}{3} \end{matrix}\middle |{x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac{8}{3}\right )} \]

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(I*pi))/(3*gamma(8/3))

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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)