\(\int \frac {1}{2+b x} \, dx\) [1530]

Optimal result

Integrand size = 7, antiderivative size = 10 \[ \int \frac {1}{2+b x} \, dx=\frac {\log (2+b x)}{b} \]

[Out]

ln(b*x+2)/b

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {31} \[ \int \frac {1}{2+b x} \, dx=\frac {\log (b x+2)}{b} \]

[In]

Int[(2 + b*x)^(-1),x]

[Out]

Log[2 + b*x]/b

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\log (2+b x)}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {1}{2+b x} \, dx=\frac {\log (2+b x)}{b} \]

[In]

Integrate[(2 + b*x)^(-1),x]

[Out]

Log[2 + b*x]/b

Maple [A] (verified)

Time = 0.51 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.10

method	result	size
default	\(\frac {\ln \left (b x +2\right )}{b}\)	\(11\)
norman	\(\frac {\ln \left (b x +2\right )}{b}\)	\(11\)
risch	\(\frac {\ln \left (b x +2\right )}{b}\)	\(11\)
parallelrisch	\(\frac {\ln \left (b x +2\right )}{b}\)	\(11\)
meijerg	\(\frac {\ln \left (\frac {b x}{2}+1\right )}{b}\)	\(12\)

[In]

int(1/(b*x+2),x,method=_RETURNVERBOSE)

[Out]

ln(b*x+2)/b

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {1}{2+b x} \, dx=\frac {\log \left (b x + 2\right )}{b} \]

[In]

integrate(1/(b*x+2),x, algorithm="fricas")

[Out]

log(b*x + 2)/b

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.70 \[ \int \frac {1}{2+b x} \, dx=\frac {\log {\left (b x + 2 \right )}}{b} \]

[In]

integrate(1/(b*x+2),x)

[Out]

log(b*x + 2)/b

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {1}{2+b x} \, dx=\frac {\log \left (b x + 2\right )}{b} \]

[In]

integrate(1/(b*x+2),x, algorithm="maxima")

[Out]

log(b*x + 2)/b

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.10 \[ \int \frac {1}{2+b x} \, dx=\frac {\log \left ({\left | b x + 2 \right |}\right )}{b} \]

[In]

integrate(1/(b*x+2),x, algorithm="giac")

[Out]

log(abs(b*x + 2))/b

Mupad [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {1}{2+b x} \, dx=\frac {\ln \left (b\,x+2\right )}{b} \]

[In]

int(1/(b*x + 2),x)

[Out]

log(b*x + 2)/b