3.17 \(\int x^3 (a+b \log (c x^n))^3 \log (1+e x) \, dx\)

Optimal. Leaf size=710 \[ \frac{3 b^2 n^2 \text{PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )}{8 e^4}+\frac{3 b^2 n^2 \text{PolyLog}(3,-e x) \left (a+b \log \left (c x^n\right )\right )}{2 e^4}-\frac{3 b n \text{PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )^2}{4 e^4}-\frac{3 b^3 n^3 \text{PolyLog}(2,-e x)}{32 e^4}-\frac{3 b^3 n^3 \text{PolyLog}(3,-e x)}{8 e^4}-\frac{3 b^3 n^3 \text{PolyLog}(4,-e x)}{2 e^4}-\frac{21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{64 e^2}+\frac{3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{32 e^3}-\frac{3 b^2 n^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{32 e^4}+\frac{3}{32} b^2 n^2 x^4 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )+\frac{37 b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )}{288 e}-\frac{9}{128} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )+\frac{15 a b^2 n^2 x}{8 e^3}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{8 e^2}+\frac{9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{32 e^2}+\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{4 e^3}-\frac{15 b n x \left (a+b \log \left (c x^n\right )\right )^2}{16 e^3}-\frac{\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3}{4 e^4}+\frac{3 b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{16 e^4}+\frac{1}{4} x^4 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3-\frac{3}{16} b n x^4 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )^3}{12 e}-\frac{7 b n x^3 \left (a+b \log \left (c x^n\right )\right )^2}{48 e}-\frac{1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )^3+\frac{3}{32} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac{15 b^3 n^2 x \log \left (c x^n\right )}{8 e^3}+\frac{45 b^3 n^3 x^2}{256 e^2}-\frac{255 b^3 n^3 x}{128 e^3}+\frac{3 b^3 n^3 \log (e x+1)}{128 e^4}-\frac{175 b^3 n^3 x^3}{3456 e}-\frac{3}{128} b^3 n^3 x^4 \log (e x+1)+\frac{3}{128} b^3 n^3 x^4 \]

[Out]

(15*a*b^2*n^2*x)/(8*e^3) - (255*b^3*n^3*x)/(128*e^3) + (45*b^3*n^3*x^2)/(256*e^2) - (175*b^3*n^3*x^3)/(3456*e)
+ (3*b^3*n^3*x^4)/128 + (15*b^3*n^2*x*Log[c*x^n])/(8*e^3) + (3*b^2*n^2*x*(a + b*Log[c*x^n]))/(32*e^3) - (21*b
^2*n^2*x^2*(a + b*Log[c*x^n]))/(64*e^2) + (37*b^2*n^2*x^3*(a + b*Log[c*x^n]))/(288*e) - (9*b^2*n^2*x^4*(a + b*
Log[c*x^n]))/128 - (15*b*n*x*(a + b*Log[c*x^n])^2)/(16*e^3) + (9*b*n*x^2*(a + b*Log[c*x^n])^2)/(32*e^2) - (7*b
*n*x^3*(a + b*Log[c*x^n])^2)/(48*e) + (3*b*n*x^4*(a + b*Log[c*x^n])^2)/32 + (x*(a + b*Log[c*x^n])^3)/(4*e^3) -
(x^2*(a + b*Log[c*x^n])^3)/(8*e^2) + (x^3*(a + b*Log[c*x^n])^3)/(12*e) - (x^4*(a + b*Log[c*x^n])^3)/16 + (3*b
^3*n^3*Log[1 + e*x])/(128*e^4) - (3*b^3*n^3*x^4*Log[1 + e*x])/128 - (3*b^2*n^2*(a + b*Log[c*x^n])*Log[1 + e*x]
)/(32*e^4) + (3*b^2*n^2*x^4*(a + b*Log[c*x^n])*Log[1 + e*x])/32 + (3*b*n*(a + b*Log[c*x^n])^2*Log[1 + e*x])/(1
6*e^4) - (3*b*n*x^4*(a + b*Log[c*x^n])^2*Log[1 + e*x])/16 - ((a + b*Log[c*x^n])^3*Log[1 + e*x])/(4*e^4) + (x^4
*(a + b*Log[c*x^n])^3*Log[1 + e*x])/4 - (3*b^3*n^3*PolyLog[2, -(e*x)])/(32*e^4) + (3*b^2*n^2*(a + b*Log[c*x^n]
)*PolyLog[2, -(e*x)])/(8*e^4) - (3*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, -(e*x)])/(4*e^4) - (3*b^3*n^3*PolyLog[3
, -(e*x)])/(8*e^4) + (3*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, -(e*x)])/(2*e^4) - (3*b^3*n^3*PolyLog[4, -(e*x)]
)/(2*e^4)

________________________________________________________________________________________

Rubi [A]  time = 0.777342, antiderivative size = 710, normalized size of antiderivative = 1., number of steps used = 29, number of rules used = 12, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546, Rules used = {2395, 43, 2377, 2296, 2295, 2305, 2304, 2374, 2383, 6589, 2376, 2391} \[ \frac{3 b^2 n^2 \text{PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )}{8 e^4}+\frac{3 b^2 n^2 \text{PolyLog}(3,-e x) \left (a+b \log \left (c x^n\right )\right )}{2 e^4}-\frac{3 b n \text{PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )^2}{4 e^4}-\frac{3 b^3 n^3 \text{PolyLog}(2,-e x)}{32 e^4}-\frac{3 b^3 n^3 \text{PolyLog}(3,-e x)}{8 e^4}-\frac{3 b^3 n^3 \text{PolyLog}(4,-e x)}{2 e^4}-\frac{21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{64 e^2}+\frac{3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{32 e^3}-\frac{3 b^2 n^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{32 e^4}+\frac{3}{32} b^2 n^2 x^4 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )+\frac{37 b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )}{288 e}-\frac{9}{128} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )+\frac{15 a b^2 n^2 x}{8 e^3}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{8 e^2}+\frac{9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{32 e^2}+\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{4 e^3}-\frac{15 b n x \left (a+b \log \left (c x^n\right )\right )^2}{16 e^3}-\frac{\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3}{4 e^4}+\frac{3 b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{16 e^4}+\frac{1}{4} x^4 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3-\frac{3}{16} b n x^4 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )^3}{12 e}-\frac{7 b n x^3 \left (a+b \log \left (c x^n\right )\right )^2}{48 e}-\frac{1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )^3+\frac{3}{32} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac{15 b^3 n^2 x \log \left (c x^n\right )}{8 e^3}+\frac{45 b^3 n^3 x^2}{256 e^2}-\frac{255 b^3 n^3 x}{128 e^3}+\frac{3 b^3 n^3 \log (e x+1)}{128 e^4}-\frac{175 b^3 n^3 x^3}{3456 e}-\frac{3}{128} b^3 n^3 x^4 \log (e x+1)+\frac{3}{128} b^3 n^3 x^4 \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*Log[c*x^n])^3*Log[1 + e*x],x]

[Out]

(15*a*b^2*n^2*x)/(8*e^3) - (255*b^3*n^3*x)/(128*e^3) + (45*b^3*n^3*x^2)/(256*e^2) - (175*b^3*n^3*x^3)/(3456*e)
+ (3*b^3*n^3*x^4)/128 + (15*b^3*n^2*x*Log[c*x^n])/(8*e^3) + (3*b^2*n^2*x*(a + b*Log[c*x^n]))/(32*e^3) - (21*b
^2*n^2*x^2*(a + b*Log[c*x^n]))/(64*e^2) + (37*b^2*n^2*x^3*(a + b*Log[c*x^n]))/(288*e) - (9*b^2*n^2*x^4*(a + b*
Log[c*x^n]))/128 - (15*b*n*x*(a + b*Log[c*x^n])^2)/(16*e^3) + (9*b*n*x^2*(a + b*Log[c*x^n])^2)/(32*e^2) - (7*b
*n*x^3*(a + b*Log[c*x^n])^2)/(48*e) + (3*b*n*x^4*(a + b*Log[c*x^n])^2)/32 + (x*(a + b*Log[c*x^n])^3)/(4*e^3) -
(x^2*(a + b*Log[c*x^n])^3)/(8*e^2) + (x^3*(a + b*Log[c*x^n])^3)/(12*e) - (x^4*(a + b*Log[c*x^n])^3)/16 + (3*b
^3*n^3*Log[1 + e*x])/(128*e^4) - (3*b^3*n^3*x^4*Log[1 + e*x])/128 - (3*b^2*n^2*(a + b*Log[c*x^n])*Log[1 + e*x]
)/(32*e^4) + (3*b^2*n^2*x^4*(a + b*Log[c*x^n])*Log[1 + e*x])/32 + (3*b*n*(a + b*Log[c*x^n])^2*Log[1 + e*x])/(1
6*e^4) - (3*b*n*x^4*(a + b*Log[c*x^n])^2*Log[1 + e*x])/16 - ((a + b*Log[c*x^n])^3*Log[1 + e*x])/(4*e^4) + (x^4
*(a + b*Log[c*x^n])^3*Log[1 + e*x])/4 - (3*b^3*n^3*PolyLog[2, -(e*x)])/(32*e^4) + (3*b^2*n^2*(a + b*Log[c*x^n]
)*PolyLog[2, -(e*x)])/(8*e^4) - (3*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, -(e*x)])/(4*e^4) - (3*b^3*n^3*PolyLog[3
, -(e*x)])/(8*e^4) + (3*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, -(e*x)])/(2*e^4) - (3*b^3*n^3*PolyLog[4, -(e*x)]
)/(2*e^4)

Rule 2395

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((f + g
*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] - Dist[(b*e*n)/(g*(q + 1)), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2377

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((g_.)*(x_))^(q_.), x_Sym
bol] :> With[{u = IntHide[(g*x)^q*Log[d*(e + f*x^m)], x]}, Dist[(a + b*Log[c*x^n])^p, u, x] - Dist[b*n*p, Int[
Dist[(a + b*Log[c*x^n])^(p - 1)/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, q}, x] && IGtQ[p, 0] &&
RationalQ[m] && RationalQ[q] && NeQ[q, -1] && (EqQ[p, 1] || (FractionQ[m] && IntegerQ[(q + 1)/m]) || (IGtQ[q,
0] && IntegerQ[(q + 1)/m] && EqQ[d*e, 1]))

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]

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^p)/m, x] + Dist[(b*n*p)/m, Int[(PolyLog[2, -(d*f*x^m)]*(a + b*Log
[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2383

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[(PolyL
og[k + 1, e*x^q]*(a + b*Log[c*x^n])^p)/q, x] - Dist[(b*n*p)/q, Int[(PolyLog[k + 1, e*x^q]*(a + b*Log[c*x^n])^(
p - 1))/x, x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 2376

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((g_.)*(x_))^(q_.), x_Sym
bol] :> With[{u = IntHide[(g*x)^q*Log[d*(e + f*x^m)^r], x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[Dist
[1/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && (IntegerQ[(q + 1)/m] || (RationalQ[m] &
& RationalQ[q])) && NeQ[q, -1]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x) \, dx &=\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{4 e^3}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{8 e^2}+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )^3}{12 e}-\frac{1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{4 e^4}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)-(3 b n) \int \left (\frac{\left (a+b \log \left (c x^n\right )\right )^2}{4 e^3}-\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}-\frac{1}{16} x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{4 e^4 x}+\frac{1}{4} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)\right ) \, dx\\ &=\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{4 e^3}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{8 e^2}+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )^3}{12 e}-\frac{1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{4 e^4}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)+\frac{1}{16} (3 b n) \int x^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx-\frac{1}{4} (3 b n) \int x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x) \, dx+\frac{(3 b n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x} \, dx}{4 e^4}-\frac{(3 b n) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{4 e^3}+\frac{(3 b n) \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{8 e^2}-\frac{(b n) \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{4 e}\\ &=-\frac{15 b n x \left (a+b \log \left (c x^n\right )\right )^2}{16 e^3}+\frac{9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{32 e^2}-\frac{7 b n x^3 \left (a+b \log \left (c x^n\right )\right )^2}{48 e}+\frac{3}{32} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{4 e^3}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{8 e^2}+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )^3}{12 e}-\frac{1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )^3+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{16 e^4}-\frac{3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{4 e^4}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2(-e x)}{4 e^4}-\frac{1}{32} \left (3 b^2 n^2\right ) \int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac{1}{2} \left (3 b^2 n^2\right ) \int \left (\frac{a+b \log \left (c x^n\right )}{4 e^3}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{8 e^2}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{12 e}-\frac{1}{16} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{4 e^4 x}+\frac{1}{4} x^3 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)\right ) \, dx+\frac{\left (3 b^2 n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)}{x} \, dx}{2 e^4}+\frac{\left (3 b^2 n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{2 e^3}-\frac{\left (3 b^2 n^2\right ) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{8 e^2}+\frac{\left (b^2 n^2\right ) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx}{6 e}\\ &=\frac{3 a b^2 n^2 x}{2 e^3}+\frac{3 b^3 n^3 x^2}{32 e^2}-\frac{b^3 n^3 x^3}{54 e}+\frac{3}{512} b^3 n^3 x^4-\frac{3 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{16 e^2}+\frac{b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )}{18 e}-\frac{3}{128} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac{15 b n x \left (a+b \log \left (c x^n\right )\right )^2}{16 e^3}+\frac{9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{32 e^2}-\frac{7 b n x^3 \left (a+b \log \left (c x^n\right )\right )^2}{48 e}+\frac{3}{32} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{4 e^3}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{8 e^2}+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )^3}{12 e}-\frac{1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )^3+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{16 e^4}-\frac{3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{4 e^4}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2(-e x)}{4 e^4}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(-e x)}{2 e^4}-\frac{1}{32} \left (3 b^2 n^2\right ) \int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac{1}{8} \left (3 b^2 n^2\right ) \int x^3 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x) \, dx-\frac{\left (3 b^2 n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x} \, dx}{8 e^4}+\frac{\left (3 b^2 n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{8 e^3}+\frac{\left (3 b^3 n^2\right ) \int \log \left (c x^n\right ) \, dx}{2 e^3}-\frac{\left (3 b^2 n^2\right ) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{16 e^2}+\frac{\left (b^2 n^2\right ) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx}{8 e}-\frac{\left (3 b^3 n^3\right ) \int \frac{\text{Li}_3(-e x)}{x} \, dx}{2 e^4}\\ &=\frac{15 a b^2 n^2 x}{8 e^3}-\frac{3 b^3 n^3 x}{2 e^3}+\frac{9 b^3 n^3 x^2}{64 e^2}-\frac{7 b^3 n^3 x^3}{216 e}+\frac{3}{256} b^3 n^3 x^4+\frac{3 b^3 n^2 x \log \left (c x^n\right )}{2 e^3}+\frac{3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{32 e^3}-\frac{21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{64 e^2}+\frac{37 b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )}{288 e}-\frac{9}{128} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac{15 b n x \left (a+b \log \left (c x^n\right )\right )^2}{16 e^3}+\frac{9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{32 e^2}-\frac{7 b n x^3 \left (a+b \log \left (c x^n\right )\right )^2}{48 e}+\frac{3}{32} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{4 e^3}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{8 e^2}+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )^3}{12 e}-\frac{1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{32 e^4}+\frac{3}{32} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{16 e^4}-\frac{3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{4 e^4}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)}{8 e^4}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2(-e x)}{4 e^4}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(-e x)}{2 e^4}-\frac{3 b^3 n^3 \text{Li}_4(-e x)}{2 e^4}+\frac{\left (3 b^3 n^2\right ) \int \log \left (c x^n\right ) \, dx}{8 e^3}-\frac{1}{8} \left (3 b^3 n^3\right ) \int \left (\frac{1}{4 e^3}-\frac{x}{8 e^2}+\frac{x^2}{12 e}-\frac{x^3}{16}-\frac{\log (1+e x)}{4 e^4 x}+\frac{1}{4} x^3 \log (1+e x)\right ) \, dx-\frac{\left (3 b^3 n^3\right ) \int \frac{\text{Li}_2(-e x)}{x} \, dx}{8 e^4}\\ &=\frac{15 a b^2 n^2 x}{8 e^3}-\frac{63 b^3 n^3 x}{32 e^3}+\frac{21 b^3 n^3 x^2}{128 e^2}-\frac{37 b^3 n^3 x^3}{864 e}+\frac{9}{512} b^3 n^3 x^4+\frac{15 b^3 n^2 x \log \left (c x^n\right )}{8 e^3}+\frac{3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{32 e^3}-\frac{21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{64 e^2}+\frac{37 b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )}{288 e}-\frac{9}{128} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac{15 b n x \left (a+b \log \left (c x^n\right )\right )^2}{16 e^3}+\frac{9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{32 e^2}-\frac{7 b n x^3 \left (a+b \log \left (c x^n\right )\right )^2}{48 e}+\frac{3}{32} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{4 e^3}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{8 e^2}+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )^3}{12 e}-\frac{1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{32 e^4}+\frac{3}{32} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{16 e^4}-\frac{3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{4 e^4}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)}{8 e^4}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2(-e x)}{4 e^4}-\frac{3 b^3 n^3 \text{Li}_3(-e x)}{8 e^4}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(-e x)}{2 e^4}-\frac{3 b^3 n^3 \text{Li}_4(-e x)}{2 e^4}-\frac{1}{32} \left (3 b^3 n^3\right ) \int x^3 \log (1+e x) \, dx+\frac{\left (3 b^3 n^3\right ) \int \frac{\log (1+e x)}{x} \, dx}{32 e^4}\\ &=\frac{15 a b^2 n^2 x}{8 e^3}-\frac{63 b^3 n^3 x}{32 e^3}+\frac{21 b^3 n^3 x^2}{128 e^2}-\frac{37 b^3 n^3 x^3}{864 e}+\frac{9}{512} b^3 n^3 x^4+\frac{15 b^3 n^2 x \log \left (c x^n\right )}{8 e^3}+\frac{3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{32 e^3}-\frac{21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{64 e^2}+\frac{37 b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )}{288 e}-\frac{9}{128} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac{15 b n x \left (a+b \log \left (c x^n\right )\right )^2}{16 e^3}+\frac{9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{32 e^2}-\frac{7 b n x^3 \left (a+b \log \left (c x^n\right )\right )^2}{48 e}+\frac{3}{32} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{4 e^3}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{8 e^2}+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )^3}{12 e}-\frac{1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac{3}{128} b^3 n^3 x^4 \log (1+e x)-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{32 e^4}+\frac{3}{32} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{16 e^4}-\frac{3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{4 e^4}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)-\frac{3 b^3 n^3 \text{Li}_2(-e x)}{32 e^4}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)}{8 e^4}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2(-e x)}{4 e^4}-\frac{3 b^3 n^3 \text{Li}_3(-e x)}{8 e^4}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(-e x)}{2 e^4}-\frac{3 b^3 n^3 \text{Li}_4(-e x)}{2 e^4}+\frac{1}{128} \left (3 b^3 e n^3\right ) \int \frac{x^4}{1+e x} \, dx\\ &=\frac{15 a b^2 n^2 x}{8 e^3}-\frac{63 b^3 n^3 x}{32 e^3}+\frac{21 b^3 n^3 x^2}{128 e^2}-\frac{37 b^3 n^3 x^3}{864 e}+\frac{9}{512} b^3 n^3 x^4+\frac{15 b^3 n^2 x \log \left (c x^n\right )}{8 e^3}+\frac{3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{32 e^3}-\frac{21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{64 e^2}+\frac{37 b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )}{288 e}-\frac{9}{128} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac{15 b n x \left (a+b \log \left (c x^n\right )\right )^2}{16 e^3}+\frac{9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{32 e^2}-\frac{7 b n x^3 \left (a+b \log \left (c x^n\right )\right )^2}{48 e}+\frac{3}{32} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{4 e^3}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{8 e^2}+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )^3}{12 e}-\frac{1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac{3}{128} b^3 n^3 x^4 \log (1+e x)-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{32 e^4}+\frac{3}{32} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{16 e^4}-\frac{3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{4 e^4}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)-\frac{3 b^3 n^3 \text{Li}_2(-e x)}{32 e^4}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)}{8 e^4}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2(-e x)}{4 e^4}-\frac{3 b^3 n^3 \text{Li}_3(-e x)}{8 e^4}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(-e x)}{2 e^4}-\frac{3 b^3 n^3 \text{Li}_4(-e x)}{2 e^4}+\frac{1}{128} \left (3 b^3 e n^3\right ) \int \left (-\frac{1}{e^4}+\frac{x}{e^3}-\frac{x^2}{e^2}+\frac{x^3}{e}+\frac{1}{e^4 (1+e x)}\right ) \, dx\\ &=\frac{15 a b^2 n^2 x}{8 e^3}-\frac{255 b^3 n^3 x}{128 e^3}+\frac{45 b^3 n^3 x^2}{256 e^2}-\frac{175 b^3 n^3 x^3}{3456 e}+\frac{3}{128} b^3 n^3 x^4+\frac{15 b^3 n^2 x \log \left (c x^n\right )}{8 e^3}+\frac{3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{32 e^3}-\frac{21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{64 e^2}+\frac{37 b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )}{288 e}-\frac{9}{128} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac{15 b n x \left (a+b \log \left (c x^n\right )\right )^2}{16 e^3}+\frac{9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{32 e^2}-\frac{7 b n x^3 \left (a+b \log \left (c x^n\right )\right )^2}{48 e}+\frac{3}{32} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{4 e^3}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{8 e^2}+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )^3}{12 e}-\frac{1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )^3+\frac{3 b^3 n^3 \log (1+e x)}{128 e^4}-\frac{3}{128} b^3 n^3 x^4 \log (1+e x)-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{32 e^4}+\frac{3}{32} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{16 e^4}-\frac{3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{4 e^4}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)-\frac{3 b^3 n^3 \text{Li}_2(-e x)}{32 e^4}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)}{8 e^4}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2(-e x)}{4 e^4}-\frac{3 b^3 n^3 \text{Li}_3(-e x)}{8 e^4}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(-e x)}{2 e^4}-\frac{3 b^3 n^3 \text{Li}_4(-e x)}{2 e^4}\\ \end{align*}

Mathematica [A]  time = 0.338973, size = 1144, normalized size = 1.61 \[ \frac{-432 a^3 e^4 x^4+162 b^3 e^4 n^3 x^4-432 b^3 e^4 \log ^3\left (c x^n\right ) x^4-486 a b^2 e^4 n^2 x^4-1296 a b^2 e^4 \log ^2\left (c x^n\right ) x^4+648 b^3 e^4 n \log ^2\left (c x^n\right ) x^4+648 a^2 b e^4 n x^4-1296 a^2 b e^4 \log \left (c x^n\right ) x^4-486 b^3 e^4 n^2 \log \left (c x^n\right ) x^4+1296 a b^2 e^4 n \log \left (c x^n\right ) x^4+1728 a^3 e^4 \log (e x+1) x^4-162 b^3 e^4 n^3 \log (e x+1) x^4+1728 b^3 e^4 \log ^3\left (c x^n\right ) \log (e x+1) x^4+648 a b^2 e^4 n^2 \log (e x+1) x^4+5184 a b^2 e^4 \log ^2\left (c x^n\right ) \log (e x+1) x^4-1296 b^3 e^4 n \log ^2\left (c x^n\right ) \log (e x+1) x^4-1296 a^2 b e^4 n \log (e x+1) x^4+5184 a^2 b e^4 \log \left (c x^n\right ) \log (e x+1) x^4+648 b^3 e^4 n^2 \log \left (c x^n\right ) \log (e x+1) x^4-2592 a b^2 e^4 n \log \left (c x^n\right ) \log (e x+1) x^4+576 a^3 e^3 x^3-350 b^3 e^3 n^3 x^3+576 b^3 e^3 \log ^3\left (c x^n\right ) x^3+888 a b^2 e^3 n^2 x^3+1728 a b^2 e^3 \log ^2\left (c x^n\right ) x^3-1008 b^3 e^3 n \log ^2\left (c x^n\right ) x^3-1008 a^2 b e^3 n x^3+1728 a^2 b e^3 \log \left (c x^n\right ) x^3+888 b^3 e^3 n^2 \log \left (c x^n\right ) x^3-2016 a b^2 e^3 n \log \left (c x^n\right ) x^3+1215 b^3 e^2 n^3 x^2-864 b^3 e^2 \log ^3\left (c x^n\right ) x^2-864 a^3 e^2 x^2-2268 a b^2 e^2 n^2 x^2-2592 a b^2 e^2 \log ^2\left (c x^n\right ) x^2+1944 b^3 e^2 n \log ^2\left (c x^n\right ) x^2+1944 a^2 b e^2 n x^2-2592 a^2 b e^2 \log \left (c x^n\right ) x^2-2268 b^3 e^2 n^2 \log \left (c x^n\right ) x^2+3888 a b^2 e^2 n \log \left (c x^n\right ) x^2-13770 b^3 e n^3 x+1728 b^3 e \log ^3\left (c x^n\right ) x+13608 a b^2 e n^2 x+5184 a b^2 e \log ^2\left (c x^n\right ) x-6480 b^3 e n \log ^2\left (c x^n\right ) x+1728 a^3 e x-6480 a^2 b e n x+13608 b^3 e n^2 \log \left (c x^n\right ) x+5184 a^2 b e \log \left (c x^n\right ) x-12960 a b^2 e n \log \left (c x^n\right ) x-1728 a^3 \log (e x+1)+162 b^3 n^3 \log (e x+1)-1728 b^3 \log ^3\left (c x^n\right ) \log (e x+1)-648 a b^2 n^2 \log (e x+1)-5184 a b^2 \log ^2\left (c x^n\right ) \log (e x+1)+1296 b^3 n \log ^2\left (c x^n\right ) \log (e x+1)+1296 a^2 b n \log (e x+1)-648 b^3 n^2 \log \left (c x^n\right ) \log (e x+1)-5184 a^2 b \log \left (c x^n\right ) \log (e x+1)+2592 a b^2 n \log \left (c x^n\right ) \log (e x+1)-648 b n \left (8 a^2-4 b n a+b^2 n^2+8 b^2 \log ^2\left (c x^n\right )-4 b (b n-4 a) \log \left (c x^n\right )\right ) \text{PolyLog}(2,-e x)+2592 b^2 n^2 \left (4 a-b n+4 b \log \left (c x^n\right )\right ) \text{PolyLog}(3,-e x)-10368 b^3 n^3 \text{PolyLog}(4,-e x)}{6912 e^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*Log[c*x^n])^3*Log[1 + e*x],x]

[Out]

(1728*a^3*e*x - 6480*a^2*b*e*n*x + 13608*a*b^2*e*n^2*x - 13770*b^3*e*n^3*x - 864*a^3*e^2*x^2 + 1944*a^2*b*e^2*
n*x^2 - 2268*a*b^2*e^2*n^2*x^2 + 1215*b^3*e^2*n^3*x^2 + 576*a^3*e^3*x^3 - 1008*a^2*b*e^3*n*x^3 + 888*a*b^2*e^3
*n^2*x^3 - 350*b^3*e^3*n^3*x^3 - 432*a^3*e^4*x^4 + 648*a^2*b*e^4*n*x^4 - 486*a*b^2*e^4*n^2*x^4 + 162*b^3*e^4*n
^3*x^4 + 5184*a^2*b*e*x*Log[c*x^n] - 12960*a*b^2*e*n*x*Log[c*x^n] + 13608*b^3*e*n^2*x*Log[c*x^n] - 2592*a^2*b*
e^2*x^2*Log[c*x^n] + 3888*a*b^2*e^2*n*x^2*Log[c*x^n] - 2268*b^3*e^2*n^2*x^2*Log[c*x^n] + 1728*a^2*b*e^3*x^3*Lo
g[c*x^n] - 2016*a*b^2*e^3*n*x^3*Log[c*x^n] + 888*b^3*e^3*n^2*x^3*Log[c*x^n] - 1296*a^2*b*e^4*x^4*Log[c*x^n] +
1296*a*b^2*e^4*n*x^4*Log[c*x^n] - 486*b^3*e^4*n^2*x^4*Log[c*x^n] + 5184*a*b^2*e*x*Log[c*x^n]^2 - 6480*b^3*e*n*
x*Log[c*x^n]^2 - 2592*a*b^2*e^2*x^2*Log[c*x^n]^2 + 1944*b^3*e^2*n*x^2*Log[c*x^n]^2 + 1728*a*b^2*e^3*x^3*Log[c*
x^n]^2 - 1008*b^3*e^3*n*x^3*Log[c*x^n]^2 - 1296*a*b^2*e^4*x^4*Log[c*x^n]^2 + 648*b^3*e^4*n*x^4*Log[c*x^n]^2 +
1728*b^3*e*x*Log[c*x^n]^3 - 864*b^3*e^2*x^2*Log[c*x^n]^3 + 576*b^3*e^3*x^3*Log[c*x^n]^3 - 432*b^3*e^4*x^4*Log[
c*x^n]^3 - 1728*a^3*Log[1 + e*x] + 1296*a^2*b*n*Log[1 + e*x] - 648*a*b^2*n^2*Log[1 + e*x] + 162*b^3*n^3*Log[1
+ e*x] + 1728*a^3*e^4*x^4*Log[1 + e*x] - 1296*a^2*b*e^4*n*x^4*Log[1 + e*x] + 648*a*b^2*e^4*n^2*x^4*Log[1 + e*x
] - 162*b^3*e^4*n^3*x^4*Log[1 + e*x] - 5184*a^2*b*Log[c*x^n]*Log[1 + e*x] + 2592*a*b^2*n*Log[c*x^n]*Log[1 + e*
x] - 648*b^3*n^2*Log[c*x^n]*Log[1 + e*x] + 5184*a^2*b*e^4*x^4*Log[c*x^n]*Log[1 + e*x] - 2592*a*b^2*e^4*n*x^4*L
og[c*x^n]*Log[1 + e*x] + 648*b^3*e^4*n^2*x^4*Log[c*x^n]*Log[1 + e*x] - 5184*a*b^2*Log[c*x^n]^2*Log[1 + e*x] +
1296*b^3*n*Log[c*x^n]^2*Log[1 + e*x] + 5184*a*b^2*e^4*x^4*Log[c*x^n]^2*Log[1 + e*x] - 1296*b^3*e^4*n*x^4*Log[c
*x^n]^2*Log[1 + e*x] - 1728*b^3*Log[c*x^n]^3*Log[1 + e*x] + 1728*b^3*e^4*x^4*Log[c*x^n]^3*Log[1 + e*x] - 648*b
*n*(8*a^2 - 4*a*b*n + b^2*n^2 - 4*b*(-4*a + b*n)*Log[c*x^n] + 8*b^2*Log[c*x^n]^2)*PolyLog[2, -(e*x)] + 2592*b^
2*n^2*(4*a - b*n + 4*b*Log[c*x^n])*PolyLog[3, -(e*x)] - 10368*b^3*n^3*PolyLog[4, -(e*x)])/(6912*e^4)

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Maple [F]  time = 0.154, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{3}\ln \left ( ex+1 \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*ln(c*x^n))^3*ln(e*x+1),x)

[Out]

int(x^3*(a+b*ln(c*x^n))^3*ln(e*x+1),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (3 \, b^{3} e^{4} x^{4} - 4 \, b^{3} e^{3} x^{3} + 6 \, b^{3} e^{2} x^{2} - 12 \, b^{3} e x - 12 \,{\left (b^{3} e^{4} x^{4} - b^{3}\right )} \log \left (e x + 1\right )\right )} \log \left (x^{n}\right )^{3}}{48 \, e^{4}} + \frac{\frac{1}{3} \,{\left (12 \, x^{4} \log \left (e x + 1\right ) - e{\left (\frac{3 \, e^{3} x^{4} - 4 \, e^{2} x^{3} + 6 \, e x^{2} - 12 \, x}{e^{4}} + \frac{12 \, \log \left (e x + 1\right )}{e^{5}}\right )}\right )} b^{3} e^{4} \log \left (c\right )^{3} +{\left (12 \, x^{4} \log \left (e x + 1\right ) - e{\left (\frac{3 \, e^{3} x^{4} - 4 \, e^{2} x^{3} + 6 \, e x^{2} - 12 \, x}{e^{4}} + \frac{12 \, \log \left (e x + 1\right )}{e^{5}}\right )}\right )} a b^{2} e^{4} \log \left (c\right )^{2} +{\left (12 \, x^{4} \log \left (e x + 1\right ) - e{\left (\frac{3 \, e^{3} x^{4} - 4 \, e^{2} x^{3} + 6 \, e x^{2} - 12 \, x}{e^{4}} + \frac{12 \, \log \left (e x + 1\right )}{e^{5}}\right )}\right )} a^{2} b e^{4} \log \left (c\right ) + \frac{1}{3} \,{\left (12 \, x^{4} \log \left (e x + 1\right ) - e{\left (\frac{3 \, e^{3} x^{4} - 4 \, e^{2} x^{3} + 6 \, e x^{2} - 12 \, x}{e^{4}} + \frac{12 \, \log \left (e x + 1\right )}{e^{5}}\right )}\right )} a^{3} e^{4} + \int \frac{48 \,{\left (b^{3} e^{4} \log \left (c\right )^{2} + 2 \, a b^{2} e^{4} \log \left (c\right ) + a^{2} b e^{4}\right )} x^{4} \log \left (e x + 1\right ) \log \left (x^{n}\right ) +{\left (3 \, b^{3} e^{4} n x^{4} - 4 \, b^{3} e^{3} n x^{3} + 6 \, b^{3} e^{2} n x^{2} - 12 \, b^{3} e n x + 12 \,{\left ({\left (4 \, a b^{2} e^{4} -{\left (e^{4} n - 4 \, e^{4} \log \left (c\right )\right )} b^{3}\right )} x^{4} + b^{3} n\right )} \log \left (e x + 1\right )\right )} \log \left (x^{n}\right )^{2}}{x}\,{d x}}{16 \, e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*log(c*x^n))^3*log(e*x+1),x, algorithm="maxima")

[Out]

-1/48*(3*b^3*e^4*x^4 - 4*b^3*e^3*x^3 + 6*b^3*e^2*x^2 - 12*b^3*e*x - 12*(b^3*e^4*x^4 - b^3)*log(e*x + 1))*log(x
^n)^3/e^4 + 1/16*integrate((48*(b^3*e^4*log(c)^2 + 2*a*b^2*e^4*log(c) + a^2*b*e^4)*x^4*log(e*x + 1)*log(x^n) +
16*(b^3*e^4*log(c)^3 + 3*a*b^2*e^4*log(c)^2 + 3*a^2*b*e^4*log(c) + a^3*e^4)*x^4*log(e*x + 1) + (3*b^3*e^4*n*x
^4 - 4*b^3*e^3*n*x^3 + 6*b^3*e^2*n*x^2 - 12*b^3*e*n*x + 12*((4*a*b^2*e^4 - (e^4*n - 4*e^4*log(c))*b^3)*x^4 + b
^3*n)*log(e*x + 1))*log(x^n)^2)/x, x)/e^4

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b^{3} x^{3} \log \left (c x^{n}\right )^{3} \log \left (e x + 1\right ) + 3 \, a b^{2} x^{3} \log \left (c x^{n}\right )^{2} \log \left (e x + 1\right ) + 3 \, a^{2} b x^{3} \log \left (c x^{n}\right ) \log \left (e x + 1\right ) + a^{3} x^{3} \log \left (e x + 1\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*log(c*x^n))^3*log(e*x+1),x, algorithm="fricas")

[Out]

integral(b^3*x^3*log(c*x^n)^3*log(e*x + 1) + 3*a*b^2*x^3*log(c*x^n)^2*log(e*x + 1) + 3*a^2*b*x^3*log(c*x^n)*lo
g(e*x + 1) + a^3*x^3*log(e*x + 1), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(a+b*ln(c*x**n))**3*ln(e*x+1),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x^{3} \log \left (e x + 1\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*log(c*x^n))^3*log(e*x+1),x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)^3*x^3*log(e*x + 1), x)