3.451 \(\int \frac{x^7}{\sqrt{1+x^3}} \, dx\)

Optimal. Leaf size=262 \[ \frac{80 \sqrt{x^3+1}}{91 \left (x+\sqrt{3}+1\right )}+\frac{2}{13} \sqrt{x^3+1} x^5-\frac{20}{91} \sqrt{x^3+1} x^2+\frac{80 \sqrt{2} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{91 \sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}}-\frac{40 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{91 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}} \]

[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 = 0.179624, antiderivative size = 262, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{80 \sqrt{x^3+1}}{91 \left (x+\sqrt{3}+1\right )}+\frac{2}{13} \sqrt{x^3+1} x^5-\frac{20}{91} \sqrt{x^3+1} x^2+\frac{80 \sqrt{2} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{91 \sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}}-\frac{40 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{91 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}} \]

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 = 13.6716, size = 240, normalized size = 0.92 \[ \frac{2 x^{5} \sqrt{x^{3} + 1}}{13} - \frac{20 x^{2} \sqrt{x^{3} + 1}}{91} + \frac{80 \sqrt{x^{3} + 1}}{91 \left (x + 1 + \sqrt{3}\right )} - \frac{40 \sqrt [4]{3} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (x + 1\right ) E\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{91 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{x^{3} + 1}} + \frac{80 \sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (x + 1\right ) F\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{273 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{x^{3} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**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
+ 1 + sqrt(3))) - 40*3**(1/4)*sqrt((x**2 - x + 1)/(x + 1 + sqrt(3))**2)*sqrt(-sq
rt(3) + 2)*(x + 1)*elliptic_e(asin((x - sqrt(3) + 1)/(x + 1 + sqrt(3))), -7 - 4*
sqrt(3))/(91*sqrt((x + 1)/(x + 1 + sqrt(3))**2)*sqrt(x**3 + 1)) + 80*sqrt(2)*3**
(3/4)*sqrt((x**2 - x + 1)/(x + 1 + sqrt(3))**2)*(x + 1)*elliptic_f(asin((x - sqr
t(3) + 1)/(x + 1 + sqrt(3))), -7 - 4*sqrt(3))/(273*sqrt((x + 1)/(x + 1 + sqrt(3)
)**2)*sqrt(x**3 + 1))

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Mathematica [A]  time = 0.457826, size = 145, normalized size = 0.55 \[ \frac{2 \left (3 x^2 \left (x^3+1\right ) \left (7 x^3-10\right )-40\ 3^{3/4} \sqrt{-\sqrt [6]{-1} \left (x+(-1)^{2/3}\right )} \sqrt{(-1)^{2/3} x^2+\sqrt [3]{-1} x+1} \left ((-1)^{5/6} F\left (\sin ^{-1}\left (\frac{\sqrt{-(-1)^{5/6} (x+1)}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )+\sqrt{3} E\left (\sin ^{-1}\left (\frac{\sqrt{-(-1)^{5/6} (x+1)}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )\right )}{273 \sqrt{x^3+1}} \]

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*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)])))/(273*Sqrt[1 + x^3])

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Maple [A]  time = 0.026, size = 198, normalized size = 0.8 \[{\frac{2\,{x}^{5}}{13}\sqrt{{x}^{3}+1}}-{\frac{20\,{x}^{2}}{91}\sqrt{{x}^{3}+1}}+{\frac{120-40\,i\sqrt{3}}{91}\sqrt{{\frac{1+x}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}}\sqrt{{\frac{1}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}} \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }}\sqrt{{\frac{1}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}} \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) }} \left ( \left ( -{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3} \right ){\it EllipticE} \left ( \sqrt{{\frac{1+x}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}},\sqrt{{\frac{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}} \right ) + \left ({\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ){\it EllipticF} \left ( \sqrt{{\frac{1+x}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}},\sqrt{{\frac{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}} \right ) \right ){\frac{1}{\sqrt{{x}^{3}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^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)+80/91*(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. \[ \int \frac{x^{7}}{\sqrt{x^{3} + 1}}\,{d x} \]

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 = 2.43256, size = 29, normalized size = 0.11 \[ \frac{x^{8} \Gamma \left (\frac{8}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{8}{3} \\ \frac{11}{3} \end{matrix}\middle |{x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac{11}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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)