3.1.17 \(\int (a+b \log (c (d+e x)^n))^4 \, dx\) [17]

Optimal. Leaf size=131 \[ -24 a b^3 n^3 x+24 b^4 n^4 x-\frac {24 b^4 n^3 (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac {12 b^2 n^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {4 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{e} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2436, 2333, 2332} \begin {gather*} -24 a b^3 n^3 x+\frac {12 b^2 n^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {4 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4}{e}-\frac {24 b^4 n^3 (d+e x) \log \left (c (d+e x)^n\right )}{e}+24 b^4 n^4 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d + e*x)^n])^4,x]

[Out]

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

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Mathematica [A]
time = 0.04, size = 112, normalized size = 0.85 \begin {gather*} \frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^4-4 b n \left ((d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3-3 b n \left ((d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2-2 b n \left (e (a-b n) x+b (d+e x) \log \left (c (d+e x)^n\right )\right )\right )\right )}{e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d + e*x)^n])^4,x]

[Out]

((d + e*x)*(a + b*Log[c*(d + e*x)^n])^4 - 4*b*n*((d + e*x)*(a + b*Log[c*(d + e*x)^n])^3 - 3*b*n*((d + e*x)*(a
+ b*Log[c*(d + e*x)^n])^2 - 2*b*n*(e*(a - b*n)*x + b*(d + e*x)*Log[c*(d + e*x)^n]))))/e

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 1.45, size = 15871, normalized size = 121.15

method	result	size
risch	\(\text {Expression too large to display}\)	\(15871\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 528 vs. \(2 (135) = 270\).
time = 0.31, size = 528, normalized size = 4.03 \begin {gather*} b^{4} x \log \left ({\left (x e + d\right )}^{n} c\right )^{4} + 4 \, a b^{3} x \log \left ({\left (x e + d\right )}^{n} c\right )^{3} + 4 \, {\left (d e^{\left (-2\right )} \log \left (x e + d\right ) - x e^{\left (-1\right )}\right )} a^{3} b n e + 6 \, a^{2} b^{2} x \log \left ({\left (x e + d\right )}^{n} c\right )^{2} + 4 \, a^{3} b x \log \left ({\left (x e + d\right )}^{n} c\right ) - 6 \, {\left ({\left (d \log \left (x e + d\right )^{2} - 2 \, x e + 2 \, d \log \left (x e + d\right )\right )} n^{2} e^{\left (-1\right )} - 2 \, {\left (d e^{\left (-2\right )} \log \left (x e + d\right ) - x e^{\left (-1\right )}\right )} n e \log \left ({\left (x e + d\right )}^{n} c\right )\right )} a^{2} b^{2} + 4 \, {\left (3 \, {\left (d e^{\left (-2\right )} \log \left (x e + d\right ) - x e^{\left (-1\right )}\right )} n e \log \left ({\left (x e + d\right )}^{n} c\right )^{2} + {\left ({\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 )} n^{2} e^{\left (-2\right )} - 3 \, {\left (d \log \left (x e + d\right )^{2} - 2 \, x e + 2 \, d \log \left (x e + d\right )\right )} n e^{\left (-2\right )} \log \left ({\left (x e + d\right )}^{n} c\right )\right )} n e\right )} a b^{3} + {\left (4 \, {\left (d e^{\left (-2\right )} \log \left (x e + d\right ) - x e^{\left (-1\right )}\right )} n e \log \left ({\left (x e + d\right )}^{n} c\right )^{3} - {\left (6 \, {\left (d \log \left (x e + d\right )^{2} - 2 \, x e + 2 \, d \log \left (x e + d\right )\right )} n e^{\left (-2\right )} \log \left ({\left (x e + d\right )}^{n} c\right )^{2} + {\left ({\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 )} n^{2} e^{\left (-3\right )} - 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 )} n e^{\left (-3\right )} \log \left ({\left (x e + d\right )}^{n} c\right )\right )} n e\right )} n e\right )} b^{4} + a^{4} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^4*x*log((x*e + d)^n*c)^4 + 4*a*b^3*x*log((x*e + d)^n*c)^3 + 4*(d*e^(-2)*log(x*e + d) - x*e^(-1))*a^3*b*n*e +
6*a^2*b^2*x*log((x*e + d)^n*c)^2 + 4*a^3*b*x*log((x*e + d)^n*c) - 6*((d*log(x*e + d)^2 - 2*x*e + 2*d*log(x*e
+ d))*n^2*e^(-1) - 2*(d*e^(-2)*log(x*e + d) - x*e^(-1))*n*e*log((x*e + d)^n*c))*a^2*b^2 + 4*(3*(d*e^(-2)*log(x
*e + d) - x*e^(-1))*n*e*log((x*e + d)^n*c)^2 + ((d*log(x*e + d)^3 + 3*d*log(x*e + d)^2 - 6*x*e + 6*d*log(x*e +
d))*n^2*e^(-2) - 3*(d*log(x*e + d)^2 - 2*x*e + 2*d*log(x*e + d))*n*e^(-2)*log((x*e + d)^n*c))*n*e)*a*b^3 + (4
*(d*e^(-2)*log(x*e + d) - x*e^(-1))*n*e*log((x*e + d)^n*c)^3 - (6*(d*log(x*e + d)^2 - 2*x*e + 2*d*log(x*e + d)
)*n*e^(-2)*log((x*e + d)^n*c)^2 + ((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))*n^2*e^(-3) - 4*(d*log(x*e + d)^3 + 3*d*log(x*e + d)^2 - 6*x*e + 6*d*log(x*e + d))*n*e^(-3)*log
((x*e + d)^n*c))*n*e)*n*e)*b^4 + a^4*x

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 611 vs. \(2 (135) = 270\).
time = 0.39, size = 611, normalized size = 4.66 \begin {gather*} {\left (b^{4} x e \log \left (c\right )^{4} - 4 \, {\left (b^{4} n - a b^{3}\right )} x e \log \left (c\right )^{3} + {\left (b^{4} n^{4} x e + b^{4} d n^{4}\right )} \log \left (x e + d\right )^{4} + 6 \, {\left (2 \, b^{4} n^{2} - 2 \, a b^{3} n + a^{2} b^{2}\right )} x e \log \left (c\right )^{2} - 4 \, {\left (b^{4} d n^{4} - a b^{3} d n^{3} + {\left (b^{4} n^{4} - a b^{3} n^{3}\right )} x e - {\left (b^{4} n^{3} x e + b^{4} d n^{3}\right )} \log \left (c\right )\right )} \log \left (x e + d\right )^{3} - 4 \, {\left (6 \, b^{4} n^{3} - 6 \, a b^{3} n^{2} + 3 \, a^{2} b^{2} n - a^{3} b\right )} x e \log \left (c\right ) + {\left (24 \, b^{4} n^{4} - 24 \, a b^{3} n^{3} + 12 \, a^{2} b^{2} n^{2} - 4 \, a^{3} b n + a^{4}\right )} x e + 6 \, {\left (2 \, b^{4} d n^{4} - 2 \, a b^{3} d n^{3} + a^{2} b^{2} d n^{2} + {\left (2 \, b^{4} n^{4} - 2 \, a b^{3} n^{3} + a^{2} b^{2} n^{2}\right )} x e + {\left (b^{4} n^{2} x e + b^{4} d n^{2}\right )} \log \left (c\right )^{2} - 2 \, {\left (b^{4} d n^{3} - a b^{3} d n^{2} + {\left (b^{4} n^{3} - a b^{3} n^{2}\right )} x e\right )} \log \left (c\right )\right )} \log \left (x e + d\right )^{2} - 4 \, {\left (6 \, b^{4} d n^{4} - 6 \, a b^{3} d n^{3} + 3 \, a^{2} b^{2} d n^{2} - a^{3} b d n - {\left (b^{4} n x e + b^{4} d n\right )} \log \left (c\right )^{3} + {\left (6 \, b^{4} n^{4} - 6 \, a b^{3} n^{3} + 3 \, a^{2} b^{2} n^{2} - a^{3} b n\right )} x e + 3 \, {\left (b^{4} d n^{2} - a b^{3} d n + {\left (b^{4} n^{2} - a b^{3} n\right )} x e\right )} \log \left (c\right )^{2} - 3 \, {\left (2 \, b^{4} d n^{3} - 2 \, a b^{3} d n^{2} + a^{2} b^{2} d n + {\left (2 \, b^{4} n^{3} - 2 \, a b^{3} n^{2} + a^{2} b^{2} n\right )} x e\right )} \log \left (c\right )\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))^4,x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (126) = 252\).
time = 1.01, size = 495, normalized size = 3.78 \begin {gather*} \begin {cases} a^{4} x + \frac {4 a^{3} b d \log {\left (c \left (d + e x\right )^{n} \right )}}{e} - 4 a^{3} b n x + 4 a^{3} b x \log {\left (c \left (d + e x\right )^{n} \right )} - \frac {12 a^{2} b^{2} d n \log {\left (c \left (d + e x\right )^{n} \right )}}{e} + \frac {6 a^{2} b^{2} d \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{e} + 12 a^{2} b^{2} n^{2} x - 12 a^{2} b^{2} n x \log {\left (c \left (d + e x\right )^{n} \right )} + 6 a^{2} b^{2} x \log {\left (c \left (d + e x\right )^{n} \right )}^{2} + \frac {24 a b^{3} d n^{2} \log {\left (c \left (d + e x\right )^{n} \right )}}{e} - \frac {12 a b^{3} d n \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{e} + \frac {4 a b^{3} d \log {\left (c \left (d + e x\right )^{n} \right )}^{3}}{e} - 24 a b^{3} n^{3} x + 24 a b^{3} n^{2} x \log {\left (c \left (d + e x\right )^{n} \right )} - 12 a b^{3} n x \log {\left (c \left (d + e x\right )^{n} \right )}^{2} + 4 a b^{3} x \log {\left (c \left (d + e x\right )^{n} \right )}^{3} - \frac {24 b^{4} d n^{3} \log {\left (c \left (d + e x\right )^{n} \right )}}{e} + \frac {12 b^{4} d n^{2} \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{e} - \frac {4 b^{4} d n \log {\left (c \left (d + e x\right )^{n} \right )}^{3}}{e} + \frac {b^{4} d \log {\left (c \left (d + e x\right )^{n} \right )}^{4}}{e} + 24 b^{4} n^{4} x - 24 b^{4} n^{3} x \log {\left (c \left (d + e x\right )^{n} \right )} + 12 b^{4} n^{2} x \log {\left (c \left (d + e x\right )^{n} \right )}^{2} - 4 b^{4} n x \log {\left (c \left (d + e x\right )^{n} \right )}^{3} + b^{4} x \log {\left (c \left (d + e x\right )^{n} \right )}^{4} & \text {for}\: e \neq 0 \\x \left (a + b \log {\left (c d^{n} \right )}\right )^{4} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**4*x + 4*a**3*b*d*log(c*(d + e*x)**n)/e - 4*a**3*b*n*x + 4*a**3*b*x*log(c*(d + e*x)**n) - 12*a**2
*b**2*d*n*log(c*(d + e*x)**n)/e + 6*a**2*b**2*d*log(c*(d + e*x)**n)**2/e + 12*a**2*b**2*n**2*x - 12*a**2*b**2*
n*x*log(c*(d + e*x)**n) + 6*a**2*b**2*x*log(c*(d + e*x)**n)**2 + 24*a*b**3*d*n**2*log(c*(d + e*x)**n)/e - 12*a
*b**3*d*n*log(c*(d + e*x)**n)**2/e + 4*a*b**3*d*log(c*(d + e*x)**n)**3/e - 24*a*b**3*n**3*x + 24*a*b**3*n**2*x
*log(c*(d + e*x)**n) - 12*a*b**3*n*x*log(c*(d + e*x)**n)**2 + 4*a*b**3*x*log(c*(d + e*x)**n)**3 - 24*b**4*d*n*
*3*log(c*(d + e*x)**n)/e + 12*b**4*d*n**2*log(c*(d + e*x)**n)**2/e - 4*b**4*d*n*log(c*(d + e*x)**n)**3/e + b**
4*d*log(c*(d + e*x)**n)**4/e + 24*b**4*n**4*x - 24*b**4*n**3*x*log(c*(d + e*x)**n) + 12*b**4*n**2*x*log(c*(d +
e*x)**n)**2 - 4*b**4*n*x*log(c*(d + e*x)**n)**3 + b**4*x*log(c*(d + e*x)**n)**4, Ne(e, 0)), (x*(a + b*log(c*d
**n))**4, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 778 vs. \(2 (135) = 270\).
time = 1.97, size = 778, normalized size = 5.94 \begin {gather*} {\left (x e + d\right )} b^{4} n^{4} e^{\left (-1\right )} \log \left (x e + d\right )^{4} - 4 \, {\left (x e + d\right )} b^{4} n^{4} e^{\left (-1\right )} \log \left (x e + d\right )^{3} + 4 \, {\left (x e + d\right )} b^{4} n^{3} e^{\left (-1\right )} \log \left (x e + d\right )^{3} \log \left (c\right ) + 12 \, {\left (x e + d\right )} b^{4} n^{4} e^{\left (-1\right )} \log \left (x e + d\right )^{2} + 4 \, {\left (x e + d\right )} a b^{3} n^{3} e^{\left (-1\right )} \log \left (x e + d\right )^{3} - 12 \, {\left (x e + d\right )} b^{4} n^{3} e^{\left (-1\right )} \log \left (x e + d\right )^{2} \log \left (c\right ) + 6 \, {\left (x e + d\right )} b^{4} n^{2} e^{\left (-1\right )} \log \left (x e + d\right )^{2} \log \left (c\right )^{2} - 24 \, {\left (x e + d\right )} b^{4} n^{4} e^{\left (-1\right )} \log \left (x e + d\right ) - 12 \, {\left (x e + d\right )} a b^{3} n^{3} e^{\left (-1\right )} \log \left (x e + d\right )^{2} + 24 \, {\left (x e + d\right )} b^{4} n^{3} e^{\left (-1\right )} \log \left (x e + d\right ) \log \left (c\right ) + 12 \, {\left (x e + d\right )} a b^{3} n^{2} e^{\left (-1\right )} \log \left (x e + d\right )^{2} \log \left (c\right ) - 12 \, {\left (x e + d\right )} b^{4} n^{2} e^{\left (-1\right )} \log \left (x e + d\right ) \log \left (c\right )^{2} + 4 \, {\left (x e + d\right )} b^{4} n e^{\left (-1\right )} \log \left (x e + d\right ) \log \left (c\right )^{3} + 24 \, {\left (x e + d\right )} b^{4} n^{4} e^{\left (-1\right )} + 24 \, {\left (x e + d\right )} a b^{3} n^{3} e^{\left (-1\right )} \log \left (x e + d\right ) + 6 \, {\left (x e + d\right )} a^{2} b^{2} n^{2} e^{\left (-1\right )} \log \left (x e + d\right )^{2} - 24 \, {\left (x e + d\right )} b^{4} n^{3} e^{\left (-1\right )} \log \left (c\right ) - 24 \, {\left (x e + d\right )} a b^{3} n^{2} e^{\left (-1\right )} \log \left (x e + d\right ) \log \left (c\right ) + 12 \, {\left (x e + d\right )} b^{4} n^{2} e^{\left (-1\right )} \log \left (c\right )^{2} + 12 \, {\left (x e + d\right )} a b^{3} n e^{\left (-1\right )} \log \left (x e + d\right ) \log \left (c\right )^{2} - 4 \, {\left (x e + d\right )} b^{4} n e^{\left (-1\right )} \log \left (c\right )^{3} + {\left (x e + d\right )} b^{4} e^{\left (-1\right )} \log \left (c\right )^{4} - 24 \, {\left (x e + d\right )} a b^{3} n^{3} e^{\left (-1\right )} - 12 \, {\left (x e + d\right )} a^{2} b^{2} n^{2} e^{\left (-1\right )} \log \left (x e + d\right ) + 24 \, {\left (x e + d\right )} a b^{3} n^{2} e^{\left (-1\right )} \log \left (c\right ) + 12 \, {\left (x e + d\right )} a^{2} b^{2} n e^{\left (-1\right )} \log \left (x e + d\right ) \log \left (c\right ) - 12 \, {\left (x e + d\right )} a b^{3} n e^{\left (-1\right )} \log \left (c\right )^{2} + 4 \, {\left (x e + d\right )} a b^{3} e^{\left (-1\right )} \log \left (c\right )^{3} + 12 \, {\left (x e + d\right )} a^{2} b^{2} n^{2} e^{\left (-1\right )} + 4 \, {\left (x e + d\right )} a^{3} b n e^{\left (-1\right )} \log \left (x e + d\right ) - 12 \, {\left (x e + d\right )} a^{2} b^{2} n e^{\left (-1\right )} \log \left (c\right ) + 6 \, {\left (x e + d\right )} a^{2} b^{2} e^{\left (-1\right )} \log \left (c\right )^{2} - 4 \, {\left (x e + d\right )} a^{3} b n e^{\left (-1\right )} + 4 \, {\left (x e + d\right )} a^{3} b e^{\left (-1\right )} \log \left (c\right ) + {\left (x e + d\right )} a^{4} 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))^4,x, algorithm="giac")

[Out]

(x*e + d)*b^4*n^4*e^(-1)*log(x*e + d)^4 - 4*(x*e + d)*b^4*n^4*e^(-1)*log(x*e + d)^3 + 4*(x*e + d)*b^4*n^3*e^(-
1)*log(x*e + d)^3*log(c) + 12*(x*e + d)*b^4*n^4*e^(-1)*log(x*e + d)^2 + 4*(x*e + d)*a*b^3*n^3*e^(-1)*log(x*e +
d)^3 - 12*(x*e + d)*b^4*n^3*e^(-1)*log(x*e + d)^2*log(c) + 6*(x*e + d)*b^4*n^2*e^(-1)*log(x*e + d)^2*log(c)^2
- 24*(x*e + d)*b^4*n^4*e^(-1)*log(x*e + d) - 12*(x*e + d)*a*b^3*n^3*e^(-1)*log(x*e + d)^2 + 24*(x*e + d)*b^4*
n^3*e^(-1)*log(x*e + d)*log(c) + 12*(x*e + d)*a*b^3*n^2*e^(-1)*log(x*e + d)^2*log(c) - 12*(x*e + d)*b^4*n^2*e^
(-1)*log(x*e + d)*log(c)^2 + 4*(x*e + d)*b^4*n*e^(-1)*log(x*e + d)*log(c)^3 + 24*(x*e + d)*b^4*n^4*e^(-1) + 24
*(x*e + d)*a*b^3*n^3*e^(-1)*log(x*e + d) + 6*(x*e + d)*a^2*b^2*n^2*e^(-1)*log(x*e + d)^2 - 24*(x*e + d)*b^4*n^
3*e^(-1)*log(c) - 24*(x*e + d)*a*b^3*n^2*e^(-1)*log(x*e + d)*log(c) + 12*(x*e + d)*b^4*n^2*e^(-1)*log(c)^2 + 1
2*(x*e + d)*a*b^3*n*e^(-1)*log(x*e + d)*log(c)^2 - 4*(x*e + d)*b^4*n*e^(-1)*log(c)^3 + (x*e + d)*b^4*e^(-1)*lo
g(c)^4 - 24*(x*e + d)*a*b^3*n^3*e^(-1) - 12*(x*e + d)*a^2*b^2*n^2*e^(-1)*log(x*e + d) + 24*(x*e + d)*a*b^3*n^2
*e^(-1)*log(c) + 12*(x*e + d)*a^2*b^2*n*e^(-1)*log(x*e + d)*log(c) - 12*(x*e + d)*a*b^3*n*e^(-1)*log(c)^2 + 4*
(x*e + d)*a*b^3*e^(-1)*log(c)^3 + 12*(x*e + d)*a^2*b^2*n^2*e^(-1) + 4*(x*e + d)*a^3*b*n*e^(-1)*log(x*e + d) -
12*(x*e + d)*a^2*b^2*n*e^(-1)*log(c) + 6*(x*e + d)*a^2*b^2*e^(-1)*log(c)^2 - 4*(x*e + d)*a^3*b*n*e^(-1) + 4*(x
*e + d)*a^3*b*e^(-1)*log(c) + (x*e + d)*a^4*e^(-1)

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Mupad [B]
time = 0.36, size = 275, normalized size = 2.10 \begin {gather*} {\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^2\,\left (\frac {6\,\left (d\,a^2\,b^2-2\,d\,a\,b^3\,n+2\,d\,b^4\,n^2\right )}{e}+6\,b^2\,x\,\left (a^2-2\,a\,b\,n+2\,b^2\,n^2\right )\right )+x\,\left (a^4-4\,a^3\,b\,n+12\,a^2\,b^2\,n^2-24\,a\,b^3\,n^3+24\,b^4\,n^4\right )+{\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^4\,\left (b^4\,x+\frac {b^4\,d}{e}\right )+{\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^3\,\left (\frac {4\,\left (a\,b^3\,d-b^4\,d\,n\right )}{e}+4\,b^3\,x\,\left (a-b\,n\right )\right )-\frac {\ln \left (d+e\,x\right )\,\left (-4\,d\,a^3\,b\,n+12\,d\,a^2\,b^2\,n^2-24\,d\,a\,b^3\,n^3+24\,d\,b^4\,n^4\right )}{e}+4\,b\,x\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\,\left (a^3-3\,a^2\,b\,n+6\,a\,b^2\,n^2-6\,b^3\,n^3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*(d + e*x)^n))^4,x)

[Out]

log(c*(d + e*x)^n)^2*((6*(a^2*b^2*d + 2*b^4*d*n^2 - 2*a*b^3*d*n))/e + 6*b^2*x*(a^2 + 2*b^2*n^2 - 2*a*b*n)) + x
*(a^4 + 24*b^4*n^4 - 24*a*b^3*n^3 + 12*a^2*b^2*n^2 - 4*a^3*b*n) + log(c*(d + e*x)^n)^4*(b^4*x + (b^4*d)/e) + l
og(c*(d + e*x)^n)^3*((4*(a*b^3*d - b^4*d*n))/e + 4*b^3*x*(a - b*n)) - (log(d + e*x)*(24*b^4*d*n^4 + 12*a^2*b^2
*d*n^2 - 4*a^3*b*d*n - 24*a*b^3*d*n^3))/e + 4*b*x*log(c*(d + e*x)^n)*(a^3 - 6*b^3*n^3 + 6*a*b^2*n^2 - 3*a^2*b*
n)

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