3.70 \(\int \log (c (d+e x)^n) \, dx\)

Optimal. Leaf size=24 \[ \frac{(d+e x) \log \left (c (d+e x)^n\right )}{e}-n x \]

[Out]

-(n*x) + ((d + e*x)*Log[c*(d + e*x)^n])/e

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Rubi [A]  time = 0.0086375, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2389, 2295} \[ \frac{(d+e x) \log \left (c (d+e x)^n\right )}{e}-n x \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(d + e*x)^n],x]

[Out]

-(n*x) + ((d + e*x)*Log[c*(d + e*x)^n])/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 \left (c (d+e x)^n\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e}\\ &=-n x+\frac{(d+e x) \log \left (c (d+e x)^n\right )}{e}\\ \end{align*}

Mathematica [A]  time = 0.0067945, size = 24, normalized size = 1. \[ \frac{(d+e x) \log \left (c (d+e x)^n\right )}{e}-n x \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(d + e*x)^n],x]

[Out]

-(n*x) + ((d + e*x)*Log[c*(d + e*x)^n])/e

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Maple [A]  time = 0.059, size = 30, normalized size = 1.3 \begin{align*} \ln \left ( c \left ( ex+d \right ) ^{n} \right ) x-nx+{\frac{dn\ln \left ( ex+d \right ) }{e}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.14588, size = 47, normalized size = 1.96 \begin{align*} -e n{\left (\frac{x}{e} - \frac{d \log \left (e x + d\right )}{e^{2}}\right )} + x \log \left ({\left (e x + d\right )}^{n} c\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.01158, size = 73, normalized size = 3.04 \begin{align*} -\frac{e n x - e x \log \left (c\right ) -{\left (e n x + d n\right )} \log \left (e x + d\right )}{e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.442061, size = 37, normalized size = 1.54 \begin{align*} \begin{cases} \frac{d n \log{\left (d + e x \right )}}{e} + n x \log{\left (d + e x \right )} - n x + x \log{\left (c \right )} & \text{for}\: e \neq 0 \\x \log{\left (c d^{n} \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((d*n*log(d + e*x)/e + n*x*log(d + e*x) - n*x + x*log(c), Ne(e, 0)), (x*log(c*d**n), True))

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Giac [A]  time = 1.24072, size = 54, normalized size = 2.25 \begin{align*}{\left (x e + d\right )} n e^{\left (-1\right )} \log \left (x e + d\right ) -{\left (x e + d\right )} n e^{\left (-1\right )} +{\left (x e + d\right )} e^{\left (-1\right )} \log \left (c\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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