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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, 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 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 \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 = 0.01, size = 29, normalized size = 1.00 \[ a x+\frac {b (d+e x) \log \left (c (d+e x)^n\right )}{e}-b n x \]

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

________________________________________________________________________________________

fricas [A]  time = 1.43, size = 40, normalized size = 1.38 \[ \frac {b e x \log \relax (c) - {\left (b e n - a e\right )} x + {\left (b e n x + b d n\right )} \log \left (e x + d\right )}{e} \]

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

________________________________________________________________________________________

giac [A]  time = 0.16, size = 46, normalized size = 1.59 \[ {\left ({\left (x e + d\right )} n e^{\left (-1\right )} \log \left (x e + d\right ) - {\left (x e + d\right )} n e^{\left (-1\right )} + {\left (x e + d\right )} e^{\left (-1\right )} \log \relax (c)\right )} b + a x \]

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

________________________________________________________________________________________

maple [A]  time = 0.04, size = 36, normalized size = 1.24 \[ \frac {b d n \ln \left (e x +d \right )}{e}-b n x +b x \ln \left (c \left (e x +d \right )^{n}\right )+a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.67, size = 40, normalized size = 1.38 \[ -b e n {\left (\frac {x}{e} - \frac {d \log \left (e x + d\right )}{e^{2}}\right )} + b x \log \left ({\left (e x + d\right )}^{n} c\right ) + a x \]

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

________________________________________________________________________________________

mupad [B]  time = 0.15, size = 35, normalized size = 1.21 \[ x\,\left (a-b\,n\right )+b\,x\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )+\frac {b\,d\,n\,\ln \left (d+e\,x\right )}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + b*log(c*(d + e*x)^n),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.48, size = 42, normalized size = 1.45 \[ a x + b \left (\begin {cases} \frac {d n \log {\left (d + e x \right )}}{e} + n x \log {\left (d + e x \right )} - n x + x \log {\relax (c )} & \text {for}\: e \neq 0 \\x \log {\left (c d^{n} \right )} & \text {otherwise} \end {cases}\right ) \]

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))

________________________________________________________________________________________