3.248 \(\int \frac{x}{(a-b x^2)^5} \, dx\)

Optimal. Leaf size=17 \[ \frac{1}{8 b \left (a-b x^2\right )^4} \]

[Out]

1/(8*b*(a - b*x^2)^4)

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Rubi [A]  time = 0.0027673, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {261} \[ \frac{1}{8 b \left (a-b x^2\right )^4} \]

Antiderivative was successfully verified.

[In]

Int[x/(a - b*x^2)^5,x]

[Out]

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

Mathematica [A]  time = 0.0030155, size = 17, normalized size = 1. \[ \frac{1}{8 b \left (a-b x^2\right )^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a - b*x^2)^5,x]

[Out]

1/(8*b*(a - b*x^2)^4)

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Maple [A]  time = 0., size = 17, normalized size = 1. \begin{align*}{\frac{1}{8\,b \left ( b{x}^{2}-a \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8/b/(b*x^2-a)^4

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Maxima [A]  time = 2.10508, size = 22, normalized size = 1.29 \begin{align*} \frac{1}{8 \,{\left (b x^{2} - a\right )}^{4} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-b*x^2+a)^5,x, algorithm="maxima")

[Out]

1/8/((b*x^2 - a)^4*b)

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Fricas [B]  time = 1.16939, size = 93, normalized size = 5.47 \begin{align*} \frac{1}{8 \,{\left (b^{5} x^{8} - 4 \, a b^{4} x^{6} + 6 \, a^{2} b^{3} x^{4} - 4 \, a^{3} b^{2} x^{2} + a^{4} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-b*x^2+a)^5,x, algorithm="fricas")

[Out]

1/8/(b^5*x^8 - 4*a*b^4*x^6 + 6*a^2*b^3*x^4 - 4*a^3*b^2*x^2 + a^4*b)

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Sympy [B]  time = 0.730105, size = 49, normalized size = 2.88 \begin{align*} \frac{1}{8 a^{4} b - 32 a^{3} b^{2} x^{2} + 48 a^{2} b^{3} x^{4} - 32 a b^{4} x^{6} + 8 b^{5} x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-b*x**2+a)**5,x)

[Out]

1/(8*a**4*b - 32*a**3*b**2*x**2 + 48*a**2*b**3*x**4 - 32*a*b**4*x**6 + 8*b**5*x**8)

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Giac [A]  time = 2.82081, size = 22, normalized size = 1.29 \begin{align*} \frac{1}{8 \,{\left (b x^{2} - a\right )}^{4} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-b*x^2+a)^5,x, algorithm="giac")

[Out]

1/8/((b*x^2 - a)^4*b)