3.1639 \(\int \frac{1}{(a+\frac{b}{x})^3 x^2} \, dx\)

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

[Out]

1/(2*b*(a + b/x)^2)

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Rubi [A]  time = 0.0037841, 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} \[ \frac{1}{2 b \left (a+\frac{b}{x}\right )^2} \]

Antiderivative was successfully verified.

[In]

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

Mathematica [A]  time = 0.0050624, size = 20, normalized size = 1.25 \[ -\frac{2 a x+b}{2 a^2 (a x+b)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)^3*x^2),x]

[Out]

-(b + 2*a*x)/(2*a^2*(b + a*x)^2)

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Maple [A]  time = 0.003, size = 27, normalized size = 1.7 \begin{align*}{\frac{b}{2\,{a}^{2} \left ( ax+b \right ) ^{2}}}-{\frac{1}{{a}^{2} \left ( ax+b \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.03062, size = 19, normalized size = 1.19 \begin{align*} \frac{1}{2 \,{\left (a + \frac{b}{x}\right )}^{2} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/((a + b/x)^2*b)

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Fricas [B]  time = 1.68739, size = 68, normalized size = 4.25 \begin{align*} -\frac{2 \, a x + b}{2 \,{\left (a^{4} x^{2} + 2 \, a^{3} b x + a^{2} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.32568, size = 32, normalized size = 2. \begin{align*} - \frac{2 a x + b}{2 a^{4} x^{2} + 4 a^{3} b x + 2 a^{2} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.08913, size = 19, normalized size = 1.19 \begin{align*} \frac{1}{2 \,{\left (a + \frac{b}{x}\right )}^{2} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/((a + b/x)^2*b)