3.514 \(\int \frac{1}{x (1+x)^{3/2} \left (1-x+x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=66 \[ \frac{2}{3 \sqrt{x+1} \sqrt{x^2-x+1}}-\frac{2 \sqrt{x^3+1} \tanh ^{-1}\left (\sqrt{x^3+1}\right )}{3 \sqrt{x+1} \sqrt{x^2-x+1}} \]

[Out]

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

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Rubi [A]  time = 0.0958263, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ \frac{2}{3 \sqrt{x+1} \sqrt{x^2-x+1}}-\frac{2 \sqrt{x^3+1} \tanh ^{-1}\left (\sqrt{x^3+1}\right )}{3 \sqrt{x+1} \sqrt{x^2-x+1}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(1 + x)^(3/2)*(1 - x + x^2)^(3/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 9.79367, size = 63, normalized size = 0.95 \[ \frac{2 \sqrt{x + 1} \sqrt{x^{2} - x + 1}}{3 \left (x^{3} + 1\right )} - \frac{2 \sqrt{x + 1} \sqrt{x^{2} - x + 1} \operatorname{atanh}{\left (\sqrt{x^{3} + 1} \right )}}{3 \sqrt{x^{3} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x + 1)*sqrt(x**2 - x + 1)/(3*(x**3 + 1)) - 2*sqrt(x + 1)*sqrt(x**2 - x +
1)*atanh(sqrt(x**3 + 1))/(3*sqrt(x**3 + 1))

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Mathematica [C]  time = 0.343846, size = 88, normalized size = 1.33 \[ \frac{2 \left (\frac{1}{\sqrt{x^2-x+1}}-\frac{\sqrt{3} (x+1) \Pi \left (1+\sqrt [3]{-1};\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{\frac{x+1}{1+\sqrt [3]{-1}}}}\right )}{3 \sqrt{x+1}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(1 + x)^(3/2)*(1 - x + x^2)^(3/2)),x]

[Out]

(2*(1/Sqrt[1 - x + x^2] - (Sqrt[3]*(1 + x)*EllipticPi[1 + (-1)^(1/3), ArcSin[Sqr
t[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 + x)/(1 + (-1)^(1/
3))]))/(3*Sqrt[1 + x])

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Maple [A]  time = 0.05, size = 43, normalized size = 0.7 \[ -{\frac{2}{3\,{x}^{3}+3}\sqrt{1+x}\sqrt{{x}^{2}-x+1} \left ({\it Artanh} \left ( \sqrt{{x}^{3}+1} \right ) \sqrt{{x}^{3}+1}-1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(1+x)^(3/2)/(x^2-x+1)^(3/2),x)

[Out]

-2/3*(1+x)^(1/2)*(x^2-x+1)^(1/2)*(arctanh((x^3+1)^(1/2))*(x^3+1)^(1/2)-1)/(x^3+1
)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{2} - x + 1\right )}^{\frac{3}{2}}{\left (x + 1\right )}^{\frac{3}{2}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 - x + 1)^(3/2)*(x + 1)^(3/2)*x),x, algorithm="maxima")

[Out]

integrate(1/((x^2 - x + 1)^(3/2)*(x + 1)^(3/2)*x), x)

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Fricas [A]  time = 0.287374, size = 122, normalized size = 1.85 \[ -\frac{\sqrt{x^{2} - x + 1} \sqrt{x + 1} \log \left (\sqrt{x^{2} - x + 1} \sqrt{x + 1} + 1\right ) - \sqrt{x^{2} - x + 1} \sqrt{x + 1} \log \left (\sqrt{x^{2} - x + 1} \sqrt{x + 1} - 1\right ) - 2}{3 \, \sqrt{x^{2} - x + 1} \sqrt{x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 - x + 1)^(3/2)*(x + 1)^(3/2)*x),x, algorithm="fricas")

[Out]

-1/3*(sqrt(x^2 - x + 1)*sqrt(x + 1)*log(sqrt(x^2 - x + 1)*sqrt(x + 1) + 1) - sqr
t(x^2 - x + 1)*sqrt(x + 1)*log(sqrt(x^2 - x + 1)*sqrt(x + 1) - 1) - 2)/(sqrt(x^2
- x + 1)*sqrt(x + 1))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \left (x + 1\right )^{\frac{3}{2}} \left (x^{2} - x + 1\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x*(x + 1)**(3/2)*(x**2 - x + 1)**(3/2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{2} - x + 1\right )}^{\frac{3}{2}}{\left (x + 1\right )}^{\frac{3}{2}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 - x + 1)^(3/2)*(x + 1)^(3/2)*x),x, algorithm="giac")

[Out]

integrate(1/((x^2 - x + 1)^(3/2)*(x + 1)^(3/2)*x), x)