3.1.55 \(\int (a+b \log (c (d+e x)^n))^3 \, dx\) [55]

Optimal. Leaf size=99 \[ 6 a b^2 n^2 x-6 b^3 n^3 x+\frac {6 b^3 n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac {3 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e} \]

[Out]

6*a*b^2*n^2*x-6*b^3*n^3*x+6*b^3*n^2*(e*x+d)*ln(c*(e*x+d)^n)/e-3*b*n*(e*x+d)*(a+b*ln(c*(e*x+d)^n))^2/e+(e*x+d)*
(a+b*ln(c*(e*x+d)^n))^3/e

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Rubi [A]
time = 0.04, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 5, 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*} 6 a b^2 n^2 x-\frac {3 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {6 b^3 n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-6 b^3 n^3 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d + e*x)^n])^3,x]

[Out]

6*a*b^2*n^2*x - 6*b^3*n^3*x + (6*b^3*n^2*(d + e*x)*Log[c*(d + e*x)^n])/e - (3*b*n*(d + e*x)*(a + b*Log[c*(d +
e*x)^n])^2)/e + ((d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/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 )^3 \, dx &=\frac {\text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x\right )}{e}\\ &=\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac {(3 b n) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e}\\ &=-\frac {3 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {\left (6 b^2 n^2\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e}\\ &=6 a b^2 n^2 x-\frac {3 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {\left (6 b^3 n^2\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e}\\ &=6 a b^2 n^2 x-6 b^3 n^3 x+\frac {6 b^3 n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac {3 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 85, normalized size = 0.86 \begin {gather*} \frac {(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 )}{e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d + e*x)^n])^3,x]

[Out]

((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 = 0.12, size = 4872, normalized size = 49.21

method	result	size
risch	\(\text {Expression too large to display}\)	\(4872\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(e*x+d)^n))^3,x,method=_RETURNVERBOSE)

[Out]

ln(c)^3*b^3*x+b^3*x*ln((e*x+d)^n)^3-3*ln(c)*Pi^2*b^3*x*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^4+3/4*P
i^2*b^3*n*x*csgn(I*c)^2*csgn(I*(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)^2-3/2*Pi^2*b^3*n*x*csgn(I*c)^2*csgn(I*(e*x+d)^
n)*csgn(I*c*(e*x+d)^n)^3-3/2*Pi^2*b^3*n*x*csgn(I*c)*csgn(I*(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)^3+3*Pi^2*b^3*n*x*c
sgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^4-3/4*ln(c)*Pi^2*b^3*x*csgn(I*c)^2*csgn(I*(e*x+d)^n)^2*csgn(I*c
*(e*x+d)^n)^2+3/2*ln(c)*Pi^2*b^3*x*csgn(I*c)^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^3+3/2*ln(c)*Pi^2*b^3*x*cs
gn(I*c)*csgn(I*(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)^3-3/4/e*Pi^2*ln(e*x+d)*b^3*d*n*csgn(I*c*(e*x+d)^n)^6+6/e*ln(c)
*ln(e*x+d)*a*b^2*d*n+3/2*I*ln(c)^2*Pi*b^3*x*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+3*I*Pi*a*b^2*n*x*csgn(I*c*(e*x+d)^
n)^3+3*I*ln(c)*Pi*b^3*n*x*csgn(I*c*(e*x+d)^n)^3+3*I*Pi*b^3*n^2*x*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+3*I*Pi*b^3*n^
2*x*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2+3/2*I*Pi*a^2*b*x*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+3/2*Pi^2*a*b^2*x*
csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^5+6*a*b^2*n^2*x-3*ln(c)^2*b^3*n*x+6*ln(c)*b^3*n^2*x-3/2*Pi^2*b^3*n*x*csg
n(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^5-3/4*Pi^2*a*b^2*x*csgn(I*c)^2*csgn(I*c*(e*x+d)^n)^4+3/2*Pi^2*a*b^2*x*csgn(
I*c)*csgn(I*c*(e*x+d)^n)^5-3/4*Pi^2*a*b^2*x*csgn(I*(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)^4+3/4*Pi^2*b^3*n*x*csgn(I*
c)^2*csgn(I*c*(e*x+d)^n)^4-6/e*ln(e*x+d)*a*b^2*d*n^2-6*b^3*n^3*x+a^3*x+3/2*I/e*Pi*b^3*d*n^2*csgn(I*c*(e*x+d)^n
)^3*ln(e*x+d)^2-3/2*I*ln(c)^2*Pi*b^3*x*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-3*I*ln(c)*Pi*b^3*n*x*cs
gn(I*c)*csgn(I*c*(e*x+d)^n)^2-3*I*ln(c)*Pi*b^3*n*x*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-3*I*Pi*b^3*n^2*x*cs
gn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+3*I*ln(c)*Pi*a*b^2*x*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+3*I*ln(c)*P
i*a*b^2*x*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-3*I*Pi*a*b^2*n*x*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2-3*I*Pi*a*b^
2*n*x*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-3/2*I*Pi*a^2*b*x*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)
+3*I/e*Pi*b^3*d*n^2*csgn(I*c*(e*x+d)^n)^3*ln(e*x+d)+3/4*b*(-4*I*Pi*ln(e*x+d)*b^2*d*n*csgn(I*c*(e*x+d)^n)^3+4*a
^2*e*x-Pi^2*b^2*e*x*csgn(I*c)^2*csgn(I*c*(e*x+d)^n)^4-4*I*Pi*b^2*e*n*x*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+4*I*ln(
c)*Pi*b^2*e*x*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2+4*I*ln(c)*Pi*b^2*e*x*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+4*I
*Pi*ln(e*x+d)*b^2*d*n*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-4*I*ln(c)*Pi*b^2*e*x*csgn(I*c)*csgn(I*(e*x+d)^n)
*csgn(I*c*(e*x+d)^n)+4*I*Pi*ln(e*x+d)*b^2*d*n*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2-4*I*Pi*a*b*e*x*csgn(I*c)*csgn(I*
(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+2*Pi^2*b^2*e*x*csgn(I*c)*csgn(I*c*(e*x+d)^n)^5+4*I*Pi*a*b*e*x*csgn(I*(e*x+d)^n)
*csgn(I*c*(e*x+d)^n)^2+4*I*Pi*a*b*e*x*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2-4*I*Pi*b^2*e*n*x*csgn(I*(e*x+d)^n)*csgn(
I*c*(e*x+d)^n)^2+2*Pi^2*b^2*e*x*csgn(I*c)^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^3+2*Pi^2*b^2*e*x*csgn(I*(e*x
+d)^n)*csgn(I*c*(e*x+d)^n)^5-Pi^2*b^2*e*x*csgn(I*(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)^4-4*I*ln(c)*Pi*b^2*e*x*csgn(
I*c*(e*x+d)^n)^3+4*ln(c)^2*b^2*e*x-Pi^2*b^2*e*x*csgn(I*c)^2*csgn(I*(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)^2-4*Pi^2*b
^2*e*x*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^4-4*I*Pi*ln(e*x+d)*b^2*d*n*csgn(I*c)*csgn(I*(e*x+d)^n)*
csgn(I*c*(e*x+d)^n)+4*I*Pi*b^2*e*n*x*csgn(I*c*(e*x+d)^n)^3+4*I*Pi*b^2*e*n*x*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I
*c*(e*x+d)^n)+2*Pi^2*b^2*e*x*csgn(I*c)*csgn(I*(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)^3-4*I*Pi*a*b*e*x*csgn(I*c*(e*x+
d)^n)^3-Pi^2*b^2*e*x*csgn(I*c*(e*x+d)^n)^6-8*ln(e*x+d)*b^2*d*n^2+8*b^2*e*n^2*x+8*ln(c)*a*b*e*x-8*ln(c)*b^2*e*n
*x+8*ln(c)*ln(e*x+d)*b^2*d*n-8*a*b*e*n*x-4*b^2*d*n^2*ln(e*x+d)^2+8*ln(e*x+d)*a*b*d*n)/e*ln((e*x+d)^n)-6*ln(c)*
a*b^2*n*x-3/4*ln(c)*Pi^2*b^3*x*csgn(I*c*(e*x+d)^n)^6+3/4*Pi^2*b^3*n*x*csgn(I*c*(e*x+d)^n)^6-3/4*Pi^2*a*b^2*x*c
sgn(I*c*(e*x+d)^n)^6+1/8*I*Pi^3*b^3*x*csgn(I*c*(e*x+d)^n)^9-3*I/e*ln(c)*Pi*ln(e*x+d)*b^3*d*n*csgn(I*c)*csgn(I*
(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-3*I/e*Pi*ln(e*x+d)*a*b^2*d*n*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+3/
2*b^2*(-I*Pi*b*e*x*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+I*Pi*b*e*x*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+
I*Pi*b*e*x*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*Pi*b*e*x*csgn(I*c*(e*x+d)^n)^3+2*ln(c)*b*e*x+2*ln(e*x+d)*
b*d*n-2*b*e*n*x+2*a*e*x)/e*ln((e*x+d)^n)^2+3*ln(c)^2*a*b^2*x+3*ln(c)*a^2*b*x+3/2/e*Pi^2*ln(e*x+d)*b^3*d*n*csgn
(I*c)*csgn(I*c*(e*x+d)^n)^5-3/4/e*Pi^2*ln(e*x+d)*b^3*d*n*csgn(I*c)^2*csgn(I*c*(e*x+d)^n)^4-3/4*ln(c)*Pi^2*b^3*
x*csgn(I*c)^2*csgn(I*c*(e*x+d)^n)^4+3/2*ln(c)*Pi^2*b^3*x*csgn(I*c)*csgn(I*c*(e*x+d)^n)^5-1/8*I*Pi^3*b^3*x*csgn
(I*c)^3*csgn(I*c*(e*x+d)^n)^6-6/e*ln(c)*b^3*d*n^2*ln(e*x+d)-3/e*a*b^2*d*n^2*ln(e*x+d)^2+3/e*ln(c)^2*ln(e*x+d)*
b^3*d*n+3/e*ln(e*x+d)*a^2*b*d*n-3/4*ln(c)*Pi^2*b^3*x*csgn(I*(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)^4+3/2*ln(c)*Pi^2*
b^3*x*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^5-3*a^2*b*n*x+1/e*b^3*d*n^3*ln(e*x+d)^3+3/e*b^3*d*n^3*ln(e*x+d)^2+
6/e*ln(e*x+d)*b^3*d*n^3-3/4/e*Pi^2*ln(e*x+d)*b^3*d*n*csgn(I*c)^2*csgn(I*(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)^2+3/2
/e*Pi^2*ln(e*x+d)*b^3*d*n*csgn(I*c)^2*csgn(I*(e...

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (102) = 204\).
time = 0.30, size = 294, normalized size = 2.97 \begin {gather*} b^{3} x \log \left ({\left (x e + d\right )}^{n} c\right )^{3} + 3 \, {\left (d e^{\left (-2\right )} \log \left (x e + d\right ) - x e^{\left (-1\right )}\right )} a^{2} b n e + 3 \, a b^{2} x \log \left ({\left (x e + d\right )}^{n} c\right )^{2} + 3 \, a^{2} b x \log \left ({\left (x e + d\right )}^{n} c\right ) - 3 \, {\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 b^{2} + {\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 )} b^{3} + a^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (102) = 204\).
time = 0.37, size = 326, normalized size = 3.29 \begin {gather*} {\left (b^{3} x e \log \left (c\right )^{3} - 3 \, {\left (b^{3} n - a b^{2}\right )} x e \log \left (c\right )^{2} + {\left (b^{3} n^{3} x e + b^{3} d n^{3}\right )} \log \left (x e + d\right )^{3} + 3 \, {\left (2 \, b^{3} n^{2} - 2 \, a b^{2} n + a^{2} b\right )} x e \log \left (c\right ) - {\left (6 \, b^{3} n^{3} - 6 \, a b^{2} n^{2} + 3 \, a^{2} b n - a^{3}\right )} x e - 3 \, {\left (b^{3} d n^{3} - a b^{2} d n^{2} + {\left (b^{3} n^{3} - a b^{2} n^{2}\right )} x e - {\left (b^{3} n^{2} x e + b^{3} d n^{2}\right )} \log \left (c\right )\right )} \log \left (x e + d\right )^{2} + 3 \, {\left (2 \, b^{3} d n^{3} - 2 \, a b^{2} d n^{2} + a^{2} b d n + {\left (2 \, b^{3} n^{3} - 2 \, a b^{2} n^{2} + a^{2} b n\right )} x e + {\left (b^{3} n x e + b^{3} d n\right )} \log \left (c\right )^{2} - 2 \, {\left (b^{3} d n^{2} - a b^{2} d n + {\left (b^{3} n^{2} - a 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))^3,x, algorithm="fricas")

[Out]

(b^3*x*e*log(c)^3 - 3*(b^3*n - a*b^2)*x*e*log(c)^2 + (b^3*n^3*x*e + b^3*d*n^3)*log(x*e + d)^3 + 3*(2*b^3*n^2 -
2*a*b^2*n + a^2*b)*x*e*log(c) - (6*b^3*n^3 - 6*a*b^2*n^2 + 3*a^2*b*n - a^3)*x*e - 3*(b^3*d*n^3 - a*b^2*d*n^2
+ (b^3*n^3 - a*b^2*n^2)*x*e - (b^3*n^2*x*e + b^3*d*n^2)*log(c))*log(x*e + d)^2 + 3*(2*b^3*d*n^3 - 2*a*b^2*d*n^
2 + a^2*b*d*n + (2*b^3*n^3 - 2*a*b^2*n^2 + a^2*b*n)*x*e + (b^3*n*x*e + b^3*d*n)*log(c)^2 - 2*(b^3*d*n^2 - a*b^
2*d*n + (b^3*n^2 - a*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. 294 vs. \(2 (95) = 190\).
time = 0.58, size = 294, normalized size = 2.97 \begin {gather*} \begin {cases} a^{3} x + \frac {3 a^{2} b d \log {\left (c \left (d + e x\right )^{n} \right )}}{e} - 3 a^{2} b n x + 3 a^{2} b x \log {\left (c \left (d + e x\right )^{n} \right )} - \frac {6 a b^{2} d n \log {\left (c \left (d + e x\right )^{n} \right )}}{e} + \frac {3 a b^{2} d \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{e} + 6 a b^{2} n^{2} x - 6 a b^{2} n x \log {\left (c \left (d + e x\right )^{n} \right )} + 3 a b^{2} x \log {\left (c \left (d + e x\right )^{n} \right )}^{2} + \frac {6 b^{3} d n^{2} \log {\left (c \left (d + e x\right )^{n} \right )}}{e} - \frac {3 b^{3} d n \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{e} + \frac {b^{3} d \log {\left (c \left (d + e x\right )^{n} \right )}^{3}}{e} - 6 b^{3} n^{3} x + 6 b^{3} n^{2} x \log {\left (c \left (d + e x\right )^{n} \right )} - 3 b^{3} n x \log {\left (c \left (d + e x\right )^{n} \right )}^{2} + b^{3} x \log {\left (c \left (d + e x\right )^{n} \right )}^{3} & \text {for}\: e \neq 0 \\x \left (a + b \log {\left (c d^{n} \right )}\right )^{3} & \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))**3,x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 409 vs. \(2 (102) = 204\).
time = 4.08, size = 409, normalized size = 4.13 \begin {gather*} {\left (x e + d\right )} b^{3} n^{3} e^{\left (-1\right )} \log \left (x e + d\right )^{3} - 3 \, {\left (x e + d\right )} b^{3} n^{3} e^{\left (-1\right )} \log \left (x e + d\right )^{2} + 3 \, {\left (x e + d\right )} b^{3} n^{2} e^{\left (-1\right )} \log \left (x e + d\right )^{2} \log \left (c\right ) + 6 \, {\left (x e + d\right )} b^{3} n^{3} e^{\left (-1\right )} \log \left (x e + d\right ) + 3 \, {\left (x e + d\right )} a b^{2} n^{2} e^{\left (-1\right )} \log \left (x e + d\right )^{2} - 6 \, {\left (x e + d\right )} b^{3} n^{2} e^{\left (-1\right )} \log \left (x e + d\right ) \log \left (c\right ) + 3 \, {\left (x e + d\right )} b^{3} n e^{\left (-1\right )} \log \left (x e + d\right ) \log \left (c\right )^{2} - 6 \, {\left (x e + d\right )} b^{3} n^{3} e^{\left (-1\right )} - 6 \, {\left (x e + d\right )} a b^{2} n^{2} e^{\left (-1\right )} \log \left (x e + d\right ) + 6 \, {\left (x e + d\right )} b^{3} n^{2} e^{\left (-1\right )} \log \left (c\right ) + 6 \, {\left (x e + d\right )} a b^{2} n e^{\left (-1\right )} \log \left (x e + d\right ) \log \left (c\right ) - 3 \, {\left (x e + d\right )} b^{3} n e^{\left (-1\right )} \log \left (c\right )^{2} + {\left (x e + d\right )} b^{3} e^{\left (-1\right )} \log \left (c\right )^{3} + 6 \, {\left (x e + d\right )} a b^{2} n^{2} e^{\left (-1\right )} + 3 \, {\left (x e + d\right )} a^{2} b n e^{\left (-1\right )} \log \left (x e + d\right ) - 6 \, {\left (x e + d\right )} a b^{2} n e^{\left (-1\right )} \log \left (c\right ) + 3 \, {\left (x e + d\right )} a b^{2} e^{\left (-1\right )} \log \left (c\right )^{2} - 3 \, {\left (x e + d\right )} a^{2} b n e^{\left (-1\right )} + 3 \, {\left (x e + d\right )} a^{2} b e^{\left (-1\right )} \log \left (c\right ) + {\left (x e + d\right )} a^{3} 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))^3,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.00, size = 172, normalized size = 1.74 \begin {gather*} x\,\left (a^3-3\,a^2\,b\,n+6\,a\,b^2\,n^2-6\,b^3\,n^3\right )+{\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^3\,\left (b^3\,x+\frac {b^3\,d}{e}\right )+{\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^2\,\left (\frac {3\,\left (a\,b^2\,d-b^3\,d\,n\right )}{e}+3\,b^2\,x\,\left (a-b\,n\right )\right )+\frac {\ln \left (d+e\,x\right )\,\left (3\,d\,a^2\,b\,n-6\,d\,a\,b^2\,n^2+6\,d\,b^3\,n^3\right )}{e}+3\,b\,x\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\,\left (a^2-2\,a\,b\,n+2\,b^2\,n^2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*(d + e*x)^n))^3,x)

[Out]

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

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