\(\int \log (c (d+e x)) \, dx\) [4]

Optimal result

Integrand size = 8, antiderivative size = 21 \[ \int \log (c (d+e x)) \, dx=-x+\frac {(d+e x) \log (c (d+e x))}{e} \]

[Out]

-x+(e*x+d)*ln(c*(e*x+d))/e

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2436, 2332} \[ \int \log (c (d+e x)) \, dx=\frac {(d+e x) \log (c (d+e x))}{e}-x \]

[In]

Int[Log[c*(d + e*x)],x]

[Out]

-x + ((d + e*x)*Log[c*(d + e*x)])/e

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 (c x) \, dx,x,d+e x)}{e} \\ & = -x+\frac {(d+e x) \log (c (d+e x))}{e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \log (c (d+e x)) \, dx=-x+\frac {(d+e x) \log (c (d+e x))}{e} \]

[In]

Integrate[Log[c*(d + e*x)],x]

[Out]

-x + ((d + e*x)*Log[c*(d + e*x)])/e

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24

method	result	size
risch	\(x \ln \left (c \left (e x +d \right )\right )-x +\frac {d \ln \left (e x +d \right )}{e}\)	\(26\)
norman	\(x \ln \left (c \left (e x +d \right )\right )+\frac {d \ln \left (c \left (e x +d \right )\right )}{e}-x\)	\(28\)
parallelrisch	\(\frac {\ln \left (c \left (e x +d \right )\right ) x e -e x +d \ln \left (c \left (e x +d \right )\right )+d}{e}\)	\(32\)
parts	\(x \ln \left (c \left (e x +d \right )\right )-e \left (\frac {x}{e}-\frac {d \ln \left (e x +d \right )}{e^{2}}\right )\)	\(33\)
derivativedivides	\(\frac {\left (c e x +c d \right ) \ln \left (c e x +c d \right )-c e x -c d}{c e}\)	\(36\)
default	\(\frac {\left (c e x +c d \right ) \ln \left (c e x +c d \right )-c e x -c d}{c e}\)	\(36\)

[In]

int(ln(c*(e*x+d)),x,method=_RETURNVERBOSE)

[Out]

x*ln(c*(e*x+d))-x+d/e*ln(e*x+d)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \log (c (d+e x)) \, dx=-\frac {e x - {\left (e x + d\right )} \log \left (c e x + c d\right )}{e} \]

[In]

integrate(log(c*(e*x+d)),x, algorithm="fricas")

[Out]

-(e*x - (e*x + d)*log(c*e*x + c*d))/e

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24 \[ \int \log (c (d+e x)) \, dx=- e \left (- \frac {d \log {\left (d + e x \right )}}{e^{2}} + \frac {x}{e}\right ) + x \log {\left (c \left (d + e x\right ) \right )} \]

[In]

integrate(ln(c*(e*x+d)),x)

[Out]

-e*(-d*log(d + e*x)/e**2 + x/e) + x*log(c*(d + e*x))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.48 \[ \int \log (c (d+e x)) \, dx=\frac {{\left (e x + d\right )} c \log \left ({\left (e x + d\right )} c\right ) - {\left (e x + d\right )} c}{c e} \]

[In]

integrate(log(c*(e*x+d)),x, algorithm="maxima")

[Out]

((e*x + d)*c*log((e*x + d)*c) - (e*x + d)*c)/(c*e)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.48 \[ \int \log (c (d+e x)) \, dx=\frac {{\left (e x + d\right )} c \log \left ({\left (e x + d\right )} c\right ) - {\left (e x + d\right )} c}{c e} \]

[In]

integrate(log(c*(e*x+d)),x, algorithm="giac")

[Out]

((e*x + d)*c*log((e*x + d)*c) - (e*x + d)*c)/(c*e)

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \log (c (d+e x)) \, dx=x\,\ln \left (c\,\left (d+e\,x\right )\right )-x+\frac {d\,\ln \left (d+e\,x\right )}{e} \]

[In]

int(log(c*(d + e*x)),x)

[Out]

x*log(c*(d + e*x)) - x + (d*log(d + e*x))/e