3.1623 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^2 x} \, dx\)

Optimal. Leaf size=23 \[ \frac{b}{a^2 (a x+b)}+\frac{\log (a x+b)}{a^2} \]

[Out]

b/(a^2*(b + a*x)) + Log[b + a*x]/a^2

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Rubi [A]  time = 0.0433145, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{b}{a^2 (a x+b)}+\frac{\log (a x+b)}{a^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^2*x),x]

[Out]

b/(a^2*(b + a*x)) + Log[b + a*x]/a^2

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Rubi in Sympy [A]  time = 7.56412, size = 19, normalized size = 0.83 \[ \frac{b}{a^{2} \left (a x + b\right )} + \frac{\log{\left (a x + b \right )}}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b/(a**2*(a*x + b)) + log(a*x + b)/a**2

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Mathematica [A]  time = 0.0125417, size = 20, normalized size = 0.87 \[ \frac{\frac{b}{a x+b}+\log (a x+b)}{a^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)^2*x),x]

[Out]

(b/(b + a*x) + Log[b + a*x])/a^2

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Maple [A]  time = 0.007, size = 24, normalized size = 1. \[{\frac{b}{{a}^{2} \left ( ax+b \right ) }}+{\frac{\ln \left ( ax+b \right ) }{{a}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^2/x,x)

[Out]

b/a^2/(a*x+b)+ln(a*x+b)/a^2

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Maxima [A]  time = 1.45658, size = 35, normalized size = 1.52 \[ \frac{b}{a^{3} x + a^{2} b} + \frac{\log \left (a x + b\right )}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b/(a^3*x + a^2*b) + log(a*x + b)/a^2

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Fricas [A]  time = 0.219615, size = 38, normalized size = 1.65 \[ \frac{{\left (a x + b\right )} \log \left (a x + b\right ) + b}{a^{3} x + a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((a*x + b)*log(a*x + b) + b)/(a^3*x + a^2*b)

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Sympy [A]  time = 1.16313, size = 20, normalized size = 0.87 \[ \frac{b}{a^{3} x + a^{2} b} + \frac{\log{\left (a x + b \right )}}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)**2/x,x)

[Out]

b/(a**3*x + a**2*b) + log(a*x + b)/a**2

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GIAC/XCAS [A]  time = 0.223905, size = 32, normalized size = 1.39 \[ \frac{{\rm ln}\left ({\left | a x + b \right |}\right )}{a^{2}} + \frac{b}{{\left (a x + b\right )} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

ln(abs(a*x + b))/a^2 + b/((a*x + b)*a^2)