3.423 \(\int \frac {x}{(1-x^2)^5} \, dx\)

Optimal. Leaf size=13 \[ \frac {1}{8 \left (1-x^2\right )^4} \]

[Out]

1/8/(-x^2+1)^4

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Rubi [A]  time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {261} \[ \frac {1}{8 \left (1-x^2\right )^4} \]

Antiderivative was successfully verified.

[In]

Int[x/(1 - x^2)^5,x]

[Out]

1/(8*(1 - x^2)^4)

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin {align*} \int \frac {x}{\left (1-x^2\right )^5} \, dx &=\frac {1}{8 \left (1-x^2\right )^4}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 11, normalized size = 0.85 \[ \frac {1}{8 \left (x^2-1\right )^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(1 - x^2)^5,x]

[Out]

1/(8*(-1 + x^2)^4)

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fricas [B]  time = 0.59, size = 24, normalized size = 1.85 \[ \frac {1}{8 \, {\left (x^{8} - 4 \, x^{6} + 6 \, x^{4} - 4 \, x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8/(x^8 - 4*x^6 + 6*x^4 - 4*x^2 + 1)

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giac [A]  time = 0.33, size = 9, normalized size = 0.69 \[ \frac {1}{8 \, {\left (x^{2} - 1\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8/(x^2 - 1)^4

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maple [A]  time = 0.00, size = 10, normalized size = 0.77 \[ \frac {1}{8 \left (x^{2}-1\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8/(x^2-1)^4

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maxima [A]  time = 0.88, size = 9, normalized size = 0.69 \[ \frac {1}{8 \, {\left (x^{2} - 1\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8/(x^2 - 1)^4

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mupad [B]  time = 2.32, size = 9, normalized size = 0.69 \[ \frac {1}{8\,{\left (x^2-1\right )}^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-x/(x^2 - 1)^5,x)

[Out]

1/(8*(x^2 - 1)^4)

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sympy [B]  time = 0.12, size = 22, normalized size = 1.69 \[ \frac {1}{8 x^{8} - 32 x^{6} + 48 x^{4} - 32 x^{2} + 8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(8*x**8 - 32*x**6 + 48*x**4 - 32*x**2 + 8)

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