3.1.17 \(\int x^3 (a+b \log (c x^n))^3 \log (1+e x) \, dx\) [17]

Optimal. Leaf size=710 \[ \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} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.50, antiderivative size = 710, normalized size of antiderivative = 1.00, 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 = {2442, 45, 2424, 2333, 2332, 2342, 2341, 2421, 2430, 6724, 2423, 2438} \begin {gather*} \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 {3 b^2 n^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{32 e^4}+\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 {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 {\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 {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 {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 {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 {3 b^3 n^3 \log (e x+1)}{128 e^4}-\frac {255 b^3 n^3 x}{128 e^3}+\frac {45 b^3 n^3 x^2}{256 e^2}-\frac {3}{128} b^3 n^3 x^4 \log (e x+1)-\frac {175 b^3 n^3 x^3}{3456 e}+\frac {3}{128} b^3 n^3 x^4 \end {gather*}

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 45

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 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 2341

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 2342

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 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-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*L
og[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 2423

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 2424

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 2430

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[PolyLo
g[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 2438

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]

Rule 2442

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 6724

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]

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*}

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Mathematica [A]
time = 0.21, size = 1144, normalized size = 1.61 \begin {gather*} \frac {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 \left (c x^n\right )-12960 a b^2 e n x \log \left (c x^n\right )+13608 b^3 e n^2 x \log \left (c x^n\right )-2592 a^2 b e^2 x^2 \log \left (c x^n\right )+3888 a b^2 e^2 n x^2 \log \left (c x^n\right )-2268 b^3 e^2 n^2 x^2 \log \left (c x^n\right )+1728 a^2 b e^3 x^3 \log \left (c x^n\right )-2016 a b^2 e^3 n x^3 \log \left (c x^n\right )+888 b^3 e^3 n^2 x^3 \log \left (c x^n\right )-1296 a^2 b e^4 x^4 \log \left (c x^n\right )+1296 a b^2 e^4 n x^4 \log \left (c x^n\right )-486 b^3 e^4 n^2 x^4 \log \left (c x^n\right )+5184 a b^2 e x \log ^2\left (c x^n\right )-6480 b^3 e n x \log ^2\left (c x^n\right )-2592 a b^2 e^2 x^2 \log ^2\left (c x^n\right )+1944 b^3 e^2 n x^2 \log ^2\left (c x^n\right )+1728 a b^2 e^3 x^3 \log ^2\left (c x^n\right )-1008 b^3 e^3 n x^3 \log ^2\left (c x^n\right )-1296 a b^2 e^4 x^4 \log ^2\left (c x^n\right )+648 b^3 e^4 n x^4 \log ^2\left (c x^n\right )+1728 b^3 e x \log ^3\left (c x^n\right )-864 b^3 e^2 x^2 \log ^3\left (c x^n\right )+576 b^3 e^3 x^3 \log ^3\left (c x^n\right )-432 b^3 e^4 x^4 \log ^3\left (c x^n\right )-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 \left (c x^n\right ) \log (1+e x)+2592 a b^2 n \log \left (c x^n\right ) \log (1+e x)-648 b^3 n^2 \log \left (c x^n\right ) \log (1+e x)+5184 a^2 b e^4 x^4 \log \left (c x^n\right ) \log (1+e x)-2592 a b^2 e^4 n x^4 \log \left (c x^n\right ) \log (1+e x)+648 b^3 e^4 n^2 x^4 \log \left (c x^n\right ) \log (1+e x)-5184 a b^2 \log ^2\left (c x^n\right ) \log (1+e x)+1296 b^3 n \log ^2\left (c x^n\right ) \log (1+e x)+5184 a b^2 e^4 x^4 \log ^2\left (c x^n\right ) \log (1+e x)-1296 b^3 e^4 n x^4 \log ^2\left (c x^n\right ) \log (1+e x)-1728 b^3 \log ^3\left (c x^n\right ) \log (1+e x)+1728 b^3 e^4 x^4 \log ^3\left (c x^n\right ) \log (1+e x)-648 b n \left (8 a^2-4 a b n+b^2 n^2-4 b (-4 a+b n) \log \left (c x^n\right )+8 b^2 \log ^2\left (c x^n\right )\right ) \text {Li}_2(-e x)+2592 b^2 n^2 \left (4 a-b n+4 b \log \left (c x^n\right )\right ) \text {Li}_3(-e x)-10368 b^3 n^3 \text {Li}_4(-e x)}{6912 e^4} \end {gather*}

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 = 0.02, size = 0, normalized size = 0.00 \[\int x^{3} \left (a +b \ln \left (c \,x^{n}\right )\right )^{3} \ln \left (e x +1\right )\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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(x*e + 1), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^3\,\ln \left (e\,x+1\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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