3.1625 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^2 x^3} \, dx\)

Optimal. Leaf size=29 \[ -\frac{\log (a x+b)}{b^2}+\frac{1}{b (a x+b)}+\frac{\log (x)}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.0510818, antiderivative size = 29, 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 \[ -\frac{\log (a x+b)}{b^2}+\frac{1}{b (a x+b)}+\frac{\log (x)}{b^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^2*x^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.83518, size = 24, normalized size = 0.83 \[ \frac{1}{b \left (a x + b\right )} + \frac{\log{\left (x \right )}}{b^{2}} - \frac{\log{\left (a x + b \right )}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**2/x**3,x)

[Out]

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

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Mathematica [A]  time = 0.0165118, size = 24, normalized size = 0.83 \[ \frac{\frac{b}{a x+b}-\log (a x+b)+\log (x)}{b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)^2*x^3),x]

[Out]

(b/(b + a*x) + Log[x] - Log[b + a*x])/b^2

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Maple [A]  time = 0.012, size = 30, normalized size = 1. \[{\frac{1}{b \left ( ax+b \right ) }}+{\frac{\ln \left ( x \right ) }{{b}^{2}}}-{\frac{\ln \left ( ax+b \right ) }{{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^2/x^3,x)

[Out]

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

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Maxima [A]  time = 1.4555, size = 38, normalized size = 1.31 \[ \frac{1}{a b x + b^{2}} - \frac{\log \left (a x + b\right )}{b^{2}} + \frac{\log \left (x\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.225335, size = 53, normalized size = 1.83 \[ -\frac{{\left (a x + b\right )} \log \left (a x + b\right ) -{\left (a x + b\right )} \log \left (x\right ) - b}{a b^{2} x + b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.40899, size = 22, normalized size = 0.76 \[ \frac{1}{a b x + b^{2}} + \frac{\log{\left (x \right )} - \log{\left (x + \frac{b}{a} \right )}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)**2/x**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.232606, size = 42, normalized size = 1.45 \[ -\frac{{\rm ln}\left ({\left | a x + b \right |}\right )}{b^{2}} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{b^{2}} + \frac{1}{{\left (a x + b\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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