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

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Rubi [A]  time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 2295

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

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]

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*}

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Mathematica [A]  time = 0.01, size = 24, normalized size = 1.00 \[ \frac {(d+e x) \log \left (c (d+e x)^n\right )}{e}-n x \]

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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fricas [A]  time = 0.49, size = 32, normalized size = 1.33 \[ -\frac {e n x - e x \log \relax (c) - {\left (e n x + d n\right )} \log \left (e x + d\right )}{e} \]

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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giac [A]  time = 0.16, size = 40, normalized size = 1.67 \[ {\left (x e + d\right )} n e^{\left (-1\right )} \log \left (x e + d\right ) - {\left (x e + d\right )} n e^{\left (-1\right )} + {\left (x e + d\right )} e^{\left (-1\right )} \log \relax (c) \]

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)

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maple [A]  time = 0.04, size = 30, normalized size = 1.25 \[ \frac {d n \ln \left (e x +d \right )}{e}-n x +x \ln \left (c \left (e x +d \right )^{n}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.72, size = 35, normalized size = 1.46 \[ -e n {\left (\frac {x}{e} - \frac {d \log \left (e x + d\right )}{e^{2}}\right )} + x \log \left ({\left (e x + d\right )}^{n} c\right ) \]

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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mupad [B]  time = 0.06, size = 29, normalized size = 1.21 \[ x\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )-n\,x+\frac {d\,n\,\ln \left (d+e\,x\right )}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*(d + e*x)^n),x)

[Out]

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

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sympy [A]  time = 0.45, size = 37, normalized size = 1.54 \[ \begin {cases} \frac {d n \log {\left (d + e x \right )}}{e} + n x \log {\left (d + e x \right )} - n x + x \log {\relax (c )} & \text {for}\: e \neq 0 \\x \log {\left (c d^{n} \right )} & \text {otherwise} \end {cases} \]

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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