\(\int \log (a+b x+c x) \, dx\) [67]

Optimal result

Integrand size = 9, antiderivative size = 25 \[ \int \log (a+b x+c x) \, dx=-x+\frac {(a+(b+c) x) \log (a+(b+c) x)}{b+c} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2494, 2436, 2332} \[ \int \log (a+b x+c x) \, dx=\frac {(a+x (b+c)) \log (a+x (b+c))}{b+c}-x \]

[In]

Int[Log[a + b*x + c*x],x]

[Out]

-x + ((a + (b + c)*x)*Log[a + (b + c)*x])/(b + c)

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]

Rule 2494

Int[((a_.) + Log[(c_.)*(v_)^(n_.)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Int[u*(a + b*Log[c*ExpandToSum[v, x]^n])^p
, x] /; FreeQ[{a, b, c, n, p}, x] && LinearQ[v, x] &&  !LinearMatchQ[v, x] &&  !(EqQ[n, 1] && MatchQ[c*v, (e_.
)*((f_) + (g_.)*x) /; FreeQ[{e, f, g}, x]])

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \log (a+b x+c x) \, dx=-x+\frac {(a+(b+c) x) \log (a+(b+c) x)}{b+c} \]

[In]

Integrate[Log[a + b*x + c*x],x]

[Out]

-x + ((a + (b + c)*x)*Log[a + (b + c)*x])/(b + c)

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28

method	result	size
norman	\(x \ln \left (b x +x c +a \right )+\frac {a \ln \left (b x +x c +a \right )}{b +c}-x\)	\(32\)
derivativedivides	\(\frac {\left (a +\left (b +c \right ) x \right ) \ln \left (a +\left (b +c \right ) x \right )-a -\left (b +c \right ) x}{b +c}\)	\(33\)
default	\(\frac {\left (a +\left (b +c \right ) x \right ) \ln \left (a +\left (b +c \right ) x \right )-a -\left (b +c \right ) x}{b +c}\)	\(33\)
parts	\(x \ln \left (b x +x c +a \right )-\left (b +c \right ) \left (\frac {x}{b +c}-\frac {a \ln \left (b x +x c +a \right )}{\left (b +c \right )^{2}}\right )\)	\(43\)
risch	\(x \ln \left (b x +x c +a \right )+\frac {a \ln \left (a +\left (b +c \right ) x \right )}{b +c}-\frac {b x}{b +c}-\frac {x c}{b +c}\)	\(46\)
parallelrisch	\(\frac {x \ln \left (b x +x c +a \right ) a b +x \ln \left (b x +x c +a \right ) a c -a b x -x c a +\ln \left (b x +x c +a \right ) a^{2}}{a \left (b +c \right )}\)	\(60\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20 \[ \int \log (a+b x+c x) \, dx=-\frac {{\left (b + c\right )} x - {\left ({\left (b + c\right )} x + a\right )} \log \left ({\left (b + c\right )} x + a\right )}{b + c} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \log (a+b x+c x) \, dx=x \log {\left (a + b x + c x \right )} + \left (- b - c\right ) \left (- \frac {a \log {\left (a + x \left (b + c\right ) \right )}}{\left (b + c\right )^{2}} + \frac {x}{b + c}\right ) \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \log (a+b x+c x) \, dx=-\frac {b x + c x - {\left (b x + c x + a\right )} \log \left (b x + c x + a\right ) + a}{b + c} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \log (a+b x+c x) \, dx=-\frac {b x + c x - {\left (b x + c x + a\right )} \log \left (b x + c x + a\right ) + a}{b + c} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \log (a+b x+c x) \, dx=x\,\ln \left (a+b\,x+c\,x\right )-x+\frac {a\,\ln \left (a+b\,x+c\,x\right )}{b+c} \]

[In]

int(log(a + b*x + c*x),x)

[Out]

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