3.1.1 \(\int \log ^4(c (d+e x)) \, dx\) [1]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2436, 2333, 2332} \begin {gather*} \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 \end {gather*}

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 2332

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

Rule 2333

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 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 \log ^4(c (d+e x)) \, dx &=\frac {\text {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 \text {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 \text {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 \text {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*}

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Mathematica [A]
time = 0.01, size = 74, normalized size = 0.91 \begin {gather*} \frac {24 e x-24 (d+e x) \log (c (d+e x))+12 (d+e x) \log ^2(c (d+e x))-4 (d+e x) \log ^3(c (d+e x))+(d+e x) \log ^4(c (d+e x))}{e} \end {gather*}

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 = 0.15, size = 99, normalized size = 1.22

method	result	size
risch	\(\frac {\left (e x +d \right ) \ln \left (c \left (e x +d \right )\right )^{4}}{e}-\frac {4 \left (e x +d \right ) \ln \left (c \left (e x +d \right )\right )^{3}}{e}+\frac {12 \left (e x +d \right ) \ln \left (c \left (e x +d \right )\right )^{2}}{e}-24 x \ln \left (c \left (e x +d \right )\right )+24 x -\frac {24 d \ln \left (e x +d \right )}{e}\)	\(87\)
derivativedivides	\(\frac {\ln \left (c e x +c d \right )^{4} \left (c e x +c d \right )-4 \left (c e x +c d \right ) \ln \left (c e x +c d \right )^{3}+12 \left (c e x +c d \right ) \ln \left (c e x +c d \right )^{2}-24 \left (c e x +c d \right ) \ln \left (c e x +c d \right )+24 c e x +24 c d}{c e}\)	\(99\)
default	\(\frac {\ln \left (c e x +c d \right )^{4} \left (c e x +c d \right )-4 \left (c e x +c d \right ) \ln \left (c e x +c d \right )^{3}+12 \left (c e x +c d \right ) \ln \left (c e x +c d \right )^{2}-24 \left (c e x +c d \right ) \ln \left (c e x +c d \right )+24 c e x +24 c d}{c e}\)	\(99\)
norman	\(x \ln \left (c \left (e x +d \right )\right )^{4}+\frac {d \ln \left (c \left (e x +d \right )\right )^{4}}{e}+24 x -24 x \ln \left (c \left (e x +d \right )\right )+12 x \ln \left (c \left (e x +d \right )\right )^{2}-4 x \ln \left (c \left (e x +d \right )\right )^{3}-\frac {24 d \ln \left (c \left (e x +d \right )\right )}{e}+\frac {12 d \ln \left (c \left (e x +d \right )\right )^{2}}{e}-\frac {4 d \ln \left (c \left (e x +d \right )\right )^{3}}{e}\)	\(115\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 205 vs. \(2 (85) = 170\).
time = 0.28, size = 205, normalized size = 2.53 \begin {gather*} 4 \, {\left (d e^{\left (-2\right )} \log \left (x e + d\right ) - x e^{\left (-1\right )}\right )} e \log \left ({\left (x e + d\right )} c\right )^{3} + x \log \left ({\left (x e + d\right )} c\right )^{4} - {\left (6 \, {\left (d \log \left (x e + d\right )^{2} - 2 \, x e + 2 \, d \log \left (x e + d\right )\right )} e^{\left (-2\right )} \log \left ({\left (x e + d\right )} c\right )^{2} - {\left (4 \, {\left (d \log \left (x e + d\right )^{3} + 3 \, d \log \left (x e + d\right )^{2} - 6 \, x e + 6 \, d \log \left (x e + d\right )\right )} e^{\left (-3\right )} \log \left ({\left (x e + d\right )} c\right ) - {\left (d \log \left (x e + d\right )^{4} + 4 \, d \log \left (x e + d\right )^{3} + 12 \, d \log \left (x e + d\right )^{2} - 24 \, x e + 24 \, d \log \left (x e + d\right )\right )} e^{\left (-3\right )}\right )} e\right )} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*(d*e^(-2)*log(x*e + d) - x*e^(-1))*e*log((x*e + d)*c)^3 + x*log((x*e + d)*c)^4 - (6*(d*log(x*e + d)^2 - 2*x*
e + 2*d*log(x*e + d))*e^(-2)*log((x*e + d)*c)^2 - (4*(d*log(x*e + d)^3 + 3*d*log(x*e + d)^2 - 6*x*e + 6*d*log(
x*e + d))*e^(-3)*log((x*e + d)*c) - (d*log(x*e + d)^4 + 4*d*log(x*e + d)^3 + 12*d*log(x*e + d)^2 - 24*x*e + 24
*d*log(x*e + d))*e^(-3))*e)*e

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Fricas [A]
time = 0.34, size = 86, normalized size = 1.06 \begin {gather*} {\left ({\left (x e + d\right )} \log \left (c x e + c d\right )^{4} - 4 \, {\left (x e + d\right )} \log \left (c x e + c d\right )^{3} + 12 \, {\left (x e + d\right )} \log \left (c x e + c d\right )^{2} + 24 \, x e - 24 \, {\left (x e + d\right )} \log \left (c x e + c d\right )\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.12, size = 88, normalized size = 1.09 \begin {gather*} 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 {gather*}

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 = 4.45, size = 92, normalized size = 1.14 \begin {gather*} {\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 {gather*}

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)

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Mupad [B]
time = 0.36, size = 119, normalized size = 1.47 \begin {gather*} 24\,x-24\,x\,\ln \left (c\,d+c\,e\,x\right )+12\,x\,{\ln \left (c\,d+c\,e\,x\right )}^2-4\,x\,{\ln \left (c\,d+c\,e\,x\right )}^3+x\,{\ln \left (c\,d+c\,e\,x\right )}^4+\frac {12\,d\,{\ln \left (c\,d+c\,e\,x\right )}^2}{e}-\frac {4\,d\,{\ln \left (c\,d+c\,e\,x\right )}^3}{e}+\frac {d\,{\ln \left (c\,d+c\,e\,x\right )}^4}{e}-\frac {24\,d\,\ln \left (d+e\,x\right )}{e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*(d + e*x))^4,x)

[Out]

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

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