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

Optimal. Leaf size=21 \[ -x+\frac {(d+e x) \log (c (d+e x))}{e} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2436, 2332} \begin {gather*} \frac {(d+e x) \log (c (d+e x))}{e}-x \end {gather*}

Antiderivative was successfully verified.

[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*} \int \log (c (d+e x)) \, dx &=\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*}

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Mathematica [A]
time = 0.00, size = 21, normalized size = 1.00 \begin {gather*} -x+\frac {(d+e x) \log (c (d+e x))}{e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(d + e*x)],x]

[Out]

-x + ((d + e*x)*Log[c*(d + e*x)])/e

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Maple [A]
time = 0.09, size = 36, normalized size = 1.71

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	\(x \ln \left (c \left (e x +d \right )\right )+\frac {d \ln \left (c \left (e x +d \right )\right )}{e}-x\)	\(28\)
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	\(\frac {\left (c e x +c d \right ) \ln \left (c e x +c d \right )-c e x -c d}{c e}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 33, normalized size = 1.57 \begin {gather*} \frac {{\left ({\left (x e + d\right )} c \log \left ({\left (x e + d\right )} c\right ) - {\left (x e + d\right )} c\right )} e^{\left (-1\right )}}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x*e + d)*c*log((x*e + d)*c) - (x*e + d)*c)*e^(-1)/c

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Fricas [A]
time = 0.34, size = 27, normalized size = 1.29 \begin {gather*} -{\left (x e - {\left (x e + d\right )} \log \left (c x e + c d\right )\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x*e - (x*e + d)*log(c*x*e + c*d))*e^(-1)

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Sympy [A]
time = 0.06, size = 26, normalized size = 1.24 \begin {gather*} - 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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Giac [A]
time = 3.36, size = 33, normalized size = 1.57 \begin {gather*} \frac {{\left ({\left (x e + d\right )} c \log \left ({\left (x e + d\right )} c\right ) - {\left (x e + d\right )} c\right )} e^{\left (-1\right )}}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x*e + d)*c*log((x*e + d)*c) - (x*e + d)*c)*e^(-1)/c

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Mupad [B]
time = 0.06, size = 25, normalized size = 1.19 \begin {gather*} x\,\ln \left (c\,\left (d+e\,x\right )\right )-x+\frac {d\,\ln \left (d+e\,x\right )}{e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*(d + e*x)),x)

[Out]

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

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