3.1826 \(\int \frac{(a+\frac{b}{x^2})^2}{x^3} \, dx\)

Optimal. Leaf size=16 \[ -\frac{\left (a+\frac{b}{x^2}\right )^3}{6 b} \]

[Out]

-(a + b/x^2)^3/(6*b)

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Rubi [A]  time = 0.0037366, 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} \[ -\frac{\left (a+\frac{b}{x^2}\right )^3}{6 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)^2/x^3,x]

[Out]

-(a + b/x^2)^3/(6*b)

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{\left (a+\frac{b}{x^2}\right )^2}{x^3} \, dx &=-\frac{\left (a+\frac{b}{x^2}\right )^3}{6 b}\\ \end{align*}

Mathematica [A]  time = 0.0008002, size = 30, normalized size = 1.88 \[ -\frac{a^2}{2 x^2}-\frac{a b}{2 x^4}-\frac{b^2}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)^2/x^3,x]

[Out]

-b^2/(6*x^6) - (a*b)/(2*x^4) - a^2/(2*x^2)

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Maple [A]  time = 0.004, size = 25, normalized size = 1.6 \begin{align*} -{\frac{ab}{2\,{x}^{4}}}-{\frac{{a}^{2}}{2\,{x}^{2}}}-{\frac{{b}^{2}}{6\,{x}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a*b/x^4-1/2/x^2*a^2-1/6*b^2/x^6

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Maxima [A]  time = 1.00612, size = 19, normalized size = 1.19 \begin{align*} -\frac{{\left (a + \frac{b}{x^{2}}\right )}^{3}}{6 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(a + b/x^2)^3/b

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Fricas [A]  time = 1.38843, size = 54, normalized size = 3.38 \begin{align*} -\frac{3 \, a^{2} x^{4} + 3 \, a b x^{2} + b^{2}}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(3*a^2*x^4 + 3*a*b*x^2 + b^2)/x^6

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Sympy [B]  time = 0.326129, size = 26, normalized size = 1.62 \begin{align*} - \frac{3 a^{2} x^{4} + 3 a b x^{2} + b^{2}}{6 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(3*a**2*x**4 + 3*a*b*x**2 + b**2)/(6*x**6)

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Giac [A]  time = 1.17065, size = 32, normalized size = 2. \begin{align*} -\frac{3 \, a^{2} x^{4} + 3 \, a b x^{2} + b^{2}}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(3*a^2*x^4 + 3*a*b*x^2 + b^2)/x^6