\(\int \frac {1}{(a+\frac {b}{x})^2 x^2} \, dx\) [1626]

Optimal result

Integrand size = 13, antiderivative size = 13 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^2} \, dx=\frac {1}{b \left (a+\frac {b}{x}\right )} \]

[Out]

1/b/(a+b/x)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {267} \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^2} \, dx=\frac {1}{b \left (a+\frac {b}{x}\right )} \]

[In]

Int[1/((a + b/x)^2*x^2),x]

[Out]

1/(b*(a + b/x))

Rule 267

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*} \text {integral}& = \frac {1}{b \left (a+\frac {b}{x}\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^2} \, dx=-\frac {1}{a (b+a x)} \]

[In]

Integrate[1/((a + b/x)^2*x^2),x]

[Out]

-(1/(a*(b + a*x)))

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00

method	result	size
gosper	\(-\frac {1}{\left (a x +b \right ) a}\)	\(13\)
default	\(-\frac {1}{\left (a x +b \right ) a}\)	\(13\)
norman	\(\frac {x}{b \left (a x +b \right )}\)	\(13\)
risch	\(-\frac {1}{\left (a x +b \right ) a}\)	\(13\)
parallelrisch	\(-\frac {1}{\left (a x +b \right ) a}\)	\(13\)
derivativedivides	\(\frac {1}{b \left (a +\frac {b}{x}\right )}\)	\(14\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^2} \, dx=-\frac {1}{a^{2} x + a b} \]

[In]

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

[Out]

-1/(a^2*x + a*b)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^2} \, dx=- \frac {1}{a^{2} x + a b} \]

[In]

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

[Out]

-1/(a**2*x + a*b)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^2} \, dx=\frac {1}{{\left (a + \frac {b}{x}\right )} b} \]

[In]

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

[Out]

1/((a + b/x)*b)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^2} \, dx=\frac {1}{{\left (a + \frac {b}{x}\right )} b} \]

[In]

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

[Out]

1/((a + b/x)*b)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^2} \, dx=-\frac {1}{a\,\left (b+a\,x\right )} \]

[In]

int(1/(x^2*(a + b/x)^2),x)

[Out]

-1/(a*(b + a*x))