3.455 \(\int \frac{1}{x^5 \sqrt{1+x^3}} \, dx\)

Optimal. Leaf size=262 \[ \frac{5 \sqrt{x^3+1}}{8 x}-\frac{5 \sqrt{x^3+1}}{8 \left (x+\sqrt{3}+1\right )}-\frac{\sqrt{x^3+1}}{4 x^4}-\frac{5 (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{4 \sqrt{2} \sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}}+\frac{5 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{16 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}} \]

[Out]

-Sqrt[1 + x^3]/(4*x^4) + (5*Sqrt[1 + x^3])/(8*x) - (5*Sqrt[1 + x^3])/(8*(1 + Sqr
t[3] + x)) + (5*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]]
)/(16*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3]) - (5*(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]])/(4*Sqrt[2]*3^(1/4)*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1
+ x^3])

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Rubi [A]  time = 0.180969, 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 \[ \frac{5 \sqrt{x^3+1}}{8 x}-\frac{5 \sqrt{x^3+1}}{8 \left (x+\sqrt{3}+1\right )}-\frac{\sqrt{x^3+1}}{4 x^4}-\frac{5 (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{4 \sqrt{2} \sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}}+\frac{5 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{16 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*Sqrt[1 + x^3]),x]

[Out]

-Sqrt[1 + x^3]/(4*x^4) + (5*Sqrt[1 + x^3])/(8*x) - (5*Sqrt[1 + x^3])/(8*(1 + Sqr
t[3] + x)) + (5*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]]
)/(16*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3]) - (5*(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]])/(4*Sqrt[2]*3^(1/4)*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1
+ x^3])

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Rubi in Sympy [A]  time = 14.3702, size = 236, normalized size = 0.9 \[ - \frac{5 \sqrt{x^{3} + 1}}{8 \left (x + 1 + \sqrt{3}\right )} + \frac{5 \sqrt [4]{3} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (x + 1\right ) E\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{16 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{x^{3} + 1}} - \frac{5 \sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (x + 1\right ) F\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{24 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{x^{3} + 1}} + \frac{5 \sqrt{x^{3} + 1}}{8 x} - \frac{\sqrt{x^{3} + 1}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(x**3+1)**(1/2),x)

[Out]

-5*sqrt(x**3 + 1)/(8*(x + 1 + sqrt(3))) + 5*3**(1/4)*sqrt((x**2 - x + 1)/(x + 1
+ sqrt(3))**2)*sqrt(-sqrt(3) + 2)*(x + 1)*elliptic_e(asin((x - sqrt(3) + 1)/(x +
1 + sqrt(3))), -7 - 4*sqrt(3))/(16*sqrt((x + 1)/(x + 1 + sqrt(3))**2)*sqrt(x**3
+ 1)) - 5*sqrt(2)*3**(3/4)*sqrt((x**2 - x + 1)/(x + 1 + sqrt(3))**2)*(x + 1)*el
liptic_f(asin((x - sqrt(3) + 1)/(x + 1 + sqrt(3))), -7 - 4*sqrt(3))/(24*sqrt((x
+ 1)/(x + 1 + sqrt(3))**2)*sqrt(x**3 + 1)) + 5*sqrt(x**3 + 1)/(8*x) - sqrt(x**3
+ 1)/(4*x**4)

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Mathematica [A]  time = 0.366755, size = 145, normalized size = 0.55 \[ \frac{5\ 3^{3/4} \sqrt{-\sqrt [6]{-1} \left (x+(-1)^{2/3}\right )} \sqrt{(-1)^{2/3} x^2+\sqrt [3]{-1} x+1} \left ((-1)^{5/6} F\left (\sin ^{-1}\left (\frac{\sqrt{-(-1)^{5/6} (x+1)}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )+\sqrt{3} E\left (\sin ^{-1}\left (\frac{\sqrt{-(-1)^{5/6} (x+1)}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )+\frac{3 \left (x^3+1\right ) \left (5 x^3-2\right )}{x^4}}{24 \sqrt{x^3+1}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/(x^5*Sqrt[1 + x^3]),x]

[Out]

((3*(1 + x^3)*(-2 + 5*x^3))/x^4 + 5*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)]))/(24*Sqrt[1 + x^3])

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Maple [A]  time = 0.029, size = 198, normalized size = 0.8 \[ -{\frac{1}{4\,{x}^{4}}\sqrt{{x}^{3}+1}}+{\frac{5}{8\,x}\sqrt{{x}^{3}+1}}-{\frac{{\frac{15}{2}}-{\frac{5\,i}{2}}\sqrt{3}}{8}\sqrt{{\frac{1+x}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}}\sqrt{{\frac{1}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}} \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }}\sqrt{{\frac{1}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}} \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) }} \left ( \left ( -{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3} \right ){\it EllipticE} \left ( \sqrt{{\frac{1+x}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}},\sqrt{{\frac{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}} \right ) + \left ({\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ){\it EllipticF} \left ( \sqrt{{\frac{1+x}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}},\sqrt{{\frac{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}} \right ) \right ){\frac{1}{\sqrt{{x}^{3}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(x^3+1)^(1/2),x)

[Out]

-1/4*(x^3+1)^(1/2)/x^4+5/8*(x^3+1)^(1/2)/x-5/8*(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))*E
llipticE(((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. \[ \int \frac{1}{\sqrt{x^{3} + 1} x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^3 + 1)*x^5),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(x^3 + 1)*x^5), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{x^{3} + 1} x^{5}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^3 + 1)*x^5),x, algorithm="fricas")

[Out]

integral(1/(sqrt(x^3 + 1)*x^5), x)

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Sympy [A]  time = 2.50536, size = 36, normalized size = 0.14 \[ \frac{\Gamma \left (- \frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{4}{3}, \frac{1}{2} \\ - \frac{1}{3} \end{matrix}\middle |{x^{3} e^{i \pi }} \right )}}{3 x^{4} \Gamma \left (- \frac{1}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**5/(x**3+1)**(1/2),x)

[Out]

gamma(-4/3)*hyper((-4/3, 1/2), (-1/3,), x**3*exp_polar(I*pi))/(3*x**4*gamma(-1/3
))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{3} + 1} x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^3 + 1)*x^5),x, algorithm="giac")

[Out]

integrate(1/(sqrt(x^3 + 1)*x^5), x)