3.20 \(\int (a+b \log (c (d+e x)^n)) \, dx\)

Optimal. Leaf size=29 \[ a x+\frac{b (d+e x) \log \left (c (d+e x)^n\right )}{e}-b n x \]

[Out]

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

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Rubi [A]  time = 0.0145399, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2389, 2295} \[ a x+\frac{b (d+e x) \log \left (c (d+e x)^n\right )}{e}-b n x \]

Antiderivative was successfully verified.

[In]

Int[a + b*Log[c*(d + e*x)^n],x]

[Out]

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

Mathematica [A]  time = 0.006368, size = 29, normalized size = 1. \[ a x+\frac{b (d+e x) \log \left (c (d+e x)^n\right )}{e}-b n x \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*Log[c*(d + e*x)^n],x]

[Out]

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

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Maple [A]  time = 0.059, size = 36, normalized size = 1.2 \begin{align*} ax+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) x-bnx+{\frac{\ln \left ( ex+d \right ) bdn}{e}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.1105, size = 54, normalized size = 1.86 \begin{align*} -b e n{\left (\frac{x}{e} - \frac{d \log \left (e x + d\right )}{e^{2}}\right )} + b x \log \left ({\left (e x + d\right )}^{n} c\right ) + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.06593, size = 93, normalized size = 3.21 \begin{align*} \frac{b e x \log \left (c\right ) -{\left (b e n - a e\right )} x +{\left (b e n x + b d n\right )} \log \left (e x + d\right )}{e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.567474, size = 42, normalized size = 1.45 \begin{align*} a x + b \left (\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}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x + b*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 = 1.24343, size = 62, normalized size = 2.14 \begin{align*}{\left ({\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 )\right )} b + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*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))*b + a*x