3.1867 \(\int \frac{1}{(a+\frac{b}{x^2})^2 x^3} \, dx\)

Optimal. Leaf size=16 \[ \frac{1}{2 b \left (a+\frac{b}{x^2}\right )} \]

[Out]

1/(2*b*(a + b/x^2))

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Rubi [A]  time = 0.003838, antiderivative size = 16, 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} \[ \frac{1}{2 b \left (a+\frac{b}{x^2}\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^2)^2*x^3),x]

[Out]

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

Mathematica [A]  time = 0.0020879, size = 16, normalized size = 1. \[ -\frac{1}{2 a \left (a x^2+b\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^2)^2*x^3),x]

[Out]

-1/(2*a*(b + a*x^2))

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Maple [A]  time = 0., size = 15, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,a \left ( a{x}^{2}+b \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+1/x^2*b)^2/x^3,x)

[Out]

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

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Maxima [A]  time = 0.987966, size = 19, normalized size = 1.19 \begin{align*} \frac{1}{2 \,{\left (a + \frac{b}{x^{2}}\right )} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^2/x^3,x, algorithm="maxima")

[Out]

1/2/((a + b/x^2)*b)

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Fricas [A]  time = 1.41691, size = 30, normalized size = 1.88 \begin{align*} -\frac{1}{2 \,{\left (a^{2} x^{2} + a b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^2/x^3,x, algorithm="fricas")

[Out]

-1/2/(a^2*x^2 + a*b)

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Sympy [A]  time = 0.399971, size = 15, normalized size = 0.94 \begin{align*} - \frac{1}{2 a^{2} x^{2} + 2 a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x**2)**2/x**3,x)

[Out]

-1/(2*a**2*x**2 + 2*a*b)

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Giac [A]  time = 1.14598, size = 19, normalized size = 1.19 \begin{align*} -\frac{1}{2 \,{\left (a x^{2} + b\right )} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^2/x^3,x, algorithm="giac")

[Out]

-1/2/((a*x^2 + b)*a)