3.2 \(\int \log ^3(c (d+e x)) \, dx\)

Optimal. Leaf size=61 \[ \frac{(d+e x) \log ^3(c (d+e x))}{e}-\frac{3 (d+e x) \log ^2(c (d+e x))}{e}+\frac{6 (d+e x) \log (c (d+e x))}{e}-6 x \]

[Out]

-6*x + (6*(d + e*x)*Log[c*(d + e*x)])/e - (3*(d + e*x)*Log[c*(d + e*x)]^2)/e + ((d + e*x)*Log[c*(d + e*x)]^3)/
e

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Rubi [A]  time = 0.0266686, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {2389, 2296, 2295} \[ \frac{(d+e x) \log ^3(c (d+e x))}{e}-\frac{3 (d+e x) \log ^2(c (d+e x))}{e}+\frac{6 (d+e x) \log (c (d+e x))}{e}-6 x \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(d + e*x)]^3,x]

[Out]

-6*x + (6*(d + e*x)*Log[c*(d + e*x)])/e - (3*(d + e*x)*Log[c*(d + e*x)]^2)/e + ((d + e*x)*Log[c*(d + e*x)]^3)/
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 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

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 ^3(c (d+e x)) \, dx &=\frac{\operatorname{Subst}\left (\int \log ^3(c x) \, dx,x,d+e x\right )}{e}\\ &=\frac{(d+e x) \log ^3(c (d+e x))}{e}-\frac{3 \operatorname{Subst}\left (\int \log ^2(c x) \, dx,x,d+e x\right )}{e}\\ &=-\frac{3 (d+e x) \log ^2(c (d+e x))}{e}+\frac{(d+e x) \log ^3(c (d+e x))}{e}+\frac{6 \operatorname{Subst}(\int \log (c x) \, dx,x,d+e x)}{e}\\ &=-6 x+\frac{6 (d+e x) \log (c (d+e x))}{e}-\frac{3 (d+e x) \log ^2(c (d+e x))}{e}+\frac{(d+e x) \log ^3(c (d+e x))}{e}\\ \end{align*}

Mathematica [A]  time = 0.0055952, size = 57, normalized size = 0.93 \[ \frac{(d+e x) \log ^3(c (d+e x))-3 (d+e x) \log ^2(c (d+e x))+6 (d+e x) \log (c (d+e x))-6 e x}{e} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(d + e*x)]^3,x]

[Out]

(-6*e*x + 6*(d + e*x)*Log[c*(d + e*x)] - 3*(d + e*x)*Log[c*(d + e*x)]^2 + (d + e*x)*Log[c*(d + e*x)]^3)/e

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Maple [A]  time = 0.063, size = 98, normalized size = 1.6 \begin{align*} \left ( \ln \left ( cex+cd \right ) \right ) ^{3}x+{\frac{ \left ( \ln \left ( cex+cd \right ) \right ) ^{3}d}{e}}-3\, \left ( \ln \left ( cex+cd \right ) \right ) ^{2}x-3\,{\frac{ \left ( \ln \left ( cex+cd \right ) \right ) ^{2}d}{e}}+6\,\ln \left ( cex+cd \right ) x+6\,{\frac{\ln \left ( cex+cd \right ) d}{e}}-6\,x-6\,{\frac{d}{e}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(e*x+d))^3,x)

[Out]

ln(c*e*x+c*d)^3*x+1/e*ln(c*e*x+c*d)^3*d-3*ln(c*e*x+c*d)^2*x-3/e*ln(c*e*x+c*d)^2*d+6*ln(c*e*x+c*d)*x+6/e*ln(c*e
*x+c*d)*d-6*x-6*d/e

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Maxima [B]  time = 1.12432, size = 169, normalized size = 2.77 \begin{align*} -3 \, e{\left (\frac{x}{e} - \frac{d \log \left (e x + d\right )}{e^{2}}\right )} \log \left ({\left (e x + d\right )} c\right )^{2} + x \log \left ({\left (e x + d\right )} c\right )^{3} - e{\left (\frac{3 \,{\left (d \log \left (e x + d\right )^{2} - 2 \, e x + 2 \, d \log \left (e x + d\right )\right )} \log \left ({\left (e x + d\right )} c\right )}{e^{2}} - \frac{d \log \left (e x + d\right )^{3} + 3 \, d \log \left (e x + d\right )^{2} - 6 \, e x + 6 \, d \log \left (e x + d\right )}{e^{2}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(e*x+d))^3,x, algorithm="maxima")

[Out]

-3*e*(x/e - d*log(e*x + d)/e^2)*log((e*x + d)*c)^2 + x*log((e*x + d)*c)^3 - e*(3*(d*log(e*x + d)^2 - 2*e*x + 2
*d*log(e*x + d))*log((e*x + d)*c)/e^2 - (d*log(e*x + d)^3 + 3*d*log(e*x + d)^2 - 6*e*x + 6*d*log(e*x + d))/e^2
)

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Fricas [A]  time = 1.86053, size = 143, normalized size = 2.34 \begin{align*} \frac{{\left (e x + d\right )} \log \left (c e x + c d\right )^{3} - 3 \,{\left (e x + d\right )} \log \left (c e x + c d\right )^{2} - 6 \, e x + 6 \,{\left (e x + d\right )} \log \left (c e x + c d\right )}{e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(e*x+d))^3,x, algorithm="fricas")

[Out]

((e*x + d)*log(c*e*x + c*d)^3 - 3*(e*x + d)*log(c*e*x + c*d)^2 - 6*e*x + 6*(e*x + d)*log(c*e*x + c*d))/e

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Sympy [A]  time = 0.456343, size = 68, normalized size = 1.11 \begin{align*} - 6 e \left (- \frac{d \log{\left (d + e x \right )}}{e^{2}} + \frac{x}{e}\right ) + 6 x \log{\left (c \left (d + e x\right ) \right )} + \frac{\left (- 3 d - 3 e x\right ) \log{\left (c \left (d + e x\right ) \right )}^{2}}{e} + \frac{\left (d + e x\right ) \log{\left (c \left (d + e x\right ) \right )}^{3}}{e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*(e*x+d))**3,x)

[Out]

-6*e*(-d*log(d + e*x)/e**2 + x/e) + 6*x*log(c*(d + e*x)) + (-3*d - 3*e*x)*log(c*(d + e*x))**2/e + (d + e*x)*lo
g(c*(d + e*x))**3/e

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Giac [A]  time = 1.29826, size = 96, normalized size = 1.57 \begin{align*}{\left (x e + d\right )} e^{\left (-1\right )} \log \left ({\left (x e + d\right )} c\right )^{3} - 3 \,{\left (x e + d\right )} e^{\left (-1\right )} \log \left ({\left (x e + d\right )} c\right )^{2} + 6 \,{\left (x e + d\right )} e^{\left (-1\right )} \log \left ({\left (x e + d\right )} c\right ) - 6 \,{\left (x e + d\right )} e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(e*x+d))^3,x, algorithm="giac")

[Out]

(x*e + d)*e^(-1)*log((x*e + d)*c)^3 - 3*(x*e + d)*e^(-1)*log((x*e + d)*c)^2 + 6*(x*e + d)*e^(-1)*log((x*e + d)
*c) - 6*(x*e + d)*e^(-1)