\(\int \log (a+b x) \, dx\) [64]

Optimal result

Integrand size = 6, antiderivative size = 19 \[ \int \log (a+b x) \, dx=-x+\frac {(a+b x) \log (a+b x)}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2436, 2332} \[ \int \log (a+b x) \, dx=\frac {(a+b x) \log (a+b x)}{b}-x \]

[In]

Int[Log[a + b*x],x]

[Out]

-x + ((a + b*x)*Log[a + b*x])/b

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2436

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}(\int \log (x) \, dx,x,a+b x)}{b} \\ & = -x+\frac {(a+b x) \log (a+b x)}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \log (a+b x) \, dx=-x+\frac {(a+b x) \log (a+b x)}{b} \]

[In]

Integrate[Log[a + b*x],x]

[Out]

-x + ((a + b*x)*Log[a + b*x])/b

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.26

method	result	size
norman	\(x \ln \left (b x +a \right )+\frac {a \ln \left (b x +a \right )}{b}-x\)	\(24\)
risch	\(x \ln \left (b x +a \right )+\frac {a \ln \left (b x +a \right )}{b}-x\)	\(24\)
derivativedivides	\(\frac {\left (b x +a \right ) \ln \left (b x +a \right )-b x -a}{b}\)	\(25\)
default	\(\frac {\left (b x +a \right ) \ln \left (b x +a \right )-b x -a}{b}\)	\(25\)
parallelrisch	\(\frac {\ln \left (b x +a \right ) x b -b x +a \ln \left (b x +a \right )+a}{b}\)	\(28\)
parts	\(x \ln \left (b x +a \right )-b \left (\frac {x}{b}-\frac {a \ln \left (b x +a \right )}{b^{2}}\right )\)	\(31\)

[In]

int(ln(b*x+a),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \log (a+b x) \, dx=-\frac {b x - {\left (b x + a\right )} \log \left (b x + a\right )}{b} \]

[In]

integrate(log(b*x+a),x, algorithm="fricas")

[Out]

-(b*x - (b*x + a)*log(b*x + a))/b

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.26 \[ \int \log (a+b x) \, dx=- b \left (- \frac {a \log {\left (a + b x \right )}}{b^{2}} + \frac {x}{b}\right ) + x \log {\left (a + b x \right )} \]

[In]

integrate(ln(b*x+a),x)

[Out]

-b*(-a*log(a + b*x)/b**2 + x/b) + x*log(a + b*x)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int \log (a+b x) \, dx=-\frac {b x - {\left (b x + a\right )} \log \left (b x + a\right ) + a}{b} \]

[In]

integrate(log(b*x+a),x, algorithm="maxima")

[Out]

-(b*x - (b*x + a)*log(b*x + a) + a)/b

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int \log (a+b x) \, dx=-\frac {b x - {\left (b x + a\right )} \log \left (b x + a\right ) + a}{b} \]

[In]

integrate(log(b*x+a),x, algorithm="giac")

[Out]

-(b*x - (b*x + a)*log(b*x + a) + a)/b

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int \log (a+b x) \, dx=x\,\ln \left (a+b\,x\right )-x+\frac {a\,\ln \left (a+b\,x\right )}{b} \]

[In]

int(log(a + b*x),x)

[Out]

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