3.1882 \(\int \frac{1}{(a+\frac{b}{x^2})^3 x^3} \, dx\)

Optimal. Leaf size=16 \[ \frac{1}{4 b \left (a+\frac{b}{x^2}\right )^2} \]

[Out]

1/(4*b*(a + b/x^2)^2)

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Rubi [A]  time = 0.0038042, 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} \[ \frac{1}{4 b \left (a+\frac{b}{x^2}\right )^2} \]

Antiderivative was successfully verified.

[In]

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

Mathematica [A]  time = 0.0062335, size = 24, normalized size = 1.5 \[ -\frac{2 a x^2+b}{4 a^2 \left (a x^2+b\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^2)^3*x^3),x]

[Out]

-(b + 2*a*x^2)/(4*a^2*(b + a*x^2)^2)

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Maple [B]  time = 0.006, size = 31, normalized size = 1.9 \begin{align*}{\frac{b}{4\,{a}^{2} \left ( a{x}^{2}+b \right ) ^{2}}}-{\frac{1}{2\,{a}^{2} \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)^3/x^3,x)

[Out]

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

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Maxima [A]  time = 1.03117, size = 19, normalized size = 1.19 \begin{align*} \frac{1}{4 \,{\left (a + \frac{b}{x^{2}}\right )}^{2} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/((a + b/x^2)^2*b)

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Fricas [B]  time = 1.38718, size = 73, normalized size = 4.56 \begin{align*} -\frac{2 \, a x^{2} + b}{4 \,{\left (a^{4} x^{4} + 2 \, a^{3} b x^{2} + a^{2} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.550227, size = 36, normalized size = 2.25 \begin{align*} - \frac{2 a x^{2} + b}{4 a^{4} x^{4} + 8 a^{3} b x^{2} + 4 a^{2} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.16542, size = 30, normalized size = 1.88 \begin{align*} -\frac{2 \, a x^{2} + b}{4 \,{\left (a x^{2} + b\right )}^{2} a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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