3.1610 \(\int \frac{1}{\left (a+\frac{b}{x}\right ) x} \, dx\)

Optimal. Leaf size=10 \[ \frac{\log (a x+b)}{a} \]

[Out]

Log[b + a*x]/a

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Rubi [A]  time = 0.0159448, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\log (a x+b)}{a} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)*x),x]

[Out]

Log[b + a*x]/a

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Rubi in Sympy [A]  time = 3.21303, size = 7, normalized size = 0.7 \[ \frac{\log{\left (a x + b \right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)/x,x)

[Out]

log(a*x + b)/a

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Mathematica [A]  time = 0.0018095, size = 10, normalized size = 1. \[ \frac{\log (a x+b)}{a} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)*x),x]

[Out]

Log[b + a*x]/a

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Maple [A]  time = 0.002, size = 11, normalized size = 1.1 \[{\frac{\ln \left ( ax+b \right ) }{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)/x,x)

[Out]

ln(a*x+b)/a

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Maxima [A]  time = 1.43688, size = 14, normalized size = 1.4 \[ \frac{\log \left (a x + b\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(a*x + b)/a

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Fricas [A]  time = 0.221061, size = 14, normalized size = 1.4 \[ \frac{\log \left (a x + b\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(a*x + b)/a

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Sympy [A]  time = 0.0943, size = 7, normalized size = 0.7 \[ \frac{\log{\left (a x + b \right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)/x,x)

[Out]

log(a*x + b)/a

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GIAC/XCAS [A]  time = 0.225743, size = 15, normalized size = 1.5 \[ \frac{{\rm ln}\left ({\left | a x + b \right |}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

ln(abs(a*x + b))/a