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

Optimal. Leaf size=23 \[ -\frac{2}{9 (x+1)^{3/2} \left (x^2-x+1\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0186099, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ -\frac{2}{9 (x+1)^{3/2} \left (x^2-x+1\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.2097, size = 27, normalized size = 1.17 \[ - \frac{2 \sqrt{x + 1} \sqrt{x^{2} - x + 1}}{9 \left (x^{3} + 1\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(x + 1)*sqrt(x**2 - x + 1)/(9*(x**3 + 1)**2)

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Mathematica [A]  time = 0.044109, size = 23, normalized size = 1. \[ -\frac{2}{9 (x+1)^{3/2} \left (x^2-x+1\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 18, normalized size = 0.8 \[ -{\frac{2}{9} \left ( 1+x \right ) ^{-{\frac{3}{2}}} \left ({x}^{2}-x+1 \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/9/(1+x)^(3/2)/(x^2-x+1)^(3/2)

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Maxima [A]  time = 0.771442, size = 32, normalized size = 1.39 \[ -\frac{2}{9 \,{\left (x^{3} + 1\right )} \sqrt{x^{2} - x + 1} \sqrt{x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.271938, size = 32, normalized size = 1.39 \[ -\frac{2}{9 \,{\left (x^{3} + 1\right )} \sqrt{x^{2} - x + 1} \sqrt{x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**2/((x + 1)**(5/2)*(x**2 - x + 1)**(5/2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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