3.263 \(\int \frac{1}{b x+c x^2} \, dx\)

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

[Out]

Log[x]/b - Log[b + c*x]/b

_______________________________________________________________________________________

Rubi [A]  time = 0.0195302, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ \frac{\log (x)}{b}-\frac{\log (b+c x)}{b} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^(-1),x]

[Out]

Log[x]/b - Log[b + c*x]/b

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.09515, size = 12, normalized size = 0.67 \[ \frac{\log{\left (x \right )}}{b} - \frac{\log{\left (b + c x \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x)/b - log(b + c*x)/b

_______________________________________________________________________________________

Mathematica [A]  time = 0.0051994, size = 18, normalized size = 1. \[ \frac{\log (x)}{b}-\frac{\log (b+c x)}{b} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x + c*x^2)^(-1),x]

[Out]

Log[x]/b - Log[b + c*x]/b

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 19, normalized size = 1.1 \[{\frac{\ln \left ( x \right ) }{b}}-{\frac{\ln \left ( cx+b \right ) }{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x^2+b*x),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.699161, size = 24, normalized size = 1.33 \[ -\frac{\log \left (c x + b\right )}{b} + \frac{\log \left (x\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(c*x + b)/b + log(x)/b

_______________________________________________________________________________________

Fricas [A]  time = 0.22265, size = 22, normalized size = 1.22 \[ -\frac{\log \left (c x + b\right ) - \log \left (x\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(log(c*x + b) - log(x))/b

_______________________________________________________________________________________

Sympy [A]  time = 0.329503, size = 10, normalized size = 0.56 \[ \frac{\log{\left (x \right )} - \log{\left (\frac{b}{c} + x \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c*x**2+b*x),x)

[Out]

(log(x) - log(b/c + x))/b

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.205105, size = 27, normalized size = 1.5 \[ -\frac{{\rm ln}\left ({\left | c x + b \right |}\right )}{b} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-ln(abs(c*x + b))/b + ln(abs(x))/b