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

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

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Rubi [A]  time = 0.02, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {913} \begin {gather*} -\frac {2}{9 (x+1)^{3/2} \left (x^2-x+1\right )^{3/2}} \end {gather*}

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))

Rule 913

Int[(x_)^2*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^
(m + 1)*(a + b*x + c*x^2)^(p + 1))/(c*e*(m + 2*p + 3)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*e*(m +
p + 2) + 2*c*d*(p + 1), 0] && EqQ[b*d*(p + 1) + a*e*(m + 1), 0] && NeQ[m + 2*p + 3, 0]

Rubi steps

\begin {align*} \int \frac {x^2}{(1+x)^{5/2} \left (1-x+x^2\right )^{5/2}} \, dx &=-\frac {2}{9 (1+x)^{3/2} \left (1-x+x^2\right )^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 23, normalized size = 1.00 \begin {gather*} -\frac {2}{9 (x+1)^{3/2} \left (x^2-x+1\right )^{3/2}} \end {gather*}

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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IntegrateAlgebraic [F]  time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

$Aborted

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fricas [A]  time = 0.40, size = 29, normalized size = 1.26 \begin {gather*} -\frac {2 \, \sqrt {x^{2} - x + 1} \sqrt {x + 1}}{9 \, {\left (x^{6} + 2 \, x^{3} + 1\right )}} \end {gather*}

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, algorithm="fricas")

[Out]

-2/9*sqrt(x^2 - x + 1)*sqrt(x + 1)/(x^6 + 2*x^3 + 1)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{{\left (x^{2} - x + 1\right )}^{\frac {5}{2}} {\left (x + 1\right )}^{\frac {5}{2}}}\,{d x} \end {gather*}

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, algorithm="giac")

[Out]

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

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maple [A]  time = 0.00, size = 18, normalized size = 0.78 \begin {gather*} -\frac {2}{9 \left (x +1\right )^{\frac {3}{2}} \left (x^{2}-x +1\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.97, size = 24, normalized size = 1.04 \begin {gather*} -\frac {2}{9 \, {\left (x^{3} + 1\right )} \sqrt {x^{2} - x + 1} \sqrt {x + 1}} \end {gather*}

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, algorithm="maxima")

[Out]

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

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mupad [B]  time = 2.88, size = 82, normalized size = 3.57 \begin {gather*} \frac {18\,\sqrt {x+1}\,{\left (x^2-x+1\right )}^{5/2}-18\,x\,\sqrt {x+1}\,{\left (x^2-x+1\right )}^{5/2}}{\left (x+1\right )\,\left (81\,x\,{\left (x^2-x+1\right )}^4-162\,{\left (x^2-x+1\right )}^4+81\,{\left (x^2-x+1\right )}^5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(18*(x + 1)^(1/2)*(x^2 - x + 1)^(5/2) - 18*x*(x + 1)^(1/2)*(x^2 - x + 1)^(5/2))/((x + 1)*(81*x*(x^2 - x + 1)^4
- 162*(x^2 - x + 1)^4 + 81*(x^2 - x + 1)^5))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\left (x + 1\right )^{\frac {5}{2}} \left (x^{2} - x + 1\right )^{\frac {5}{2}}}\, dx \end {gather*}

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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