3.163 \(\int \frac{1}{x (a+b x)} \, dx\)

Optimal. Leaf size=18 \[ \frac{\log (x)}{a}-\frac{\log (a+b x)}{a} \]

[Out]

Log[x]/a - Log[a + b*x]/a

_______________________________________________________________________________________

Rubi [A]  time = 0.0135164, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{\log (x)}{a}-\frac{\log (a+b x)}{a} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a + b*x)),x]

[Out]

Log[x]/a - Log[a + b*x]/a

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 3.45017, size = 12, normalized size = 0.67 \[ \frac{\log{\left (x \right )}}{a} - \frac{\log{\left (a + b x \right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(b*x+a),x)

[Out]

log(x)/a - log(a + b*x)/a

_______________________________________________________________________________________

Mathematica [A]  time = 0.00518244, size = 18, normalized size = 1. \[ \frac{\log (x)}{a}-\frac{\log (a+b x)}{a} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(a + b*x)),x]

[Out]

Log[x]/a - Log[a + b*x]/a

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 19, normalized size = 1.1 \[{\frac{\ln \left ( x \right ) }{a}}-{\frac{\ln \left ( bx+a \right ) }{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x+a),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.34008, size = 24, normalized size = 1.33 \[ -\frac{\log \left (b x + a\right )}{a} + \frac{\log \left (x\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(b*x + a)/a + log(x)/a

_______________________________________________________________________________________

Fricas [A]  time = 0.19851, size = 22, normalized size = 1.22 \[ -\frac{\log \left (b x + a\right ) - \log \left (x\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(log(b*x + a) - log(x))/a

_______________________________________________________________________________________

Sympy [A]  time = 0.310256, size = 10, normalized size = 0.56 \[ \frac{\log{\left (x \right )} - \log{\left (\frac{a}{b} + x \right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(b*x+a),x)

[Out]

(log(x) - log(a/b + x))/a

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.216505, size = 27, normalized size = 1.5 \[ -\frac{{\rm ln}\left ({\left | b x + a \right |}\right )}{a} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-ln(abs(b*x + a))/a + ln(abs(x))/a