3.1612 \(\int \frac{1}{\left (a+\frac{b}{x}\right ) x^3} \, dx\)

Optimal. Leaf size=28 \[ -\frac{a \log (x)}{b^2}+\frac{a \log (a x+b)}{b^2}-\frac{1}{b x} \]

[Out]

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

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Rubi [A]  time = 0.046779, antiderivative size = 28, 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 \[ -\frac{a \log (x)}{b^2}+\frac{a \log (a x+b)}{b^2}-\frac{1}{b x} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)*x^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.22525, size = 24, normalized size = 0.86 \[ - \frac{a \log{\left (x \right )}}{b^{2}} + \frac{a \log{\left (a x + b \right )}}{b^{2}} - \frac{1}{b x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00746488, size = 28, normalized size = 1. \[ -\frac{a \log (x)}{b^2}+\frac{a \log (a x+b)}{b^2}-\frac{1}{b x} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)*x^3),x]

[Out]

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

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Maple [A]  time = 0.012, size = 29, normalized size = 1. \[ -{\frac{1}{bx}}-{\frac{a\ln \left ( x \right ) }{{b}^{2}}}+{\frac{a\ln \left ( ax+b \right ) }{{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)/x^3,x)

[Out]

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

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Maxima [A]  time = 1.43579, size = 38, normalized size = 1.36 \[ \frac{a \log \left (a x + b\right )}{b^{2}} - \frac{a \log \left (x\right )}{b^{2}} - \frac{1}{b x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.227891, size = 35, normalized size = 1.25 \[ \frac{a x \log \left (a x + b\right ) - a x \log \left (x\right ) - b}{b^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.28784, size = 19, normalized size = 0.68 \[ \frac{a \left (- \log{\left (x \right )} + \log{\left (x + \frac{b}{a} \right )}\right )}{b^{2}} - \frac{1}{b x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)/x**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.223476, size = 41, normalized size = 1.46 \[ \frac{a{\rm ln}\left ({\left | a x + b \right |}\right )}{b^{2}} - \frac{a{\rm ln}\left ({\left | x \right |}\right )}{b^{2}} - \frac{1}{b x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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