3.521 \(\int \frac{1}{x (1+x)^{5/2} \left (1-x+x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=96 \[ \frac{2}{3 \sqrt{x+1} \sqrt{x^2-x+1}}+\frac{2}{9 \sqrt{x+1} \sqrt{x^2-x+1} \left (x^3+1\right )}-\frac{2 \sqrt{x^3+1} \tanh ^{-1}\left (\sqrt{x^3+1}\right )}{3 \sqrt{x+1} \sqrt{x^2-x+1}} \]

[Out]

2/(3*Sqrt[1 + x]*Sqrt[1 - x + x^2]) + 2/(9*Sqrt[1 + x]*Sqrt[1 - x + x^2]*(1 + x^
3)) - (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 = 0.112605, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ \frac{2}{3 \sqrt{x+1} \sqrt{x^2-x+1}}+\frac{2}{9 \sqrt{x+1} \sqrt{x^2-x+1} \left (x^3+1\right )}-\frac{2 \sqrt{x^3+1} \tanh ^{-1}\left (\sqrt{x^3+1}\right )}{3 \sqrt{x+1} \sqrt{x^2-x+1}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(1 + x)^(5/2)*(1 - x + x^2)^(5/2)),x]

[Out]

2/(3*Sqrt[1 + x]*Sqrt[1 - x + x^2]) + 2/(9*Sqrt[1 + x]*Sqrt[1 - x + x^2]*(1 + x^
3)) - (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 = 10.8541, size = 90, normalized size = 0.94 \[ \frac{2 \sqrt{x + 1} \sqrt{x^{2} - x + 1}}{3 \left (x^{3} + 1\right )} + \frac{2 \sqrt{x + 1} \sqrt{x^{2} - x + 1}}{9 \left (x^{3} + 1\right )^{2}} - \frac{2 \sqrt{x + 1} \sqrt{x^{2} - x + 1} \operatorname{atanh}{\left (\sqrt{x^{3} + 1} \right )}}{3 \sqrt{x^{3} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x + 1)*sqrt(x**2 - x + 1)/(3*(x**3 + 1)) + 2*sqrt(x + 1)*sqrt(x**2 - x +
1)/(9*(x**3 + 1)**2) - 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 = 0.369333, size = 98, normalized size = 1.02 \[ \frac{2 \left (\frac{3 x^3+4}{\left (x^2-x+1\right )^{3/2}}-\frac{3 \sqrt{3} (x+1)^2 \Pi \left (1+\sqrt [3]{-1};\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{\frac{x+1}{1+\sqrt [3]{-1}}}}\right )}{9 (x+1)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(1 + x)^(5/2)*(1 - x + x^2)^(5/2)),x]

[Out]

(2*((4 + 3*x^3)/(1 - x + x^2)^(3/2) - (3*Sqrt[3]*(1 + x)^2*EllipticPi[1 + (-1)^(
1/3), ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 +
x)/(1 + (-1)^(1/3))]))/(9*(1 + x)^(3/2))

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Maple [A]  time = 0.071, size = 69, normalized size = 0.7 \[ -{\frac{2}{9\,{x}^{3}+9} \left ( 3\,{\it Artanh} \left ( \sqrt{{x}^{3}+1} \right ) \sqrt{{x}^{3}+1}{x}^{3}-3\,{x}^{3}+3\,{\it Artanh} \left ( \sqrt{{x}^{3}+1} \right ) \sqrt{{x}^{3}+1}-4 \right ){\frac{1}{\sqrt{1+x}}}{\frac{1}{\sqrt{{x}^{2}-x+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(1+x)^(5/2)/(x^2-x+1)^(5/2),x)

[Out]

-2/9*(3*arctanh((x^3+1)^(1/2))*(x^3+1)^(1/2)*x^3-3*x^3+3*arctanh((x^3+1)^(1/2))*
(x^3+1)^(1/2)-4)/(x^3+1)/(x^2-x+1)^(1/2)/(1+x)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{2} - x + 1\right )}^{\frac{5}{2}}{\left (x + 1\right )}^{\frac{5}{2}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 - x + 1)^(5/2)*(x + 1)^(5/2)*x),x, algorithm="maxima")

[Out]

integrate(1/((x^2 - x + 1)^(5/2)*(x + 1)^(5/2)*x), x)

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Fricas [A]  time = 0.277998, size = 153, normalized size = 1.59 \[ \frac{6 \, x^{3} - 3 \,{\left (x^{3} + 1\right )} \sqrt{x^{2} - x + 1} \sqrt{x + 1} \log \left (\sqrt{x^{2} - x + 1} \sqrt{x + 1} + 1\right ) + 3 \,{\left (x^{3} + 1\right )} \sqrt{x^{2} - x + 1} \sqrt{x + 1} \log \left (\sqrt{x^{2} - x + 1} \sqrt{x + 1} - 1\right ) + 8}{9 \,{\left (x^{3} + 1\right )} \sqrt{x^{2} - x + 1} \sqrt{x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 - x + 1)^(5/2)*(x + 1)^(5/2)*x),x, algorithm="fricas")

[Out]

1/9*(6*x^3 - 3*(x^3 + 1)*sqrt(x^2 - x + 1)*sqrt(x + 1)*log(sqrt(x^2 - x + 1)*sqr
t(x + 1) + 1) + 3*(x^3 + 1)*sqrt(x^2 - x + 1)*sqrt(x + 1)*log(sqrt(x^2 - x + 1)*
sqrt(x + 1) - 1) + 8)/((x^3 + 1)*sqrt(x^2 - x + 1)*sqrt(x + 1))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{2} - x + 1\right )}^{\frac{5}{2}}{\left (x + 1\right )}^{\frac{5}{2}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 - x + 1)^(5/2)*(x + 1)^(5/2)*x),x, algorithm="giac")

[Out]

integrate(1/((x^2 - x + 1)^(5/2)*(x + 1)^(5/2)*x), x)