3.18 \(\int x^2 (a+b \log (c x^n))^3 \log (1+e x) \, dx\)

Optimal. Leaf size=615 \[ -\frac{2 b^2 n^2 \text{PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac{2 b^2 n^2 \text{PolyLog}(3,-e x) \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac{b n \text{PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )^2}{e^3}+\frac{2 b^3 n^3 \text{PolyLog}(2,-e x)}{9 e^3}+\frac{2 b^3 n^3 \text{PolyLog}(3,-e x)}{3 e^3}+\frac{2 b^3 n^3 \text{PolyLog}(4,-e x)}{e^3}-\frac{2 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{9 e^2}+\frac{2 b^2 n^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{9 e^3}+\frac{19 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{36 e}+\frac{2}{9} b^2 n^2 x^3 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )-\frac{2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{8 a b^2 n^2 x}{3 e^2}+\frac{4 b n x \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2}-\frac{b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{3 e^3}-\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{3 e^2}+\frac{\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3}{3 e^3}-\frac{5 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}-\frac{1}{3} b n x^3 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 e}+\frac{1}{3} x^3 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3+\frac{2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^3-\frac{8 b^3 n^2 x \log \left (c x^n\right )}{3 e^2}+\frac{80 b^3 n^3 x}{27 e^2}-\frac{2 b^3 n^3 \log (e x+1)}{27 e^3}-\frac{65 b^3 n^3 x^2}{216 e}-\frac{2}{27} b^3 n^3 x^3 \log (e x+1)+\frac{8}{81} b^3 n^3 x^3 \]

[Out]

(-8*a*b^2*n^2*x)/(3*e^2) + (80*b^3*n^3*x)/(27*e^2) - (65*b^3*n^3*x^2)/(216*e) + (8*b^3*n^3*x^3)/81 - (8*b^3*n^
2*x*Log[c*x^n])/(3*e^2) - (2*b^2*n^2*x*(a + b*Log[c*x^n]))/(9*e^2) + (19*b^2*n^2*x^2*(a + b*Log[c*x^n]))/(36*e
) - (2*b^2*n^2*x^3*(a + b*Log[c*x^n]))/9 + (4*b*n*x*(a + b*Log[c*x^n])^2)/(3*e^2) - (5*b*n*x^2*(a + b*Log[c*x^
n])^2)/(12*e) + (2*b*n*x^3*(a + b*Log[c*x^n])^2)/9 - (x*(a + b*Log[c*x^n])^3)/(3*e^2) + (x^2*(a + b*Log[c*x^n]
)^3)/(6*e) - (x^3*(a + b*Log[c*x^n])^3)/9 - (2*b^3*n^3*Log[1 + e*x])/(27*e^3) - (2*b^3*n^3*x^3*Log[1 + e*x])/2
7 + (2*b^2*n^2*(a + b*Log[c*x^n])*Log[1 + e*x])/(9*e^3) + (2*b^2*n^2*x^3*(a + b*Log[c*x^n])*Log[1 + e*x])/9 -
(b*n*(a + b*Log[c*x^n])^2*Log[1 + e*x])/(3*e^3) - (b*n*x^3*(a + b*Log[c*x^n])^2*Log[1 + e*x])/3 + ((a + b*Log[
c*x^n])^3*Log[1 + e*x])/(3*e^3) + (x^3*(a + b*Log[c*x^n])^3*Log[1 + e*x])/3 + (2*b^3*n^3*PolyLog[2, -(e*x)])/(
9*e^3) - (2*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, -(e*x)])/(3*e^3) + (b*n*(a + b*Log[c*x^n])^2*PolyLog[2, -(e*
x)])/e^3 + (2*b^3*n^3*PolyLog[3, -(e*x)])/(3*e^3) - (2*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, -(e*x)])/e^3 + (2
*b^3*n^3*PolyLog[4, -(e*x)])/e^3

________________________________________________________________________________________

Rubi [A]  time = 0.637563, antiderivative size = 615, normalized size of antiderivative = 1., number of steps used = 26, 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} \[ -\frac{2 b^2 n^2 \text{PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac{2 b^2 n^2 \text{PolyLog}(3,-e x) \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac{b n \text{PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )^2}{e^3}+\frac{2 b^3 n^3 \text{PolyLog}(2,-e x)}{9 e^3}+\frac{2 b^3 n^3 \text{PolyLog}(3,-e x)}{3 e^3}+\frac{2 b^3 n^3 \text{PolyLog}(4,-e x)}{e^3}-\frac{2 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{9 e^2}+\frac{2 b^2 n^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{9 e^3}+\frac{19 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{36 e}+\frac{2}{9} b^2 n^2 x^3 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )-\frac{2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{8 a b^2 n^2 x}{3 e^2}+\frac{4 b n x \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2}-\frac{b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{3 e^3}-\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{3 e^2}+\frac{\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3}{3 e^3}-\frac{5 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}-\frac{1}{3} b n x^3 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 e}+\frac{1}{3} x^3 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3+\frac{2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^3-\frac{8 b^3 n^2 x \log \left (c x^n\right )}{3 e^2}+\frac{80 b^3 n^3 x}{27 e^2}-\frac{2 b^3 n^3 \log (e x+1)}{27 e^3}-\frac{65 b^3 n^3 x^2}{216 e}-\frac{2}{27} b^3 n^3 x^3 \log (e x+1)+\frac{8}{81} b^3 n^3 x^3 \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*Log[c*x^n])^3*Log[1 + e*x],x]

[Out]

(-8*a*b^2*n^2*x)/(3*e^2) + (80*b^3*n^3*x)/(27*e^2) - (65*b^3*n^3*x^2)/(216*e) + (8*b^3*n^3*x^3)/81 - (8*b^3*n^
2*x*Log[c*x^n])/(3*e^2) - (2*b^2*n^2*x*(a + b*Log[c*x^n]))/(9*e^2) + (19*b^2*n^2*x^2*(a + b*Log[c*x^n]))/(36*e
) - (2*b^2*n^2*x^3*(a + b*Log[c*x^n]))/9 + (4*b*n*x*(a + b*Log[c*x^n])^2)/(3*e^2) - (5*b*n*x^2*(a + b*Log[c*x^
n])^2)/(12*e) + (2*b*n*x^3*(a + b*Log[c*x^n])^2)/9 - (x*(a + b*Log[c*x^n])^3)/(3*e^2) + (x^2*(a + b*Log[c*x^n]
)^3)/(6*e) - (x^3*(a + b*Log[c*x^n])^3)/9 - (2*b^3*n^3*Log[1 + e*x])/(27*e^3) - (2*b^3*n^3*x^3*Log[1 + e*x])/2
7 + (2*b^2*n^2*(a + b*Log[c*x^n])*Log[1 + e*x])/(9*e^3) + (2*b^2*n^2*x^3*(a + b*Log[c*x^n])*Log[1 + e*x])/9 -
(b*n*(a + b*Log[c*x^n])^2*Log[1 + e*x])/(3*e^3) - (b*n*x^3*(a + b*Log[c*x^n])^2*Log[1 + e*x])/3 + ((a + b*Log[
c*x^n])^3*Log[1 + e*x])/(3*e^3) + (x^3*(a + b*Log[c*x^n])^3*Log[1 + e*x])/3 + (2*b^3*n^3*PolyLog[2, -(e*x)])/(
9*e^3) - (2*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, -(e*x)])/(3*e^3) + (b*n*(a + b*Log[c*x^n])^2*PolyLog[2, -(e*
x)])/e^3 + (2*b^3*n^3*PolyLog[3, -(e*x)])/(3*e^3) - (2*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, -(e*x)])/e^3 + (2
*b^3*n^3*PolyLog[4, -(e*x)])/e^3

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^2 \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}{3 e^2}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 e}-\frac{1}{9} x^3 \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)}{3 e^3}+\frac{1}{3} x^3 \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}{3 e^2}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{6 e}-\frac{1}{9} x^2 \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)}{3 e^3 x}+\frac{1}{3} x^2 \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}{3 e^2}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 e}-\frac{1}{9} x^3 \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)}{3 e^3}+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)+\frac{1}{3} (b n) \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx-(b n) \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x) \, dx-\frac{(b n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x} \, dx}{e^3}+\frac{(b n) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{e^2}-\frac{(b n) \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{2 e}\\ &=\frac{4 b n x \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2}-\frac{5 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}+\frac{2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{3 e^2}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 e}-\frac{1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^3-\frac{b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{3 e^3}-\frac{1}{3} b n x^3 \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)}{3 e^3}+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)+\frac{b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2(-e x)}{e^3}-\frac{1}{9} \left (2 b^2 n^2\right ) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx+\left (2 b^2 n^2\right ) \int \left (-\frac{a+b \log \left (c x^n\right )}{3 e^2}+\frac{x \left (a+b \log \left (c x^n\right )\right )}{6 e}-\frac{1}{9} x^2 \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)}{3 e^3 x}+\frac{1}{3} x^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)\right ) \, dx-\frac{\left (2 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^3}-\frac{\left (2 b^2 n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^2}+\frac{\left (b^2 n^2\right ) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{2 e}\\ &=-\frac{2 a b^2 n^2 x}{e^2}-\frac{b^3 n^3 x^2}{8 e}+\frac{2}{81} b^3 n^3 x^3+\frac{b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{4 e}-\frac{2}{27} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{4 b n x \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2}-\frac{5 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}+\frac{2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{3 e^2}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 e}-\frac{1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^3-\frac{b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{3 e^3}-\frac{1}{3} b n x^3 \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)}{3 e^3}+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)+\frac{b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2(-e x)}{e^3}-\frac{2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(-e x)}{e^3}-\frac{1}{9} \left (2 b^2 n^2\right ) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac{1}{3} \left (2 b^2 n^2\right ) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x) \, dx+\frac{\left (2 b^2 n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x} \, dx}{3 e^3}-\frac{\left (2 b^2 n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{3 e^2}-\frac{\left (2 b^3 n^2\right ) \int \log \left (c x^n\right ) \, dx}{e^2}+\frac{\left (b^2 n^2\right ) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{3 e}+\frac{\left (2 b^3 n^3\right ) \int \frac{\text{Li}_3(-e x)}{x} \, dx}{e^3}\\ &=-\frac{8 a b^2 n^2 x}{3 e^2}+\frac{2 b^3 n^3 x}{e^2}-\frac{5 b^3 n^3 x^2}{24 e}+\frac{4}{81} b^3 n^3 x^3-\frac{2 b^3 n^2 x \log \left (c x^n\right )}{e^2}-\frac{2 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{9 e^2}+\frac{19 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{36 e}-\frac{2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{4 b n x \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2}-\frac{5 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}+\frac{2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{3 e^2}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 e}-\frac{1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^3+\frac{2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{9 e^3}+\frac{2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac{b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{3 e^3}-\frac{1}{3} b n x^3 \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)}{3 e^3}+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)-\frac{2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)}{3 e^3}+\frac{b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2(-e x)}{e^3}-\frac{2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(-e x)}{e^3}+\frac{2 b^3 n^3 \text{Li}_4(-e x)}{e^3}-\frac{\left (2 b^3 n^2\right ) \int \log \left (c x^n\right ) \, dx}{3 e^2}-\frac{1}{3} \left (2 b^3 n^3\right ) \int \left (-\frac{1}{3 e^2}+\frac{x}{6 e}-\frac{x^2}{9}+\frac{\log (1+e x)}{3 e^3 x}+\frac{1}{3} x^2 \log (1+e x)\right ) \, dx+\frac{\left (2 b^3 n^3\right ) \int \frac{\text{Li}_2(-e x)}{x} \, dx}{3 e^3}\\ &=-\frac{8 a b^2 n^2 x}{3 e^2}+\frac{26 b^3 n^3 x}{9 e^2}-\frac{19 b^3 n^3 x^2}{72 e}+\frac{2}{27} b^3 n^3 x^3-\frac{8 b^3 n^2 x \log \left (c x^n\right )}{3 e^2}-\frac{2 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{9 e^2}+\frac{19 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{36 e}-\frac{2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{4 b n x \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2}-\frac{5 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}+\frac{2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{3 e^2}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 e}-\frac{1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^3+\frac{2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{9 e^3}+\frac{2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac{b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{3 e^3}-\frac{1}{3} b n x^3 \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)}{3 e^3}+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)-\frac{2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)}{3 e^3}+\frac{b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2(-e x)}{e^3}+\frac{2 b^3 n^3 \text{Li}_3(-e x)}{3 e^3}-\frac{2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(-e x)}{e^3}+\frac{2 b^3 n^3 \text{Li}_4(-e x)}{e^3}-\frac{1}{9} \left (2 b^3 n^3\right ) \int x^2 \log (1+e x) \, dx-\frac{\left (2 b^3 n^3\right ) \int \frac{\log (1+e x)}{x} \, dx}{9 e^3}\\ &=-\frac{8 a b^2 n^2 x}{3 e^2}+\frac{26 b^3 n^3 x}{9 e^2}-\frac{19 b^3 n^3 x^2}{72 e}+\frac{2}{27} b^3 n^3 x^3-\frac{8 b^3 n^2 x \log \left (c x^n\right )}{3 e^2}-\frac{2 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{9 e^2}+\frac{19 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{36 e}-\frac{2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{4 b n x \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2}-\frac{5 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}+\frac{2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{3 e^2}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 e}-\frac{1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^3-\frac{2}{27} b^3 n^3 x^3 \log (1+e x)+\frac{2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{9 e^3}+\frac{2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac{b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{3 e^3}-\frac{1}{3} b n x^3 \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)}{3 e^3}+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)+\frac{2 b^3 n^3 \text{Li}_2(-e x)}{9 e^3}-\frac{2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)}{3 e^3}+\frac{b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2(-e x)}{e^3}+\frac{2 b^3 n^3 \text{Li}_3(-e x)}{3 e^3}-\frac{2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(-e x)}{e^3}+\frac{2 b^3 n^3 \text{Li}_4(-e x)}{e^3}+\frac{1}{27} \left (2 b^3 e n^3\right ) \int \frac{x^3}{1+e x} \, dx\\ &=-\frac{8 a b^2 n^2 x}{3 e^2}+\frac{26 b^3 n^3 x}{9 e^2}-\frac{19 b^3 n^3 x^2}{72 e}+\frac{2}{27} b^3 n^3 x^3-\frac{8 b^3 n^2 x \log \left (c x^n\right )}{3 e^2}-\frac{2 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{9 e^2}+\frac{19 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{36 e}-\frac{2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{4 b n x \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2}-\frac{5 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}+\frac{2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{3 e^2}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 e}-\frac{1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^3-\frac{2}{27} b^3 n^3 x^3 \log (1+e x)+\frac{2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{9 e^3}+\frac{2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac{b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{3 e^3}-\frac{1}{3} b n x^3 \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)}{3 e^3}+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)+\frac{2 b^3 n^3 \text{Li}_2(-e x)}{9 e^3}-\frac{2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)}{3 e^3}+\frac{b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2(-e x)}{e^3}+\frac{2 b^3 n^3 \text{Li}_3(-e x)}{3 e^3}-\frac{2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(-e x)}{e^3}+\frac{2 b^3 n^3 \text{Li}_4(-e x)}{e^3}+\frac{1}{27} \left (2 b^3 e n^3\right ) \int \left (\frac{1}{e^3}-\frac{x}{e^2}+\frac{x^2}{e}-\frac{1}{e^3 (1+e x)}\right ) \, dx\\ &=-\frac{8 a b^2 n^2 x}{3 e^2}+\frac{80 b^3 n^3 x}{27 e^2}-\frac{65 b^3 n^3 x^2}{216 e}+\frac{8}{81} b^3 n^3 x^3-\frac{8 b^3 n^2 x \log \left (c x^n\right )}{3 e^2}-\frac{2 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{9 e^2}+\frac{19 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{36 e}-\frac{2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{4 b n x \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2}-\frac{5 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}+\frac{2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac{x \left (a+b \log \left (c x^n\right )\right )^3}{3 e^2}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 e}-\frac{1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^3-\frac{2 b^3 n^3 \log (1+e x)}{27 e^3}-\frac{2}{27} b^3 n^3 x^3 \log (1+e x)+\frac{2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{9 e^3}+\frac{2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac{b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{3 e^3}-\frac{1}{3} b n x^3 \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)}{3 e^3}+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)+\frac{2 b^3 n^3 \text{Li}_2(-e x)}{9 e^3}-\frac{2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)}{3 e^3}+\frac{b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2(-e x)}{e^3}+\frac{2 b^3 n^3 \text{Li}_3(-e x)}{3 e^3}-\frac{2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(-e x)}{e^3}+\frac{2 b^3 n^3 \text{Li}_4(-e x)}{e^3}\\ \end{align*}

Mathematica [A]  time = 0.279207, size = 975, normalized size = 1.59 \[ \frac{-72 e^3 x^3 a^3+108 e^2 x^2 a^3-216 e x a^3+216 e^3 x^3 \log (e x+1) a^3+216 \log (e x+1) a^3+144 b e^3 n x^3 a^2-270 b e^2 n x^2 a^2+864 b e n x a^2-216 b e^3 x^3 \log \left (c x^n\right ) a^2+324 b e^2 x^2 \log \left (c x^n\right ) a^2-648 b e x \log \left (c x^n\right ) a^2-216 b e^3 n x^3 \log (e x+1) a^2-216 b n \log (e x+1) a^2+648 b e^3 x^3 \log \left (c x^n\right ) \log (e x+1) a^2+648 b \log \left (c x^n\right ) \log (e x+1) a^2-144 b^2 e^3 n^2 x^3 a+342 b^2 e^2 n^2 x^2 a-216 b^2 e^3 x^3 \log ^2\left (c x^n\right ) a+324 b^2 e^2 x^2 \log ^2\left (c x^n\right ) a-648 b^2 e x \log ^2\left (c x^n\right ) a-1872 b^2 e n^2 x a+288 b^2 e^3 n x^3 \log \left (c x^n\right ) a-540 b^2 e^2 n x^2 \log \left (c x^n\right ) a+1728 b^2 e n x \log \left (c x^n\right ) a+144 b^2 e^3 n^2 x^3 \log (e x+1) a+144 b^2 n^2 \log (e x+1) a+648 b^2 e^3 x^3 \log ^2\left (c x^n\right ) \log (e x+1) a+648 b^2 \log ^2\left (c x^n\right ) \log (e x+1) a-432 b^2 e^3 n x^3 \log \left (c x^n\right ) \log (e x+1) a-432 b^2 n \log \left (c x^n\right ) \log (e x+1) a+64 b^3 e^3 n^3 x^3-72 b^3 e^3 x^3 \log ^3\left (c x^n\right )+108 b^3 e^2 x^2 \log ^3\left (c x^n\right )-216 b^3 e x \log ^3\left (c x^n\right )-195 b^3 e^2 n^3 x^2+144 b^3 e^3 n x^3 \log ^2\left (c x^n\right )-270 b^3 e^2 n x^2 \log ^2\left (c x^n\right )+864 b^3 e n x \log ^2\left (c x^n\right )+1920 b^3 e n^3 x-144 b^3 e^3 n^2 x^3 \log \left (c x^n\right )+342 b^3 e^2 n^2 x^2 \log \left (c x^n\right )-1872 b^3 e n^2 x \log \left (c x^n\right )-48 b^3 n^3 \log (e x+1)-48 b^3 e^3 n^3 x^3 \log (e x+1)+216 b^3 \log ^3\left (c x^n\right ) \log (e x+1)+216 b^3 e^3 x^3 \log ^3\left (c x^n\right ) \log (e x+1)-216 b^3 e^3 n x^3 \log ^2\left (c x^n\right ) \log (e x+1)-216 b^3 n \log ^2\left (c x^n\right ) \log (e x+1)+144 b^3 e^3 n^2 x^3 \log \left (c x^n\right ) \log (e x+1)+144 b^3 n^2 \log \left (c x^n\right ) \log (e x+1)+72 b n \left (9 a^2-6 b n a+2 b^2 n^2+9 b^2 \log ^2\left (c x^n\right )-6 b (b n-3 a) \log \left (c x^n\right )\right ) \text{PolyLog}(2,-e x)+432 b^2 n^2 \left (-3 a+b n-3 b \log \left (c x^n\right )\right ) \text{PolyLog}(3,-e x)+1296 b^3 n^3 \text{PolyLog}(4,-e x)}{648 e^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*Log[c*x^n])^3*Log[1 + e*x],x]

[Out]

(-216*a^3*e*x + 864*a^2*b*e*n*x - 1872*a*b^2*e*n^2*x + 1920*b^3*e*n^3*x + 108*a^3*e^2*x^2 - 270*a^2*b*e^2*n*x^
2 + 342*a*b^2*e^2*n^2*x^2 - 195*b^3*e^2*n^3*x^2 - 72*a^3*e^3*x^3 + 144*a^2*b*e^3*n*x^3 - 144*a*b^2*e^3*n^2*x^3
+ 64*b^3*e^3*n^3*x^3 - 648*a^2*b*e*x*Log[c*x^n] + 1728*a*b^2*e*n*x*Log[c*x^n] - 1872*b^3*e*n^2*x*Log[c*x^n] +
324*a^2*b*e^2*x^2*Log[c*x^n] - 540*a*b^2*e^2*n*x^2*Log[c*x^n] + 342*b^3*e^2*n^2*x^2*Log[c*x^n] - 216*a^2*b*e^
3*x^3*Log[c*x^n] + 288*a*b^2*e^3*n*x^3*Log[c*x^n] - 144*b^3*e^3*n^2*x^3*Log[c*x^n] - 648*a*b^2*e*x*Log[c*x^n]^
2 + 864*b^3*e*n*x*Log[c*x^n]^2 + 324*a*b^2*e^2*x^2*Log[c*x^n]^2 - 270*b^3*e^2*n*x^2*Log[c*x^n]^2 - 216*a*b^2*e
^3*x^3*Log[c*x^n]^2 + 144*b^3*e^3*n*x^3*Log[c*x^n]^2 - 216*b^3*e*x*Log[c*x^n]^3 + 108*b^3*e^2*x^2*Log[c*x^n]^3
- 72*b^3*e^3*x^3*Log[c*x^n]^3 + 216*a^3*Log[1 + e*x] - 216*a^2*b*n*Log[1 + e*x] + 144*a*b^2*n^2*Log[1 + e*x]
- 48*b^3*n^3*Log[1 + e*x] + 216*a^3*e^3*x^3*Log[1 + e*x] - 216*a^2*b*e^3*n*x^3*Log[1 + e*x] + 144*a*b^2*e^3*n^
2*x^3*Log[1 + e*x] - 48*b^3*e^3*n^3*x^3*Log[1 + e*x] + 648*a^2*b*Log[c*x^n]*Log[1 + e*x] - 432*a*b^2*n*Log[c*x
^n]*Log[1 + e*x] + 144*b^3*n^2*Log[c*x^n]*Log[1 + e*x] + 648*a^2*b*e^3*x^3*Log[c*x^n]*Log[1 + e*x] - 432*a*b^2
*e^3*n*x^3*Log[c*x^n]*Log[1 + e*x] + 144*b^3*e^3*n^2*x^3*Log[c*x^n]*Log[1 + e*x] + 648*a*b^2*Log[c*x^n]^2*Log[
1 + e*x] - 216*b^3*n*Log[c*x^n]^2*Log[1 + e*x] + 648*a*b^2*e^3*x^3*Log[c*x^n]^2*Log[1 + e*x] - 216*b^3*e^3*n*x
^3*Log[c*x^n]^2*Log[1 + e*x] + 216*b^3*Log[c*x^n]^3*Log[1 + e*x] + 216*b^3*e^3*x^3*Log[c*x^n]^3*Log[1 + e*x] +
72*b*n*(9*a^2 - 6*a*b*n + 2*b^2*n^2 - 6*b*(-3*a + b*n)*Log[c*x^n] + 9*b^2*Log[c*x^n]^2)*PolyLog[2, -(e*x)] +
432*b^2*n^2*(-3*a + b*n - 3*b*Log[c*x^n])*PolyLog[3, -(e*x)] + 1296*b^3*n^3*PolyLog[4, -(e*x)])/(648*e^3)

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Maple [F]  time = 0.131, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \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^2*(a+b*ln(c*x^n))^3*ln(e*x+1),x)

[Out]

int(x^2*(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 (2 \, b^{3} e^{3} x^{3} - 3 \, b^{3} e^{2} x^{2} + 6 \, b^{3} e x - 6 \,{\left (b^{3} e^{3} x^{3} + b^{3}\right )} \log \left (e x + 1\right )\right )} \log \left (x^{n}\right )^{3}}{18 \, e^{3}} + \frac{\frac{1}{3} \,{\left (6 \, x^{3} \log \left (e x + 1\right ) - e{\left (\frac{2 \, e^{2} x^{3} - 3 \, e x^{2} + 6 \, x}{e^{3}} - \frac{6 \, \log \left (e x + 1\right )}{e^{4}}\right )}\right )} b^{3} e^{3} \log \left (c\right )^{3} +{\left (6 \, x^{3} \log \left (e x + 1\right ) - e{\left (\frac{2 \, e^{2} x^{3} - 3 \, e x^{2} + 6 \, x}{e^{3}} - \frac{6 \, \log \left (e x + 1\right )}{e^{4}}\right )}\right )} a b^{2} e^{3} \log \left (c\right )^{2} +{\left (6 \, x^{3} \log \left (e x + 1\right ) - e{\left (\frac{2 \, e^{2} x^{3} - 3 \, e x^{2} + 6 \, x}{e^{3}} - \frac{6 \, \log \left (e x + 1\right )}{e^{4}}\right )}\right )} a^{2} b e^{3} \log \left (c\right ) + \frac{1}{3} \,{\left (6 \, x^{3} \log \left (e x + 1\right ) - e{\left (\frac{2 \, e^{2} x^{3} - 3 \, e x^{2} + 6 \, x}{e^{3}} - \frac{6 \, \log \left (e x + 1\right )}{e^{4}}\right )}\right )} a^{3} e^{3} + \int \frac{18 \,{\left (b^{3} e^{3} \log \left (c\right )^{2} + 2 \, a b^{2} e^{3} \log \left (c\right ) + a^{2} b e^{3}\right )} x^{3} \log \left (e x + 1\right ) \log \left (x^{n}\right ) +{\left (2 \, b^{3} e^{3} n x^{3} - 3 \, b^{3} e^{2} n x^{2} + 6 \, b^{3} e n x - 6 \,{\left (b^{3} n -{\left (3 \, a b^{2} e^{3} -{\left (e^{3} n - 3 \, e^{3} \log \left (c\right )\right )} b^{3}\right )} x^{3}\right )} \log \left (e x + 1\right )\right )} \log \left (x^{n}\right )^{2}}{x}\,{d x}}{6 \, e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*x^n))^3*log(e*x+1),x, algorithm="maxima")

[Out]

-1/18*(2*b^3*e^3*x^3 - 3*b^3*e^2*x^2 + 6*b^3*e*x - 6*(b^3*e^3*x^3 + b^3)*log(e*x + 1))*log(x^n)^3/e^3 + 1/6*in
tegrate((18*(b^3*e^3*log(c)^2 + 2*a*b^2*e^3*log(c) + a^2*b*e^3)*x^3*log(e*x + 1)*log(x^n) + 6*(b^3*e^3*log(c)^
3 + 3*a*b^2*e^3*log(c)^2 + 3*a^2*b*e^3*log(c) + a^3*e^3)*x^3*log(e*x + 1) + (2*b^3*e^3*n*x^3 - 3*b^3*e^2*n*x^2
+ 6*b^3*e*n*x - 6*(b^3*n - (3*a*b^2*e^3 - (e^3*n - 3*e^3*log(c))*b^3)*x^3)*log(e*x + 1))*log(x^n)^2)/x, x)/e^
3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*x^n))^3*log(e*x+1),x, algorithm="fricas")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*x^n))^3*log(e*x+1),x, algorithm="giac")

[Out]

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