3.193 \(\int \frac{x}{(1-x^2)^{9/8}} \, dx\)

Optimal. Leaf size=13 \[ \frac{4}{\sqrt [8]{1-x^2}} \]

[Out]

4/(1 - x^2)^(1/8)

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Rubi [A]  time = 0.0024579, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {261} \[ \frac{4}{\sqrt [8]{1-x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x/(1 - x^2)^(9/8),x]

[Out]

4/(1 - x^2)^(1/8)

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 )^{9/8}} \, dx &=\frac{4}{\sqrt [8]{1-x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0021493, size = 13, normalized size = 1. \[ \frac{4}{\sqrt [8]{1-x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(1 - x^2)^(9/8),x]

[Out]

4/(1 - x^2)^(1/8)

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Maple [A]  time = 0.002, size = 18, normalized size = 1.4 \begin{align*} -4\,{ \left ( -1+x \right ) \left ( 1+x \right ) \left ( -{x}^{2}+1 \right ) ^{-{\frac{9}{8}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(-x^2+1)^(9/8),x)

[Out]

-4*(-1+x)*(1+x)/(-x^2+1)^(9/8)

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Maxima [A]  time = 0.957097, size = 15, normalized size = 1.15 \begin{align*} \frac{4}{{\left (-x^{2} + 1\right )}^{\frac{1}{8}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x^2+1)^(9/8),x, algorithm="maxima")

[Out]

4/(-x^2 + 1)^(1/8)

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Fricas [A]  time = 1.52099, size = 42, normalized size = 3.23 \begin{align*} -\frac{4 \,{\left (-x^{2} + 1\right )}^{\frac{7}{8}}}{x^{2} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x^2+1)^(9/8),x, algorithm="fricas")

[Out]

-4*(-x^2 + 1)^(7/8)/(x^2 - 1)

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Sympy [A]  time = 1.07254, size = 8, normalized size = 0.62 \begin{align*} \frac{4}{\sqrt [8]{1 - x^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x**2+1)**(9/8),x)

[Out]

4/(1 - x**2)**(1/8)

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Giac [A]  time = 1.08615, size = 15, normalized size = 1.15 \begin{align*} \frac{4}{{\left (-x^{2} + 1\right )}^{\frac{1}{8}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x^2+1)^(9/8),x, algorithm="giac")

[Out]

4/(-x^2 + 1)^(1/8)