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

Optimal. Leaf size=530 \[ \frac{3 b^2 n^2 \text{PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac{3 b^2 n^2 \text{PolyLog}(3,-e x) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{3 b n \text{PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}-\frac{3 b^3 n^3 \text{PolyLog}(2,-e x)}{4 e^2}-\frac{3 b^3 n^3 \text{PolyLog}(3,-e x)}{2 e^2}-\frac{3 b^3 n^3 \text{PolyLog}(4,-e x)}{e^2}-\frac{3 b^2 n^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{4 e^2}+\frac{3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{4 e}+\frac{3}{4} b^2 n^2 x^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )-\frac{9}{8} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac{9 a b^2 n^2 x}{2 e}+\frac{3 b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{4 e^2}-\frac{\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2}-\frac{9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{4 e}-\frac{3}{4} b n x^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2+\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{2 e}+\frac{1}{2} x^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3+\frac{3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )^3+\frac{9 b^3 n^2 x \log \left (c x^n\right )}{2 e}+\frac{3 b^3 n^3 \log (e x+1)}{8 e^2}-\frac{3}{8} b^3 n^3 x^2 \log (e x+1)-\frac{45 b^3 n^3 x}{8 e}+\frac{3}{4} b^3 n^3 x^2 \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.493278, antiderivative size = 530, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 12, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {2395, 43, 2377, 2296, 2295, 2305, 2304, 2374, 2383, 6589, 2376, 2391} \[ \frac{3 b^2 n^2 \text{PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac{3 b^2 n^2 \text{PolyLog}(3,-e x) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{3 b n \text{PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}-\frac{3 b^3 n^3 \text{PolyLog}(2,-e x)}{4 e^2}-\frac{3 b^3 n^3 \text{PolyLog}(3,-e x)}{2 e^2}-\frac{3 b^3 n^3 \text{PolyLog}(4,-e x)}{e^2}-\frac{3 b^2 n^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{4 e^2}+\frac{3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{4 e}+\frac{3}{4} b^2 n^2 x^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )-\frac{9}{8} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac{9 a b^2 n^2 x}{2 e}+\frac{3 b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{4 e^2}-\frac{\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2}-\frac{9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{4 e}-\frac{3}{4} b n x^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2+\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{2 e}+\frac{1}{2} x^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3+\frac{3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )^3+\frac{9 b^3 n^2 x \log \left (c x^n\right )}{2 e}+\frac{3 b^3 n^3 \log (e x+1)}{8 e^2}-\frac{3}{8} b^3 n^3 x^2 \log (e x+1)-\frac{45 b^3 n^3 x}{8 e}+\frac{3}{4} b^3 n^3 x^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 \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}{2 e}-\frac{1}{4} x^2 \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)}{2 e^2}+\frac{1}{2} x^2 \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}{2 e}-\frac{1}{4} x \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)}{2 e^2 x}+\frac{1}{2} x \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}{2 e}-\frac{1}{4} x^2 \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)}{2 e^2}+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)+\frac{1}{4} (3 b n) \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx-\frac{1}{2} (3 b n) \int x \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}{2 e^2}-\frac{(3 b n) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{2 e}\\ &=-\frac{9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{4 e}+\frac{3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{2 e}-\frac{1}{4} x^2 \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)}{4 e^2}-\frac{3}{4} b n x^2 \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)}{2 e^2}+\frac{1}{2} x^2 \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)}{2 e^2}-\frac{1}{4} \left (3 b^2 n^2\right ) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx+\left (3 b^2 n^2\right ) \int \left (\frac{a+b \log \left (c x^n\right )}{2 e}-\frac{1}{4} x \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)}{2 e^2 x}+\frac{1}{2} x \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}{e^2}+\frac{\left (3 b^2 n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{e}\\ &=\frac{3 a b^2 n^2 x}{e}+\frac{3}{16} b^3 n^3 x^2-\frac{3}{8} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{4 e}+\frac{3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{2 e}-\frac{1}{4} x^2 \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)}{4 e^2}-\frac{3}{4} b n x^2 \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)}{2 e^2}+\frac{1}{2} x^2 \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)}{2 e^2}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(-e x)}{e^2}-\frac{1}{4} \left (3 b^2 n^2\right ) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac{1}{2} \left (3 b^2 n^2\right ) \int x \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}{2 e^2}+\frac{\left (3 b^2 n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{2 e}+\frac{\left (3 b^3 n^2\right ) \int \log \left (c x^n\right ) \, dx}{e}-\frac{\left (3 b^3 n^3\right ) \int \frac{\text{Li}_3(-e x)}{x} \, dx}{e^2}\\ &=\frac{9 a b^2 n^2 x}{2 e}-\frac{3 b^3 n^3 x}{e}+\frac{3}{8} b^3 n^3 x^2+\frac{3 b^3 n^2 x \log \left (c x^n\right )}{e}+\frac{3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{4 e}-\frac{9}{8} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{4 e}+\frac{3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{2 e}-\frac{1}{4} x^2 \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)}{4 e^2}+\frac{3}{4} b^2 n^2 x^2 \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)}{4 e^2}-\frac{3}{4} b n x^2 \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)}{2 e^2}+\frac{1}{2} x^2 \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)}{2 e^2}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2(-e x)}{2 e^2}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(-e x)}{e^2}-\frac{3 b^3 n^3 \text{Li}_4(-e x)}{e^2}+\frac{\left (3 b^3 n^2\right ) \int \log \left (c x^n\right ) \, dx}{2 e}-\frac{1}{2} \left (3 b^3 n^3\right ) \int \left (\frac{1}{2 e}-\frac{x}{4}-\frac{\log (1+e x)}{2 e^2 x}+\frac{1}{2} x \log (1+e x)\right ) \, dx-\frac{\left (3 b^3 n^3\right ) \int \frac{\text{Li}_2(-e x)}{x} \, dx}{2 e^2}\\ &=\frac{9 a b^2 n^2 x}{2 e}-\frac{21 b^3 n^3 x}{4 e}+\frac{9}{16} b^3 n^3 x^2+\frac{9 b^3 n^2 x \log \left (c x^n\right )}{2 e}+\frac{3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{4 e}-\frac{9}{8} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{4 e}+\frac{3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{2 e}-\frac{1}{4} x^2 \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)}{4 e^2}+\frac{3}{4} b^2 n^2 x^2 \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)}{4 e^2}-\frac{3}{4} b n x^2 \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)}{2 e^2}+\frac{1}{2} x^2 \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)}{2 e^2}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2(-e x)}{2 e^2}-\frac{3 b^3 n^3 \text{Li}_3(-e x)}{2 e^2}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(-e x)}{e^2}-\frac{3 b^3 n^3 \text{Li}_4(-e x)}{e^2}-\frac{1}{4} \left (3 b^3 n^3\right ) \int x \log (1+e x) \, dx+\frac{\left (3 b^3 n^3\right ) \int \frac{\log (1+e x)}{x} \, dx}{4 e^2}\\ &=\frac{9 a b^2 n^2 x}{2 e}-\frac{21 b^3 n^3 x}{4 e}+\frac{9}{16} b^3 n^3 x^2+\frac{9 b^3 n^2 x \log \left (c x^n\right )}{2 e}+\frac{3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{4 e}-\frac{9}{8} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{4 e}+\frac{3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{2 e}-\frac{1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac{3}{8} b^3 n^3 x^2 \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 e^2}+\frac{3}{4} b^2 n^2 x^2 \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)}{4 e^2}-\frac{3}{4} b n x^2 \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)}{2 e^2}+\frac{1}{2} x^2 \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)}{4 e^2}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)}{2 e^2}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2(-e x)}{2 e^2}-\frac{3 b^3 n^3 \text{Li}_3(-e x)}{2 e^2}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(-e x)}{e^2}-\frac{3 b^3 n^3 \text{Li}_4(-e x)}{e^2}+\frac{1}{8} \left (3 b^3 e n^3\right ) \int \frac{x^2}{1+e x} \, dx\\ &=\frac{9 a b^2 n^2 x}{2 e}-\frac{21 b^3 n^3 x}{4 e}+\frac{9}{16} b^3 n^3 x^2+\frac{9 b^3 n^2 x \log \left (c x^n\right )}{2 e}+\frac{3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{4 e}-\frac{9}{8} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{4 e}+\frac{3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{2 e}-\frac{1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac{3}{8} b^3 n^3 x^2 \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 e^2}+\frac{3}{4} b^2 n^2 x^2 \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)}{4 e^2}-\frac{3}{4} b n x^2 \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)}{2 e^2}+\frac{1}{2} x^2 \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)}{4 e^2}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)}{2 e^2}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2(-e x)}{2 e^2}-\frac{3 b^3 n^3 \text{Li}_3(-e x)}{2 e^2}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(-e x)}{e^2}-\frac{3 b^3 n^3 \text{Li}_4(-e x)}{e^2}+\frac{1}{8} \left (3 b^3 e n^3\right ) \int \left (-\frac{1}{e^2}+\frac{x}{e}+\frac{1}{e^2 (1+e x)}\right ) \, dx\\ &=\frac{9 a b^2 n^2 x}{2 e}-\frac{45 b^3 n^3 x}{8 e}+\frac{3}{4} b^3 n^3 x^2+\frac{9 b^3 n^2 x \log \left (c x^n\right )}{2 e}+\frac{3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{4 e}-\frac{9}{8} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{4 e}+\frac{3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{2 e}-\frac{1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )^3+\frac{3 b^3 n^3 \log (1+e x)}{8 e^2}-\frac{3}{8} b^3 n^3 x^2 \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 e^2}+\frac{3}{4} b^2 n^2 x^2 \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)}{4 e^2}-\frac{3}{4} b n x^2 \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)}{2 e^2}+\frac{1}{2} x^2 \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)}{4 e^2}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)}{2 e^2}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2(-e x)}{2 e^2}-\frac{3 b^3 n^3 \text{Li}_3(-e x)}{2 e^2}+\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(-e x)}{e^2}-\frac{3 b^3 n^3 \text{Li}_4(-e x)}{e^2}\\ \end{align*}

Mathematica [A]  time = 0.247528, size = 806, normalized size = 1.52 \[ \frac{-2 e^2 x^2 a^3+4 e x a^3+4 e^2 x^2 \log (e x+1) a^3-4 \log (e x+1) a^3+6 b e^2 n x^2 a^2-18 b e n x a^2-6 b e^2 x^2 \log \left (c x^n\right ) a^2+12 b e x \log \left (c x^n\right ) a^2-6 b e^2 n x^2 \log (e x+1) a^2+6 b n \log (e x+1) a^2+12 b e^2 x^2 \log \left (c x^n\right ) \log (e x+1) a^2-12 b \log \left (c x^n\right ) \log (e x+1) a^2-9 b^2 e^2 n^2 x^2 a-6 b^2 e^2 x^2 \log ^2\left (c x^n\right ) a+12 b^2 e x \log ^2\left (c x^n\right ) a+42 b^2 e n^2 x a+12 b^2 e^2 n x^2 \log \left (c x^n\right ) a-36 b^2 e n x \log \left (c x^n\right ) a-6 b^2 n^2 \log (e x+1) a+6 b^2 e^2 n^2 x^2 \log (e x+1) a-12 b^2 \log ^2\left (c x^n\right ) \log (e x+1) a+12 b^2 e^2 x^2 \log ^2\left (c x^n\right ) \log (e x+1) a-12 b^2 e^2 n x^2 \log \left (c x^n\right ) \log (e x+1) a+12 b^2 n \log \left (c x^n\right ) \log (e x+1) a-2 b^3 e^2 x^2 \log ^3\left (c x^n\right )+4 b^3 e x \log ^3\left (c x^n\right )+6 b^3 e^2 n^3 x^2+6 b^3 e^2 n x^2 \log ^2\left (c x^n\right )-18 b^3 e n x \log ^2\left (c x^n\right )-45 b^3 e n^3 x-9 b^3 e^2 n^2 x^2 \log \left (c x^n\right )+42 b^3 e n^2 x \log \left (c x^n\right )+3 b^3 n^3 \log (e x+1)-4 b^3 \log ^3\left (c x^n\right ) \log (e x+1)+4 b^3 e^2 x^2 \log ^3\left (c x^n\right ) \log (e x+1)-3 b^3 e^2 n^3 x^2 \log (e x+1)-6 b^3 e^2 n x^2 \log ^2\left (c x^n\right ) \log (e x+1)+6 b^3 n \log ^2\left (c x^n\right ) \log (e x+1)-6 b^3 n^2 \log \left (c x^n\right ) \log (e x+1)+6 b^3 e^2 n^2 x^2 \log \left (c x^n\right ) \log (e x+1)-6 b n \left (2 a^2-2 b n a+b^2 n^2+2 b^2 \log ^2\left (c x^n\right )-2 b (b n-2 a) \log \left (c x^n\right )\right ) \text{PolyLog}(2,-e x)+12 b^2 n^2 \left (2 a-b n+2 b \log \left (c x^n\right )\right ) \text{PolyLog}(3,-e x)-24 b^3 n^3 \text{PolyLog}(4,-e x)}{8 e^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*Log[c*x^n])^3*Log[1 + e*x],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 - 2*a^3*e^2*x^2 + 6*a^2*b*e^2*n*x^2 - 9*a*b^2*
e^2*n^2*x^2 + 6*b^3*e^2*n^3*x^2 + 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] - 6*a^2*b*e^2*x^2*Log[c*x^n] + 12*a*b^2*e^2*n*x^2*Log[c*x^n] - 9*b^3*e^2*n^2*x^2*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 - 6*a*b^2*e^2*x^2*Log[c*x^n]^2 + 6*b^3*e^2*n*x^2*Log[c*x^n]^2 + 4*b^
3*e*x*Log[c*x^n]^3 - 2*b^3*e^2*x^2*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*Lo
g[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*n*(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*n^2*(2*a -
b*n + 2*b*Log[c*x^n])*PolyLog[3, -(e*x)] - 24*b^3*n^3*PolyLog[4, -(e*x)])/(8*e^2)

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Maple [F]  time = 0.129, size = 0, normalized size = 0. \begin{align*} \int x \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*(a+b*ln(c*x^n))^3*ln(e*x+1),x)

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(b^3*x*log(c*x^n)^3*log(e*x + 1) + 3*a*b^2*x*log(c*x^n)^2*log(e*x + 1) + 3*a^2*b*x*log(c*x^n)*log(e*x
+ 1) + a^3*x*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*(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 \log \left (e x + 1\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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