3.1.20 \(\int (a+b \log (c (d+e x)^n)) \, dx\) [20]

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

[Out]

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

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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 = {2436, 2332} \begin {gather*} a x+\frac {b (d+e x) \log \left (c (d+e x)^n\right )}{e}-b n x \end {gather*}

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 2332

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

Rule 2436

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 \text {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*}

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Mathematica [A]
time = 0.01, size = 29, normalized size = 1.00 \begin {gather*} a x-b n x+\frac {b (d+e x) \log \left (c (d+e x)^n\right )}{e} \end {gather*}

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 = 0.20, size = 36, normalized size = 1.24

method	result	size
default	\(a x +b \ln \left (c \left (e x +d \right )^{n}\right ) x -b n x +\frac {b n d \ln \left (e x +d \right )}{e}\)	\(36\)
norman	\(\left (-b n +a \right ) x +b x \ln \left (c \,{\mathrm e}^{n \ln \left (e x +d \right )}\right )+\frac {b n d \ln \left (e x +d \right )}{e}\)	\(38\)
risch	\(a x +b x \ln \left (\left (e x +d \right )^{n}\right )-\frac {i b \pi x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )}{2}+\frac {i b \pi x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{2}+\frac {i b \pi x \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{2}-\frac {i b \pi x \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}}{2}+\frac {b n d \ln \left (e x +d \right )}{e}+\ln \left (c \right ) b x -b n x\)	\(149\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.30, size = 40, normalized size = 1.38 \begin {gather*} {\left (d e^{\left (-2\right )} \log \left (x e + d\right ) - x e^{\left (-1\right )}\right )} b n e + b x \log \left ({\left (x e + d\right )}^{n} c\right ) + a x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d*e^(-2)*log(x*e + d) - x*e^(-1))*b*n*e + b*x*log((x*e + d)^n*c) + a*x

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Fricas [A]
time = 0.37, size = 42, normalized size = 1.45 \begin {gather*} {\left (b x e \log \left (c\right ) - {\left (b n - a\right )} x e + {\left (b n x e + b d n\right )} \log \left (x e + d\right )\right )} e^{\left (-1\right )} \end {gather*}

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

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Sympy [A]
time = 0.14, size = 41, normalized size = 1.41 \begin {gather*} a x + b \left (\begin {cases} \frac {d \log {\left (c \left (d + e x\right )^{n} \right )}}{e} - n x + x \log {\left (c \left (d + e x\right )^{n} \right )} & \text {for}\: e \neq 0 \\x \log {\left (c d^{n} \right )} & \text {otherwise} \end {cases}\right ) \end {gather*}

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

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Giac [A]
time = 3.57, size = 46, normalized size = 1.59 \begin {gather*} {\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 {gather*}

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

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Mupad [B]
time = 0.15, size = 35, normalized size = 1.21 \begin {gather*} 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} \end {gather*}

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

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