3.4 \(\int \log (c (d+e x)) \, dx\)

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

[Out]

-x + ((d + e*x)*Log[c*(d + e*x)])/e

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Rubi [A]  time = 0.0080155, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2389, 2295} \[ \frac{(d+e x) \log (c (d+e x))}{e}-x \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(d + e*x)],x]

[Out]

-x + ((d + e*x)*Log[c*(d + e*x)])/e

Rule 2389

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 2295

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

Rubi steps

\begin{align*} \int \log (c (d+e x)) \, dx &=\frac{\operatorname{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]  time = 0.0044039, size = 21, normalized size = 1. \[ \frac{(d+e x) \log (c (d+e x))}{e}-x \]

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 = 0.064, size = 36, normalized size = 1.7 \begin{align*} \ln \left ( cex+cd \right ) x+{\frac{\ln \left ( cex+cd \right ) d}{e}}-x-{\frac{d}{e}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.12578, size = 42, normalized size = 2. \begin{align*} \frac{{\left (e x + d\right )} c \log \left ({\left (e x + d\right )} c\right ) -{\left (e x + d\right )} c}{c e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.8341, size = 53, normalized size = 2.52 \begin{align*} -\frac{e x -{\left (e x + d\right )} \log \left (c e x + c d\right )}{e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(e*x - (e*x + d)*log(c*e*x + c*d))/e

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Sympy [A]  time = 0.372492, size = 26, normalized size = 1.24 \begin{align*} - 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{align*}

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 = 1.17423, size = 45, normalized size = 2.14 \begin{align*} \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{align*}

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