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 = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 1.00 \[ -\frac {1}{2 a \left (a x^2+b\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^2)^2*x^3),x]

[Out]

-1/2*1/(a*(b + a*x^2))

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fricas [A]  time = 0.58, size = 15, normalized size = 0.94 \[ -\frac {1}{2 \, {\left (a^{2} x^{2} + a b\right )}} \]

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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giac [A]  time = 0.15, size = 14, normalized size = 0.88 \[ -\frac {1}{2 \, {\left (a x^{2} + b\right )} a} \]

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)

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maple [A]  time = 0.00, size = 15, normalized size = 0.94 \[ -\frac {1}{2 \left (a \,x^{2}+b \right ) a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.84, size = 14, normalized size = 0.88 \[ \frac {1}{2 \, {\left (a + \frac {b}{x^{2}}\right )} b} \]

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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mupad [B]  time = 0.02, size = 14, normalized size = 0.88 \[ -\frac {1}{2\,a\,\left (a\,x^2+b\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(a + b/x^2)^2),x)

[Out]

-1/(2*a*(b + a*x^2))

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sympy [A]  time = 0.17, size = 15, normalized size = 0.94 \[ - \frac {1}{2 a^{2} x^{2} + 2 a b} \]

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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