3.1 \(\int \log ^4(c (d+e x)) \, dx\)

Optimal. Leaf size=81 \[ \frac{(d+e x) \log ^4(c (d+e x))}{e}-\frac{4 (d+e x) \log ^3(c (d+e x))}{e}+\frac{12 (d+e x) \log ^2(c (d+e x))}{e}-\frac{24 (d+e x) \log (c (d+e x))}{e}+24 x \]

[Out]

24*x - (24*(d + e*x)*Log[c*(d + e*x)])/e + (12*(d + e*x)*Log[c*(d + e*x)]^2)/e - (4*(d + e*x)*Log[c*(d + e*x)]
^3)/e + ((d + e*x)*Log[c*(d + e*x)]^4)/e

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Rubi [A]  time = 0.0350419, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {2389, 2296, 2295} \[ \frac{(d+e x) \log ^4(c (d+e x))}{e}-\frac{4 (d+e x) \log ^3(c (d+e x))}{e}+\frac{12 (d+e x) \log ^2(c (d+e x))}{e}-\frac{24 (d+e x) \log (c (d+e x))}{e}+24 x \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(d + e*x)]^4,x]

[Out]

24*x - (24*(d + e*x)*Log[c*(d + e*x)])/e + (12*(d + e*x)*Log[c*(d + e*x)]^2)/e - (4*(d + e*x)*Log[c*(d + e*x)]
^3)/e + ((d + e*x)*Log[c*(d + e*x)]^4)/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 ^4(c (d+e x)) \, dx &=\frac{\operatorname{Subst}\left (\int \log ^4(c x) \, dx,x,d+e x\right )}{e}\\ &=\frac{(d+e x) \log ^4(c (d+e x))}{e}-\frac{4 \operatorname{Subst}\left (\int \log ^3(c x) \, dx,x,d+e x\right )}{e}\\ &=-\frac{4 (d+e x) \log ^3(c (d+e x))}{e}+\frac{(d+e x) \log ^4(c (d+e x))}{e}+\frac{12 \operatorname{Subst}\left (\int \log ^2(c x) \, dx,x,d+e x\right )}{e}\\ &=\frac{12 (d+e x) \log ^2(c (d+e x))}{e}-\frac{4 (d+e x) \log ^3(c (d+e x))}{e}+\frac{(d+e x) \log ^4(c (d+e x))}{e}-\frac{24 \operatorname{Subst}(\int \log (c x) \, dx,x,d+e x)}{e}\\ &=24 x-\frac{24 (d+e x) \log (c (d+e x))}{e}+\frac{12 (d+e x) \log ^2(c (d+e x))}{e}-\frac{4 (d+e x) \log ^3(c (d+e x))}{e}+\frac{(d+e x) \log ^4(c (d+e x))}{e}\\ \end{align*}

Mathematica [A]  time = 0.0077568, size = 74, normalized size = 0.91 \[ \frac{(d+e x) \log ^4(c (d+e x))-4 (d+e x) \log ^3(c (d+e x))+12 (d+e x) \log ^2(c (d+e x))-24 (d+e x) \log (c (d+e x))+24 e x}{e} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(d + e*x)]^4,x]

[Out]

(24*e*x - 24*(d + e*x)*Log[c*(d + e*x)] + 12*(d + e*x)*Log[c*(d + e*x)]^2 - 4*(d + e*x)*Log[c*(d + e*x)]^3 + (
d + e*x)*Log[c*(d + e*x)]^4)/e

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Maple [A]  time = 0.06, size = 129, normalized size = 1.6 \begin{align*} \left ( \ln \left ( cex+cd \right ) \right ) ^{4}x+{\frac{ \left ( \ln \left ( cex+cd \right ) \right ) ^{4}d}{e}}-4\, \left ( \ln \left ( cex+cd \right ) \right ) ^{3}x-4\,{\frac{ \left ( \ln \left ( cex+cd \right ) \right ) ^{3}d}{e}}+12\, \left ( \ln \left ( cex+cd \right ) \right ) ^{2}x+12\,{\frac{ \left ( \ln \left ( cex+cd \right ) \right ) ^{2}d}{e}}-24\,\ln \left ( cex+cd \right ) x-24\,{\frac{\ln \left ( cex+cd \right ) d}{e}}+24\,x+24\,{\frac{d}{e}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(e*x+d))^4,x)

[Out]

ln(c*e*x+c*d)^4*x+1/e*ln(c*e*x+c*d)^4*d-4*ln(c*e*x+c*d)^3*x-4/e*ln(c*e*x+c*d)^3*d+12*ln(c*e*x+c*d)^2*x+12/e*ln
(c*e*x+c*d)^2*d-24*ln(c*e*x+c*d)*x-24/e*ln(c*e*x+c*d)*d+24*x+24*d/e

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Maxima [B]  time = 1.25579, size = 254, normalized size = 3.14 \begin{align*} -4 \, 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 )^{3} + x \log \left ({\left (e x + d\right )} c\right )^{4} +{\left (e{\left (\frac{4 \,{\left (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 )\right )} \log \left ({\left (e x + d\right )} c\right )}{e^{3}} - \frac{d \log \left (e x + d\right )^{4} + 4 \, d \log \left (e x + d\right )^{3} + 12 \, d \log \left (e x + d\right )^{2} - 24 \, e x + 24 \, d \log \left (e x + d\right )}{e^{3}}\right )} - \frac{6 \,{\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 )^{2}}{e^{2}}\right )} e \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(e*x+d))^4,x, algorithm="maxima")

[Out]

-4*e*(x/e - d*log(e*x + d)/e^2)*log((e*x + d)*c)^3 + x*log((e*x + d)*c)^4 + (e*(4*(d*log(e*x + d)^3 + 3*d*log(
e*x + d)^2 - 6*e*x + 6*d*log(e*x + d))*log((e*x + d)*c)/e^3 - (d*log(e*x + d)^4 + 4*d*log(e*x + d)^3 + 12*d*lo
g(e*x + d)^2 - 24*e*x + 24*d*log(e*x + d))/e^3) - 6*(d*log(e*x + d)^2 - 2*e*x + 2*d*log(e*x + d))*log((e*x + d
)*c)^2/e^2)*e

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Fricas [A]  time = 1.84186, size = 192, normalized size = 2.37 \begin{align*} \frac{{\left (e x + d\right )} \log \left (c e x + c d\right )^{4} - 4 \,{\left (e x + d\right )} \log \left (c e x + c d\right )^{3} + 12 \,{\left (e x + d\right )} \log \left (c e x + c d\right )^{2} + 24 \, e x - 24 \,{\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))^4,x, algorithm="fricas")

[Out]

((e*x + d)*log(c*e*x + c*d)^4 - 4*(e*x + d)*log(c*e*x + c*d)^3 + 12*(e*x + d)*log(c*e*x + c*d)^2 + 24*e*x - 24
*(e*x + d)*log(c*e*x + c*d))/e

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Sympy [A]  time = 0.62818, size = 88, normalized size = 1.09 \begin{align*} 24 e \left (- \frac{d \log{\left (d + e x \right )}}{e^{2}} + \frac{x}{e}\right ) - 24 x \log{\left (c \left (d + e x\right ) \right )} + \frac{\left (- 4 d - 4 e x\right ) \log{\left (c \left (d + e x\right ) \right )}^{3}}{e} + \frac{\left (d + e x\right ) \log{\left (c \left (d + e x\right ) \right )}^{4}}{e} + \frac{\left (12 d + 12 e x\right ) \log{\left (c \left (d + e x\right ) \right )}^{2}}{e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*(e*x+d))**4,x)

[Out]

24*e*(-d*log(d + e*x)/e**2 + x/e) - 24*x*log(c*(d + e*x)) + (-4*d - 4*e*x)*log(c*(d + e*x))**3/e + (d + e*x)*l
og(c*(d + e*x))**4/e + (12*d + 12*e*x)*log(c*(d + e*x))**2/e

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Giac [A]  time = 1.30065, size = 124, normalized size = 1.53 \begin{align*}{\left (x e + d\right )} e^{\left (-1\right )} \log \left ({\left (x e + d\right )} c\right )^{4} - 4 \,{\left (x e + d\right )} e^{\left (-1\right )} \log \left ({\left (x e + d\right )} c\right )^{3} + 12 \,{\left (x e + d\right )} e^{\left (-1\right )} \log \left ({\left (x e + d\right )} c\right )^{2} - 24 \,{\left (x e + d\right )} e^{\left (-1\right )} \log \left ({\left (x e + d\right )} c\right ) + 24 \,{\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))^4,x, algorithm="giac")

[Out]

(x*e + d)*e^(-1)*log((x*e + d)*c)^4 - 4*(x*e + d)*e^(-1)*log((x*e + d)*c)^3 + 12*(x*e + d)*e^(-1)*log((x*e + d
)*c)^2 - 24*(x*e + d)*e^(-1)*log((x*e + d)*c) + 24*(x*e + d)*e^(-1)