3.446 \(\int \frac{x^6}{\sqrt{1+x^3}} \, dx\)

Optimal. Leaf size=136 \[ -\frac{16}{55} \sqrt{x^3+1} x+\frac{2}{11} \sqrt{x^3+1} x^4+\frac{32 \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{55 \sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}} \]

[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 = 0.0971414, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{16}{55} \sqrt{x^3+1} x+\frac{2}{11} \sqrt{x^3+1} x^4+\frac{32 \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{55 \sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}} \]

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 = 5.70981, size = 124, normalized size = 0.91 \[ \frac{2 x^{4} \sqrt{x^{3} + 1}}{11} - \frac{16 x \sqrt{x^{3} + 1}}{55} + \frac{32 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (x + 1\right ) F\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{165 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{x^{3} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Mathematica [A]  time = 0.23599, size = 108, normalized size = 0.79 \[ \frac{2 \left (16 \sqrt [6]{-1} 3^{3/4} \sqrt{-\sqrt [6]{-1} \left (x+(-1)^{2/3}\right )} \sqrt{(-1)^{2/3} x^2+\sqrt [3]{-1} x+1} F\left (\sin ^{-1}\left (\frac{\sqrt{-(-1)^{5/6} (x+1)}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )+3 x \left (5 x^6-3 x^3-8\right )\right )}{165 \sqrt{x^3+1}} \]

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

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Maple [A]  time = 0.025, size = 139, normalized size = 1. \[{\frac{2\,{x}^{4}}{11}\sqrt{{x}^{3}+1}}-{\frac{16\,x}{55}\sqrt{{x}^{3}+1}}+{\frac{48-16\,i\sqrt{3}}{55}\sqrt{{\frac{1+x}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}}\sqrt{{\frac{1}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}} \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }}\sqrt{{\frac{1}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}} \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) }}{\it EllipticF} \left ( \sqrt{{\frac{1+x}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}},\sqrt{{\frac{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{{x}^{3}+1}}}} \]

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)+32/55*(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)*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. \[ \int \frac{x^{6}}{\sqrt{x^{3} + 1}}\,{d x} \]

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 = 2.21569, size = 29, normalized size = 0.21 \[ \frac{x^{7} \Gamma \left (\frac{7}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{7}{3} \\ \frac{10}{3} \end{matrix}\middle |{x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac{10}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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)