3.1626 \(\int \frac{1}{(a+\frac{b}{x})^2 x^2} \, dx\)

Optimal. Leaf size=13 \[ \frac{1}{b \left (a+\frac{b}{x}\right )} \]

[Out]

1/(b*(a + b/x))

________________________________________________________________________________________

Rubi [A]  time = 0.0035431, antiderivative size = 13, 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} \[ \frac{1}{b \left (a+\frac{b}{x}\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)^2*x^2),x]

[Out]

1/(b*(a + b/x))

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

Mathematica [A]  time = 0.0025463, size = 12, normalized size = 0.92 \[ -\frac{1}{a (a x+b)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)^2*x^2),x]

[Out]

-(1/(a*(b + a*x)))

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 13, normalized size = 1. \begin{align*} -{\frac{1}{a \left ( ax+b \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.973919, size = 18, normalized size = 1.38 \begin{align*} \frac{1}{{\left (a + \frac{b}{x}\right )} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/((a + b/x)*b)

________________________________________________________________________________________

Fricas [A]  time = 1.37836, size = 24, normalized size = 1.85 \begin{align*} -\frac{1}{a^{2} x + a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(a^2*x + a*b)

________________________________________________________________________________________

Sympy [A]  time = 0.269841, size = 10, normalized size = 0.77 \begin{align*} - \frac{1}{a^{2} x + a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(a**2*x + a*b)

________________________________________________________________________________________

Giac [A]  time = 1.08801, size = 18, normalized size = 1.38 \begin{align*} \frac{1}{{\left (a + \frac{b}{x}\right )} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/((a + b/x)*b)