3.23 \(\int \frac{(a+b \log (c x^n))^3 \log (1+e x)}{x^3} \, dx\)

Optimal. Leaf size=470 \[ -\frac{3}{2} b^2 e^2 n^2 \text{PolyLog}\left (2,-\frac{1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )-3 b^2 e^2 n^2 \text{PolyLog}\left (3,-\frac{1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3}{2} b e^2 n \text{PolyLog}\left (2,-\frac{1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{3}{4} b^3 e^2 n^3 \text{PolyLog}\left (2,-\frac{1}{e x}\right )-\frac{3}{2} b^3 e^2 n^3 \text{PolyLog}\left (3,-\frac{1}{e x}\right )-3 b^3 e^2 n^3 \text{PolyLog}\left (4,-\frac{1}{e x}\right )+\frac{3}{4} b^2 e^2 n^2 \log \left (\frac{1}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{21 b^2 e n^2 \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac{3 b^2 n^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}+\frac{3}{4} b e^2 n \log \left (\frac{1}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{2} e^2 \log \left (\frac{1}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac{9 b e n \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac{3 b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac{e \left (a+b \log \left (c x^n\right )\right )^3}{2 x}-\frac{\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac{3}{8} b^3 e^2 n^3 \log (x)+\frac{3}{8} b^3 e^2 n^3 \log (e x+1)-\frac{3 b^3 n^3 \log (e x+1)}{8 x^2}-\frac{45 b^3 e n^3}{8 x} \]

[Out]

(-45*b^3*e*n^3)/(8*x) - (3*b^3*e^2*n^3*Log[x])/8 - (21*b^2*e*n^2*(a + b*Log[c*x^n]))/(4*x) + (3*b^2*e^2*n^2*Lo
g[1 + 1/(e*x)]*(a + b*Log[c*x^n]))/4 - (9*b*e*n*(a + b*Log[c*x^n])^2)/(4*x) + (3*b*e^2*n*Log[1 + 1/(e*x)]*(a +
b*Log[c*x^n])^2)/4 - (e*(a + b*Log[c*x^n])^3)/(2*x) + (e^2*Log[1 + 1/(e*x)]*(a + b*Log[c*x^n])^3)/2 + (3*b^3*
e^2*n^3*Log[1 + e*x])/8 - (3*b^3*n^3*Log[1 + e*x])/(8*x^2) - (3*b^2*n^2*(a + b*Log[c*x^n])*Log[1 + e*x])/(4*x^
2) - (3*b*n*(a + b*Log[c*x^n])^2*Log[1 + e*x])/(4*x^2) - ((a + b*Log[c*x^n])^3*Log[1 + e*x])/(2*x^2) - (3*b^3*
e^2*n^3*PolyLog[2, -(1/(e*x))])/4 - (3*b^2*e^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, -(1/(e*x))])/2 - (3*b*e^2*n*(
a + b*Log[c*x^n])^2*PolyLog[2, -(1/(e*x))])/2 - (3*b^3*e^2*n^3*PolyLog[3, -(1/(e*x))])/2 - 3*b^2*e^2*n^2*(a +
b*Log[c*x^n])*PolyLog[3, -(1/(e*x))] - 3*b^3*e^2*n^3*PolyLog[4, -(1/(e*x))]

________________________________________________________________________________________

Rubi [A]  time = 0.82018, antiderivative size = 499, normalized size of antiderivative = 1.06, number of steps used = 30, number of rules used = 14, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.636, Rules used = {2305, 2304, 2378, 44, 2351, 2301, 2317, 2391, 2353, 2302, 30, 2374, 6589, 2383} \[ \frac{3}{2} b^2 e^2 n^2 \text{PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )-3 b^2 e^2 n^2 \text{PolyLog}(3,-e x) \left (a+b \log \left (c x^n\right )\right )+\frac{3}{2} b e^2 n \text{PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )^2+\frac{3}{4} b^3 e^2 n^3 \text{PolyLog}(2,-e x)-\frac{3}{2} b^3 e^2 n^3 \text{PolyLog}(3,-e x)+3 b^3 e^2 n^3 \text{PolyLog}(4,-e x)+\frac{3}{4} b^2 e^2 n^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )-\frac{3 b^2 n^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac{21 b^2 e n^2 \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac{e^2 \left (a+b \log \left (c x^n\right )\right )^4}{8 b n}-\frac{1}{4} e^2 \left (a+b \log \left (c x^n\right )\right )^3+\frac{1}{2} e^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3-\frac{3}{8} b e^2 n \left (a+b \log \left (c x^n\right )\right )^2+\frac{3}{4} b e^2 n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2-\frac{\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac{e \left (a+b \log \left (c x^n\right )\right )^3}{2 x}-\frac{3 b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac{9 b e n \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac{3}{8} b^3 e^2 n^3 \log (x)+\frac{3}{8} b^3 e^2 n^3 \log (e x+1)-\frac{3 b^3 n^3 \log (e x+1)}{8 x^2}-\frac{45 b^3 e n^3}{8 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-45*b^3*e*n^3)/(8*x) - (3*b^3*e^2*n^3*Log[x])/8 - (21*b^2*e*n^2*(a + b*Log[c*x^n]))/(4*x) - (3*b*e^2*n*(a + b
*Log[c*x^n])^2)/8 - (9*b*e*n*(a + b*Log[c*x^n])^2)/(4*x) - (e^2*(a + b*Log[c*x^n])^3)/4 - (e*(a + b*Log[c*x^n]
)^3)/(2*x) - (e^2*(a + b*Log[c*x^n])^4)/(8*b*n) + (3*b^3*e^2*n^3*Log[1 + e*x])/8 - (3*b^3*n^3*Log[1 + e*x])/(8
*x^2) + (3*b^2*e^2*n^2*(a + b*Log[c*x^n])*Log[1 + e*x])/4 - (3*b^2*n^2*(a + b*Log[c*x^n])*Log[1 + e*x])/(4*x^2
) + (3*b*e^2*n*(a + b*Log[c*x^n])^2*Log[1 + e*x])/4 - (3*b*n*(a + b*Log[c*x^n])^2*Log[1 + e*x])/(4*x^2) + (e^2
*(a + b*Log[c*x^n])^3*Log[1 + e*x])/2 - ((a + b*Log[c*x^n])^3*Log[1 + e*x])/(2*x^2) + (3*b^3*e^2*n^3*PolyLog[2
, -(e*x)])/4 + (3*b^2*e^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, -(e*x)])/2 + (3*b*e^2*n*(a + b*Log[c*x^n])^2*PolyL
og[2, -(e*x)])/2 - (3*b^3*e^2*n^3*PolyLog[3, -(e*x)])/2 - 3*b^2*e^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, -(e*x)]
+ 3*b^3*e^2*n^3*PolyLog[4, -(e*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 2378

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

Rule 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 2351

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c,
d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && IntegerQ[r]))

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2317

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

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]

Rule 2353

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[
{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r
]))

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, 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 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 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]

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x^3} \, dx &=-\frac{3 b^3 n^3 \log (1+e x)}{8 x^2}-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{4 x^2}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{4 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{2 x^2}-e \int \left (-\frac{3 b^3 n^3}{8 x^2 (1+e x)}-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^2 (1+e x)}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2 (1+e x)}-\frac{\left (a+b \log \left (c x^n\right )\right )^3}{2 x^2 (1+e x)}\right ) \, dx\\ &=-\frac{3 b^3 n^3 \log (1+e x)}{8 x^2}-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{4 x^2}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{4 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{2 x^2}+\frac{1}{2} e \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{x^2 (1+e x)} \, dx+\frac{1}{4} (3 b e n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (1+e x)} \, dx+\frac{1}{4} \left (3 b^2 e n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{x^2 (1+e x)} \, dx+\frac{1}{8} \left (3 b^3 e n^3\right ) \int \frac{1}{x^2 (1+e x)} \, dx\\ &=-\frac{3 b^3 n^3 \log (1+e x)}{8 x^2}-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{4 x^2}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{4 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{2 x^2}+\frac{1}{2} e \int \left (\frac{\left (a+b \log \left (c x^n\right )\right )^3}{x^2}-\frac{e \left (a+b \log \left (c x^n\right )\right )^3}{x}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )^3}{1+e x}\right ) \, dx+\frac{1}{4} (3 b e n) \int \left (\frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^2}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )^2}{1+e x}\right ) \, dx+\frac{1}{4} \left (3 b^2 e n^2\right ) \int \left (\frac{a+b \log \left (c x^n\right )}{x^2}-\frac{e \left (a+b \log \left (c x^n\right )\right )}{x}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )}{1+e x}\right ) \, dx+\frac{1}{8} \left (3 b^3 e n^3\right ) \int \left (\frac{1}{x^2}-\frac{e}{x}+\frac{e^2}{1+e x}\right ) \, dx\\ &=-\frac{3 b^3 e n^3}{8 x}-\frac{3}{8} b^3 e^2 n^3 \log (x)+\frac{3}{8} b^3 e^2 n^3 \log (1+e x)-\frac{3 b^3 n^3 \log (1+e x)}{8 x^2}-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{4 x^2}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{4 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{2 x^2}+\frac{1}{2} e \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{x^2} \, dx-\frac{1}{2} e^2 \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{x} \, dx+\frac{1}{2} e^3 \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{1+e x} \, dx+\frac{1}{4} (3 b e n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx-\frac{1}{4} \left (3 b e^2 n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx+\frac{1}{4} \left (3 b e^3 n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{1+e x} \, dx+\frac{1}{4} \left (3 b^2 e n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{x^2} \, dx-\frac{1}{4} \left (3 b^2 e^2 n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{x} \, dx+\frac{1}{4} \left (3 b^2 e^3 n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{1+e x} \, dx\\ &=-\frac{9 b^3 e n^3}{8 x}-\frac{3}{8} b^3 e^2 n^3 \log (x)-\frac{3 b^2 e n^2 \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac{3}{8} b e^2 n \left (a+b \log \left (c x^n\right )\right )^2-\frac{3 b e n \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac{e \left (a+b \log \left (c x^n\right )\right )^3}{2 x}+\frac{3}{8} b^3 e^2 n^3 \log (1+e x)-\frac{3 b^3 n^3 \log (1+e x)}{8 x^2}+\frac{3}{4} b^2 e^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{4 x^2}+\frac{3}{4} b e^2 n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{4 x^2}+\frac{1}{2} e^2 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{2 x^2}-\frac{1}{4} \left (3 e^2\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )-\frac{e^2 \operatorname{Subst}\left (\int x^3 \, dx,x,a+b \log \left (c x^n\right )\right )}{2 b n}+\frac{1}{2} (3 b e n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx-\frac{1}{2} \left (3 b e^2 n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x} \, dx+\frac{1}{2} \left (3 b^2 e n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{x^2} \, dx-\frac{1}{2} \left (3 b^2 e^2 n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x} \, dx-\frac{1}{4} \left (3 b^3 e^2 n^3\right ) \int \frac{\log (1+e x)}{x} \, dx\\ &=-\frac{21 b^3 e n^3}{8 x}-\frac{3}{8} b^3 e^2 n^3 \log (x)-\frac{9 b^2 e n^2 \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac{3}{8} b e^2 n \left (a+b \log \left (c x^n\right )\right )^2-\frac{9 b e n \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac{1}{4} e^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac{e \left (a+b \log \left (c x^n\right )\right )^3}{2 x}-\frac{e^2 \left (a+b \log \left (c x^n\right )\right )^4}{8 b n}+\frac{3}{8} b^3 e^2 n^3 \log (1+e x)-\frac{3 b^3 n^3 \log (1+e x)}{8 x^2}+\frac{3}{4} b^2 e^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{4 x^2}+\frac{3}{4} b e^2 n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{4 x^2}+\frac{1}{2} e^2 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{2 x^2}+\frac{3}{4} b^3 e^2 n^3 \text{Li}_2(-e x)+\frac{3}{2} b^2 e^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)+\frac{3}{2} b e^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2(-e x)+\left (3 b^2 e n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{x^2} \, dx-\left (3 b^2 e^2 n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)}{x} \, dx-\frac{1}{2} \left (3 b^3 e^2 n^3\right ) \int \frac{\text{Li}_2(-e x)}{x} \, dx\\ &=-\frac{45 b^3 e n^3}{8 x}-\frac{3}{8} b^3 e^2 n^3 \log (x)-\frac{21 b^2 e n^2 \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac{3}{8} b e^2 n \left (a+b \log \left (c x^n\right )\right )^2-\frac{9 b e n \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac{1}{4} e^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac{e \left (a+b \log \left (c x^n\right )\right )^3}{2 x}-\frac{e^2 \left (a+b \log \left (c x^n\right )\right )^4}{8 b n}+\frac{3}{8} b^3 e^2 n^3 \log (1+e x)-\frac{3 b^3 n^3 \log (1+e x)}{8 x^2}+\frac{3}{4} b^2 e^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{4 x^2}+\frac{3}{4} b e^2 n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{4 x^2}+\frac{1}{2} e^2 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{2 x^2}+\frac{3}{4} b^3 e^2 n^3 \text{Li}_2(-e x)+\frac{3}{2} b^2 e^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)+\frac{3}{2} b e^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2(-e x)-\frac{3}{2} b^3 e^2 n^3 \text{Li}_3(-e x)-3 b^2 e^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(-e x)+\left (3 b^3 e^2 n^3\right ) \int \frac{\text{Li}_3(-e x)}{x} \, dx\\ &=-\frac{45 b^3 e n^3}{8 x}-\frac{3}{8} b^3 e^2 n^3 \log (x)-\frac{21 b^2 e n^2 \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac{3}{8} b e^2 n \left (a+b \log \left (c x^n\right )\right )^2-\frac{9 b e n \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac{1}{4} e^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac{e \left (a+b \log \left (c x^n\right )\right )^3}{2 x}-\frac{e^2 \left (a+b \log \left (c x^n\right )\right )^4}{8 b n}+\frac{3}{8} b^3 e^2 n^3 \log (1+e x)-\frac{3 b^3 n^3 \log (1+e x)}{8 x^2}+\frac{3}{4} b^2 e^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{4 x^2}+\frac{3}{4} b e^2 n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{4 x^2}+\frac{1}{2} e^2 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{2 x^2}+\frac{3}{4} b^3 e^2 n^3 \text{Li}_2(-e x)+\frac{3}{2} b^2 e^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)+\frac{3}{2} b e^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2(-e x)-\frac{3}{2} b^3 e^2 n^3 \text{Li}_3(-e x)-3 b^2 e^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(-e x)+3 b^3 e^2 n^3 \text{Li}_4(-e x)\\ \end{align*}

Mathematica [B]  time = 0.377461, size = 1047, normalized size = 2.23 \[ -\frac{-b^3 e^2 n^3 x^2 \log ^4(x)+2 b^3 e^2 n^3 x^2 \log ^3(x)+4 a b^2 e^2 n^2 x^2 \log ^3(x)+4 b^3 e^2 n^2 x^2 \log \left (c x^n\right ) \log ^3(x)-3 b^3 e^2 n^3 x^2 \log ^2(x)-6 a b^2 e^2 n^2 x^2 \log ^2(x)-6 a^2 b e^2 n x^2 \log ^2(x)-6 b^3 e^2 n x^2 \log ^2\left (c x^n\right ) \log ^2(x)-6 b^3 e^2 n^2 x^2 \log \left (c x^n\right ) \log ^2(x)-12 a b^2 e^2 n x^2 \log \left (c x^n\right ) \log ^2(x)+4 b^3 e^2 x^2 \log ^3\left (c x^n\right ) \log (x)+3 b^3 e^2 n^3 x^2 \log (x)+4 a^3 e^2 x^2 \log (x)+6 a b^2 e^2 n^2 x^2 \log (x)+6 a^2 b e^2 n x^2 \log (x)+12 a b^2 e^2 x^2 \log ^2\left (c x^n\right ) \log (x)+6 b^3 e^2 n x^2 \log ^2\left (c x^n\right ) \log (x)+12 a^2 b e^2 x^2 \log \left (c x^n\right ) \log (x)+6 b^3 e^2 n^2 x^2 \log \left (c x^n\right ) \log (x)+12 a b^2 e^2 n x^2 \log \left (c x^n\right ) \log (x)+4 b^3 e x \log ^3\left (c x^n\right )+12 a b^2 e x \log ^2\left (c x^n\right )+18 b^3 e n x \log ^2\left (c x^n\right )+45 b^3 e n^3 x+42 a b^2 e n^2 x+4 a^3 e x+18 a^2 b e n x+42 b^3 e n^2 x \log \left (c x^n\right )+12 a^2 b e x \log \left (c x^n\right )+36 a b^2 e n x \log \left (c x^n\right )+4 a^3 \log (e x+1)+3 b^3 n^3 \log (e x+1)+4 b^3 \log ^3\left (c x^n\right ) \log (e x+1)-4 b^3 e^2 x^2 \log ^3\left (c x^n\right ) \log (e x+1)+6 a b^2 n^2 \log (e x+1)-3 b^3 e^2 n^3 x^2 \log (e x+1)-4 a^3 e^2 x^2 \log (e x+1)-6 a b^2 e^2 n^2 x^2 \log (e x+1)-6 a^2 b e^2 n x^2 \log (e x+1)+12 a b^2 \log ^2\left (c x^n\right ) \log (e x+1)-12 a b^2 e^2 x^2 \log ^2\left (c x^n\right ) \log (e x+1)-6 b^3 e^2 n x^2 \log ^2\left (c x^n\right ) \log (e x+1)+6 b^3 n \log ^2\left (c x^n\right ) \log (e x+1)+6 a^2 b n \log (e x+1)+6 b^3 n^2 \log \left (c x^n\right ) \log (e x+1)-12 a^2 b e^2 x^2 \log \left (c x^n\right ) \log (e x+1)-6 b^3 e^2 n^2 x^2 \log \left (c x^n\right ) \log (e x+1)-12 a b^2 e^2 n x^2 \log \left (c x^n\right ) \log (e x+1)+12 a^2 b \log \left (c x^n\right ) \log (e x+1)+12 a b^2 n \log \left (c x^n\right ) \log (e x+1)-6 b e^2 n x^2 \left (2 a^2+2 b n a+b^2 n^2+2 b^2 \log ^2\left (c x^n\right )+2 b (2 a+b n) \log \left (c x^n\right )\right ) \text{PolyLog}(2,-e x)+12 b^2 e^2 n^2 x^2 \left (2 a+b n+2 b \log \left (c x^n\right )\right ) \text{PolyLog}(3,-e x)-24 b^3 e^2 n^3 x^2 \text{PolyLog}(4,-e x)}{8 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(4*a^3*e*x + 18*a^2*b*e*n*x + 42*a*b^2*e*n^2*x + 45*b^3*e*n^3*x + 4*a^3*e^2*x^2*Log[x] + 6*a^2*b*e^2*n*x^2*Lo
g[x] + 6*a*b^2*e^2*n^2*x^2*Log[x] + 3*b^3*e^2*n^3*x^2*Log[x] - 6*a^2*b*e^2*n*x^2*Log[x]^2 - 6*a*b^2*e^2*n^2*x^
2*Log[x]^2 - 3*b^3*e^2*n^3*x^2*Log[x]^2 + 4*a*b^2*e^2*n^2*x^2*Log[x]^3 + 2*b^3*e^2*n^3*x^2*Log[x]^3 - b^3*e^2*
n^3*x^2*Log[x]^4 + 12*a^2*b*e*x*Log[c*x^n] + 36*a*b^2*e*n*x*Log[c*x^n] + 42*b^3*e*n^2*x*Log[c*x^n] + 12*a^2*b*
e^2*x^2*Log[x]*Log[c*x^n] + 12*a*b^2*e^2*n*x^2*Log[x]*Log[c*x^n] + 6*b^3*e^2*n^2*x^2*Log[x]*Log[c*x^n] - 12*a*
b^2*e^2*n*x^2*Log[x]^2*Log[c*x^n] - 6*b^3*e^2*n^2*x^2*Log[x]^2*Log[c*x^n] + 4*b^3*e^2*n^2*x^2*Log[x]^3*Log[c*x
^n] + 12*a*b^2*e*x*Log[c*x^n]^2 + 18*b^3*e*n*x*Log[c*x^n]^2 + 12*a*b^2*e^2*x^2*Log[x]*Log[c*x^n]^2 + 6*b^3*e^2
*n*x^2*Log[x]*Log[c*x^n]^2 - 6*b^3*e^2*n*x^2*Log[x]^2*Log[c*x^n]^2 + 4*b^3*e*x*Log[c*x^n]^3 + 4*b^3*e^2*x^2*Lo
g[x]*Log[c*x^n]^3 + 4*a^3*Log[1 + e*x] + 6*a^2*b*n*Log[1 + e*x] + 6*a*b^2*n^2*Log[1 + e*x] + 3*b^3*n^3*Log[1 +
e*x] - 4*a^3*e^2*x^2*Log[1 + e*x] - 6*a^2*b*e^2*n*x^2*Log[1 + e*x] - 6*a*b^2*e^2*n^2*x^2*Log[1 + e*x] - 3*b^3
*e^2*n^3*x^2*Log[1 + e*x] + 12*a^2*b*Log[c*x^n]*Log[1 + e*x] + 12*a*b^2*n*Log[c*x^n]*Log[1 + e*x] + 6*b^3*n^2*
Log[c*x^n]*Log[1 + e*x] - 12*a^2*b*e^2*x^2*Log[c*x^n]*Log[1 + e*x] - 12*a*b^2*e^2*n*x^2*Log[c*x^n]*Log[1 + e*x
] - 6*b^3*e^2*n^2*x^2*Log[c*x^n]*Log[1 + e*x] + 12*a*b^2*Log[c*x^n]^2*Log[1 + e*x] + 6*b^3*n*Log[c*x^n]^2*Log[
1 + e*x] - 12*a*b^2*e^2*x^2*Log[c*x^n]^2*Log[1 + e*x] - 6*b^3*e^2*n*x^2*Log[c*x^n]^2*Log[1 + e*x] + 4*b^3*Log[
c*x^n]^3*Log[1 + e*x] - 4*b^3*e^2*x^2*Log[c*x^n]^3*Log[1 + e*x] - 6*b*e^2*n*x^2*(2*a^2 + 2*a*b*n + b^2*n^2 + 2
*b*(2*a + b*n)*Log[c*x^n] + 2*b^2*Log[c*x^n]^2)*PolyLog[2, -(e*x)] + 12*b^2*e^2*n^2*x^2*(2*a + b*n + 2*b*Log[c
*x^n])*PolyLog[3, -(e*x)] - 24*b^3*e^2*n^3*x^2*PolyLog[4, -(e*x)])/(8*x^2)

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Maple [F]  time = 0.155, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{3}\ln \left ( ex+1 \right ) }{{x}^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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