3.887 \(\int \frac{b+2 c x}{x (b+c x)} \, dx\)

Optimal. Leaf size=8 \[ \log (x (b+c x)) \]

[Out]

Log[x*(b + c*x)]

_______________________________________________________________________________________

Rubi [A]  time = 0.0265935, antiderivative size = 9, normalized size of antiderivative = 1.12, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \log (b+c x)+\log (x) \]

Antiderivative was successfully verified.

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

[Out]

Log[x] + Log[b + c*x]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.3167, size = 8, normalized size = 1. \[ \log{\left (x \right )} + \log{\left (b + c x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x) + log(b + c*x)

_______________________________________________________________________________________

Mathematica [A]  time = 0.00814933, size = 9, normalized size = 1.12 \[ \log (b+c x)+\log (x) \]

Antiderivative was successfully verified.

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

[Out]

Log[x] + Log[b + c*x]

_______________________________________________________________________________________

Maple [A]  time = 0.002, size = 9, normalized size = 1.1 \[ \ln \left ( x \left ( cx+b \right ) \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*x+b)/x/(c*x+b),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.40142, size = 12, normalized size = 1.5 \[ \log \left (c x + b\right ) + \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(c*x + b) + log(x)

_______________________________________________________________________________________

Fricas [A]  time = 0.208297, size = 14, normalized size = 1.75 \[ \log \left (c x^{2} + b x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(c*x^2 + b*x)

_______________________________________________________________________________________

Sympy [A]  time = 0.560487, size = 8, normalized size = 1. \[ \log{\left (b x + c x^{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(b*x + c*x**2)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.206336, size = 15, normalized size = 1.88 \[{\rm ln}\left ({\left | c x + b \right |}\right ) +{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

ln(abs(c*x + b)) + ln(abs(x))