3.1624 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^2 x^2} \, dx\)

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

[Out]

1/(b*(a + b/x))

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Rubi [A]  time = 0.0169857, 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 \[ \frac{1}{b \left (a+\frac{b}{x}\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^2*x^2),x]

[Out]

1/(b*(a + b/x))

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Rubi in Sympy [A]  time = 2.25561, size = 7, normalized size = 0.54 \[ \frac{1}{b \left (a + \frac{b}{x}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**2/x**2,x)

[Out]

1/(b*(a + b/x))

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Mathematica [A]  time = 0.00548419, 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)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^2/x^2,x)

[Out]

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

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Maxima [A]  time = 1.42094, size = 18, normalized size = 1.38 \[ \frac{1}{{\left (a + \frac{b}{x}\right )} b} \]

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)

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Fricas [A]  time = 0.211075, size = 18, normalized size = 1.38 \[ -\frac{1}{a^{2} x + a b} \]

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)

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Sympy [A]  time = 1.11713, size = 10, normalized size = 0.77 \[ - \frac{1}{a^{2} x + a b} \]

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)

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GIAC/XCAS [A]  time = 0.222367, size = 18, normalized size = 1.38 \[ \frac{1}{{\left (a + \frac{b}{x}\right )} b} \]

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)