3.1884 \(\int \frac{1}{(a+\frac{b}{x^2})^3 x^5} \, dx\)

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

[Out]

-1/(4*a*(b + a*x^2)^2)

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Rubi [A]  time = 0.0050669, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {263, 261} \[ -\frac{1}{4 a \left (a x^2+b\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^2)^3*x^5),x]

[Out]

-1/(4*a*(b + a*x^2)^2)

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

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^5} \, dx &=\int \frac{x}{\left (b+a x^2\right )^3} \, dx\\ &=-\frac{1}{4 a \left (b+a x^2\right )^2}\\ \end{align*}

Mathematica [A]  time = 0.0024111, size = 16, normalized size = 1. \[ -\frac{1}{4 a \left (a x^2+b\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^2)^3*x^5),x]

[Out]

-1/(4*a*(b + a*x^2)^2)

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Maple [A]  time = 0.001, size = 15, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,a \left ( a{x}^{2}+b \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+1/x^2*b)^3/x^5,x)

[Out]

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

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Maxima [A]  time = 0.994418, size = 35, normalized size = 2.19 \begin{align*} -\frac{1}{4 \,{\left (a^{3} x^{4} + 2 \, a^{2} b x^{2} + a b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^3/x^5,x, algorithm="maxima")

[Out]

-1/4/(a^3*x^4 + 2*a^2*b*x^2 + a*b^2)

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Fricas [A]  time = 1.37883, size = 51, normalized size = 3.19 \begin{align*} -\frac{1}{4 \,{\left (a^{3} x^{4} + 2 \, a^{2} b x^{2} + a b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^3/x^5,x, algorithm="fricas")

[Out]

-1/4/(a^3*x^4 + 2*a^2*b*x^2 + a*b^2)

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Sympy [A]  time = 0.514413, size = 27, normalized size = 1.69 \begin{align*} - \frac{1}{4 a^{3} x^{4} + 8 a^{2} b x^{2} + 4 a b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x**2)**3/x**5,x)

[Out]

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

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Giac [A]  time = 1.16098, size = 19, normalized size = 1.19 \begin{align*} -\frac{1}{4 \,{\left (a x^{2} + b\right )}^{2} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^3/x^5,x, algorithm="giac")

[Out]

-1/4/((a*x^2 + b)^2*a)