3.241 \(\int \frac{x}{(a-b x^2)^3} \, dx\)

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

[Out]

1/(4*b*(a - b*x^2)^2)

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Rubi [A]  time = 0.0030325, 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} \[ \frac{1}{4 b \left (a-b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[x/(a - b*x^2)^3,x]

[Out]

1/(4*b*(a - b*x^2)^2)

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 )^3} \, dx &=\frac{1}{4 b \left (a-b x^2\right )^2}\\ \end{align*}

Mathematica [A]  time = 0.0020358, size = 17, normalized size = 1. \[ \frac{1}{4 b \left (a-b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a - b*x^2)^3,x]

[Out]

1/(4*b*(a - b*x^2)^2)

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Maple [A]  time = 0.001, size = 17, normalized size = 1. \begin{align*}{\frac{1}{4\,b \left ( b{x}^{2}-a \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(-b*x^2+a)^3,x)

[Out]

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

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Maxima [A]  time = 1.51478, size = 22, normalized size = 1.29 \begin{align*} \frac{1}{4 \,{\left (b x^{2} - a\right )}^{2} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-b*x^2+a)^3,x, algorithm="maxima")

[Out]

1/4/((b*x^2 - a)^2*b)

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Fricas [A]  time = 1.1873, size = 50, normalized size = 2.94 \begin{align*} \frac{1}{4 \,{\left (b^{3} x^{4} - 2 \, a b^{2} x^{2} + a^{2} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-b*x^2+a)^3,x, algorithm="fricas")

[Out]

1/4/(b^3*x^4 - 2*a*b^2*x^2 + a^2*b)

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Sympy [B]  time = 0.389559, size = 26, normalized size = 1.53 \begin{align*} \frac{1}{4 a^{2} b - 8 a b^{2} x^{2} + 4 b^{3} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-b*x**2+a)**3,x)

[Out]

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

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Giac [A]  time = 2.89824, size = 22, normalized size = 1.29 \begin{align*} \frac{1}{4 \,{\left (b x^{2} - a\right )}^{2} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-b*x^2+a)^3,x, algorithm="giac")

[Out]

1/4/((b*x^2 - a)^2*b)