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

Optimal. Leaf size=456 \[ -\frac{b n \text{PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )}{2 e^4}+\frac{b^2 n^2 \text{PolyLog}(2,-e x)}{8 e^4}+\frac{b^2 n^2 \text{PolyLog}(3,-e x)}{2 e^4}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}+\frac{3 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{16 e^2}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 e^3}-\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{8 e^3}-\frac{\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{4 e^4}+\frac{b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{8 e^4}+\frac{1}{4} x^4 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{8} b n x^4 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}-\frac{7 b n x^3 \left (a+b \log \left (c x^n\right )\right )}{72 e}-\frac{1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )-\frac{a b n x}{2 e^3}-\frac{b^2 n x \log \left (c x^n\right )}{2 e^3}-\frac{7 b^2 n^2 x^2}{64 e^2}+\frac{21 b^2 n^2 x}{32 e^3}-\frac{b^2 n^2 \log (e x+1)}{32 e^4}+\frac{37 b^2 n^2 x^3}{864 e}+\frac{1}{32} b^2 n^2 x^4 \log (e x+1)-\frac{3}{128} b^2 n^2 x^4 \]

[Out]

-(a*b*n*x)/(2*e^3) + (21*b^2*n^2*x)/(32*e^3) - (7*b^2*n^2*x^2)/(64*e^2) + (37*b^2*n^2*x^3)/(864*e) - (3*b^2*n^
2*x^4)/128 - (b^2*n*x*Log[c*x^n])/(2*e^3) - (b*n*x*(a + b*Log[c*x^n]))/(8*e^3) + (3*b*n*x^2*(a + b*Log[c*x^n])
)/(16*e^2) - (7*b*n*x^3*(a + b*Log[c*x^n]))/(72*e) + (b*n*x^4*(a + b*Log[c*x^n]))/16 + (x*(a + b*Log[c*x^n])^2
)/(4*e^3) - (x^2*(a + b*Log[c*x^n])^2)/(8*e^2) + (x^3*(a + b*Log[c*x^n])^2)/(12*e) - (x^4*(a + b*Log[c*x^n])^2
)/16 - (b^2*n^2*Log[1 + e*x])/(32*e^4) + (b^2*n^2*x^4*Log[1 + e*x])/32 + (b*n*(a + b*Log[c*x^n])*Log[1 + e*x])
/(8*e^4) - (b*n*x^4*(a + b*Log[c*x^n])*Log[1 + e*x])/8 - ((a + b*Log[c*x^n])^2*Log[1 + e*x])/(4*e^4) + (x^4*(a
+ b*Log[c*x^n])^2*Log[1 + e*x])/4 + (b^2*n^2*PolyLog[2, -(e*x)])/(8*e^4) - (b*n*(a + b*Log[c*x^n])*PolyLog[2,
-(e*x)])/(2*e^4) + (b^2*n^2*PolyLog[3, -(e*x)])/(2*e^4)

________________________________________________________________________________________

Rubi [A]  time = 0.330808, antiderivative size = 456, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {2395, 43, 2377, 2295, 2304, 2374, 6589, 2376, 2391} \[ -\frac{b n \text{PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )}{2 e^4}+\frac{b^2 n^2 \text{PolyLog}(2,-e x)}{8 e^4}+\frac{b^2 n^2 \text{PolyLog}(3,-e x)}{2 e^4}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}+\frac{3 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{16 e^2}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 e^3}-\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{8 e^3}-\frac{\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{4 e^4}+\frac{b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{8 e^4}+\frac{1}{4} x^4 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{8} b n x^4 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}-\frac{7 b n x^3 \left (a+b \log \left (c x^n\right )\right )}{72 e}-\frac{1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )-\frac{a b n x}{2 e^3}-\frac{b^2 n x \log \left (c x^n\right )}{2 e^3}-\frac{7 b^2 n^2 x^2}{64 e^2}+\frac{21 b^2 n^2 x}{32 e^3}-\frac{b^2 n^2 \log (e x+1)}{32 e^4}+\frac{37 b^2 n^2 x^3}{864 e}+\frac{1}{32} b^2 n^2 x^4 \log (e x+1)-\frac{3}{128} b^2 n^2 x^4 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*b*n*x)/(2*e^3) + (21*b^2*n^2*x)/(32*e^3) - (7*b^2*n^2*x^2)/(64*e^2) + (37*b^2*n^2*x^3)/(864*e) - (3*b^2*n^
2*x^4)/128 - (b^2*n*x*Log[c*x^n])/(2*e^3) - (b*n*x*(a + b*Log[c*x^n]))/(8*e^3) + (3*b*n*x^2*(a + b*Log[c*x^n])
)/(16*e^2) - (7*b*n*x^3*(a + b*Log[c*x^n]))/(72*e) + (b*n*x^4*(a + b*Log[c*x^n]))/16 + (x*(a + b*Log[c*x^n])^2
)/(4*e^3) - (x^2*(a + b*Log[c*x^n])^2)/(8*e^2) + (x^3*(a + b*Log[c*x^n])^2)/(12*e) - (x^4*(a + b*Log[c*x^n])^2
)/16 - (b^2*n^2*Log[1 + e*x])/(32*e^4) + (b^2*n^2*x^4*Log[1 + e*x])/32 + (b*n*(a + b*Log[c*x^n])*Log[1 + e*x])
/(8*e^4) - (b*n*x^4*(a + b*Log[c*x^n])*Log[1 + e*x])/8 - ((a + b*Log[c*x^n])^2*Log[1 + e*x])/(4*e^4) + (x^4*(a
+ b*Log[c*x^n])^2*Log[1 + e*x])/4 + (b^2*n^2*PolyLog[2, -(e*x)])/(8*e^4) - (b*n*(a + b*Log[c*x^n])*PolyLog[2,
-(e*x)])/(2*e^4) + (b^2*n^2*PolyLog[3, -(e*x)])/(2*e^4)

Rule 2395

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

Rule 43

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

Rule 2377

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

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

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 6589

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

Rule 2376

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

Rule 2391

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

Rubi steps

\begin{align*} \int x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x) \, dx &=\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 e^3}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}-\frac{1}{16} x^4 \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}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-(2 b n) \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{x \left (a+b \log \left (c x^n\right )\right )^2}{4 e^3}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}-\frac{1}{16} x^4 \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}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)+\frac{1}{8} (b n) \int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx-\frac{1}{2} (b n) \int x^3 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x) \, dx+\frac{(b n) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x} \, dx}{2 e^4}-\frac{(b n) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{2 e^3}+\frac{(b n) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{4 e^2}-\frac{(b n) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx}{6 e}\\ &=-\frac{a b n x}{2 e^3}-\frac{b^2 n^2 x^2}{16 e^2}+\frac{b^2 n^2 x^3}{54 e}-\frac{1}{128} b^2 n^2 x^4-\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{8 e^3}+\frac{3 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{16 e^2}-\frac{7 b n x^3 \left (a+b \log \left (c x^n\right )\right )}{72 e}+\frac{1}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 e^3}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}-\frac{1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{8 e^4}-\frac{1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{4 e^4}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)}{2 e^4}-\frac{\left (b^2 n\right ) \int \log \left (c x^n\right ) \, dx}{2 e^3}+\frac{1}{2} \left (b^2 n^2\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 (b^2 n^2\right ) \int \frac{\text{Li}_2(-e x)}{x} \, dx}{2 e^4}\\ &=-\frac{a b n x}{2 e^3}+\frac{5 b^2 n^2 x}{8 e^3}-\frac{3 b^2 n^2 x^2}{32 e^2}+\frac{7 b^2 n^2 x^3}{216 e}-\frac{1}{64} b^2 n^2 x^4-\frac{b^2 n x \log \left (c x^n\right )}{2 e^3}-\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{8 e^3}+\frac{3 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{16 e^2}-\frac{7 b n x^3 \left (a+b \log \left (c x^n\right )\right )}{72 e}+\frac{1}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 e^3}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}-\frac{1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{8 e^4}-\frac{1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{4 e^4}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)}{2 e^4}+\frac{b^2 n^2 \text{Li}_3(-e x)}{2 e^4}+\frac{1}{8} \left (b^2 n^2\right ) \int x^3 \log (1+e x) \, dx-\frac{\left (b^2 n^2\right ) \int \frac{\log (1+e x)}{x} \, dx}{8 e^4}\\ &=-\frac{a b n x}{2 e^3}+\frac{5 b^2 n^2 x}{8 e^3}-\frac{3 b^2 n^2 x^2}{32 e^2}+\frac{7 b^2 n^2 x^3}{216 e}-\frac{1}{64} b^2 n^2 x^4-\frac{b^2 n x \log \left (c x^n\right )}{2 e^3}-\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{8 e^3}+\frac{3 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{16 e^2}-\frac{7 b n x^3 \left (a+b \log \left (c x^n\right )\right )}{72 e}+\frac{1}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 e^3}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}-\frac{1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{32} b^2 n^2 x^4 \log (1+e x)+\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{8 e^4}-\frac{1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{4 e^4}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)+\frac{b^2 n^2 \text{Li}_2(-e x)}{8 e^4}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)}{2 e^4}+\frac{b^2 n^2 \text{Li}_3(-e x)}{2 e^4}-\frac{1}{32} \left (b^2 e n^2\right ) \int \frac{x^4}{1+e x} \, dx\\ &=-\frac{a b n x}{2 e^3}+\frac{5 b^2 n^2 x}{8 e^3}-\frac{3 b^2 n^2 x^2}{32 e^2}+\frac{7 b^2 n^2 x^3}{216 e}-\frac{1}{64} b^2 n^2 x^4-\frac{b^2 n x \log \left (c x^n\right )}{2 e^3}-\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{8 e^3}+\frac{3 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{16 e^2}-\frac{7 b n x^3 \left (a+b \log \left (c x^n\right )\right )}{72 e}+\frac{1}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 e^3}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}-\frac{1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{32} b^2 n^2 x^4 \log (1+e x)+\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{8 e^4}-\frac{1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{4 e^4}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)+\frac{b^2 n^2 \text{Li}_2(-e x)}{8 e^4}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)}{2 e^4}+\frac{b^2 n^2 \text{Li}_3(-e x)}{2 e^4}-\frac{1}{32} \left (b^2 e n^2\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{a b n x}{2 e^3}+\frac{21 b^2 n^2 x}{32 e^3}-\frac{7 b^2 n^2 x^2}{64 e^2}+\frac{37 b^2 n^2 x^3}{864 e}-\frac{3}{128} b^2 n^2 x^4-\frac{b^2 n x \log \left (c x^n\right )}{2 e^3}-\frac{b n x \left (a+b \log \left (c x^n\right )\right )}{8 e^3}+\frac{3 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{16 e^2}-\frac{7 b n x^3 \left (a+b \log \left (c x^n\right )\right )}{72 e}+\frac{1}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 e^3}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}-\frac{1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )^2-\frac{b^2 n^2 \log (1+e x)}{32 e^4}+\frac{1}{32} b^2 n^2 x^4 \log (1+e x)+\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{8 e^4}-\frac{1}{8} b n x^4 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{4 e^4}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)+\frac{b^2 n^2 \text{Li}_2(-e x)}{8 e^4}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)}{2 e^4}+\frac{b^2 n^2 \text{Li}_3(-e x)}{2 e^4}\\ \end{align*}

Mathematica [A]  time = 0.192894, size = 594, normalized size = 1.3 \[ \frac{432 b n \text{PolyLog}(2,-e x) \left (-4 a-4 b \log \left (c x^n\right )+b n\right )+1728 b^2 n^2 \text{PolyLog}(3,-e x)-216 a^2 e^4 x^4+288 a^2 e^3 x^3-432 a^2 e^2 x^2+864 a^2 e^4 x^4 \log (e x+1)+864 a^2 e x-864 a^2 \log (e x+1)-432 a b e^4 x^4 \log \left (c x^n\right )+1728 a b e^4 x^4 \log (e x+1) \log \left (c x^n\right )+576 a b e^3 x^3 \log \left (c x^n\right )-864 a b e^2 x^2 \log \left (c x^n\right )+1728 a b e x \log \left (c x^n\right )-1728 a b \log (e x+1) \log \left (c x^n\right )+216 a b e^4 n x^4-336 a b e^3 n x^3+648 a b e^2 n x^2-432 a b e^4 n x^4 \log (e x+1)-2160 a b e n x+432 a b n \log (e x+1)-216 b^2 e^4 x^4 \log ^2\left (c x^n\right )+864 b^2 e^4 x^4 \log (e x+1) \log ^2\left (c x^n\right )+288 b^2 e^3 x^3 \log ^2\left (c x^n\right )-432 b^2 e^2 x^2 \log ^2\left (c x^n\right )+216 b^2 e^4 n x^4 \log \left (c x^n\right )-432 b^2 e^4 n x^4 \log (e x+1) \log \left (c x^n\right )-336 b^2 e^3 n x^3 \log \left (c x^n\right )+648 b^2 e^2 n x^2 \log \left (c x^n\right )+864 b^2 e x \log ^2\left (c x^n\right )-864 b^2 \log (e x+1) \log ^2\left (c x^n\right )-2160 b^2 e n x \log \left (c x^n\right )+432 b^2 n \log (e x+1) \log \left (c x^n\right )-81 b^2 e^4 n^2 x^4+148 b^2 e^3 n^2 x^3-378 b^2 e^2 n^2 x^2+108 b^2 e^4 n^2 x^4 \log (e x+1)+2268 b^2 e n^2 x-108 b^2 n^2 \log (e x+1)}{3456 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(864*a^2*e*x - 2160*a*b*e*n*x + 2268*b^2*e*n^2*x - 432*a^2*e^2*x^2 + 648*a*b*e^2*n*x^2 - 378*b^2*e^2*n^2*x^2 +
288*a^2*e^3*x^3 - 336*a*b*e^3*n*x^3 + 148*b^2*e^3*n^2*x^3 - 216*a^2*e^4*x^4 + 216*a*b*e^4*n*x^4 - 81*b^2*e^4*
n^2*x^4 + 1728*a*b*e*x*Log[c*x^n] - 2160*b^2*e*n*x*Log[c*x^n] - 864*a*b*e^2*x^2*Log[c*x^n] + 648*b^2*e^2*n*x^2
*Log[c*x^n] + 576*a*b*e^3*x^3*Log[c*x^n] - 336*b^2*e^3*n*x^3*Log[c*x^n] - 432*a*b*e^4*x^4*Log[c*x^n] + 216*b^2
*e^4*n*x^4*Log[c*x^n] + 864*b^2*e*x*Log[c*x^n]^2 - 432*b^2*e^2*x^2*Log[c*x^n]^2 + 288*b^2*e^3*x^3*Log[c*x^n]^2
- 216*b^2*e^4*x^4*Log[c*x^n]^2 - 864*a^2*Log[1 + e*x] + 432*a*b*n*Log[1 + e*x] - 108*b^2*n^2*Log[1 + e*x] + 8
64*a^2*e^4*x^4*Log[1 + e*x] - 432*a*b*e^4*n*x^4*Log[1 + e*x] + 108*b^2*e^4*n^2*x^4*Log[1 + e*x] - 1728*a*b*Log
[c*x^n]*Log[1 + e*x] + 432*b^2*n*Log[c*x^n]*Log[1 + e*x] + 1728*a*b*e^4*x^4*Log[c*x^n]*Log[1 + e*x] - 432*b^2*
e^4*n*x^4*Log[c*x^n]*Log[1 + e*x] - 864*b^2*Log[c*x^n]^2*Log[1 + e*x] + 864*b^2*e^4*x^4*Log[c*x^n]^2*Log[1 + e
*x] + 432*b*n*(-4*a + b*n - 4*b*Log[c*x^n])*PolyLog[2, -(e*x)] + 1728*b^2*n^2*PolyLog[3, -(e*x)])/(3456*e^4)

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Maple [F]  time = 0.189, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}\ln \left ( ex+1 \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (3 \, b^{2} e^{4} x^{4} - 4 \, b^{2} e^{3} x^{3} + 6 \, b^{2} e^{2} x^{2} - 12 \, b^{2} e x - 12 \,{\left (b^{2} e^{4} x^{4} - b^{2}\right )} \log \left (e x + 1\right )\right )} \log \left (x^{n}\right )^{2}}{48 \, e^{4}} + \frac{-\frac{3}{16} \, b^{2} e^{4} n^{2} x^{4} + \frac{3}{4} \, b^{2} e^{4} n x^{4} \log \left (x^{n}\right ) + \frac{4}{9} \, b^{2} e^{3} n^{2} x^{3} - \frac{4}{3} \, b^{2} e^{3} n x^{3} \log \left (x^{n}\right ) + \frac{1}{2} \,{\left (12 \, x^{4} \log \left (e x + 1\right ) - e{\left (\frac{3 \, e^{3} x^{4} - 4 \, e^{2} x^{3} + 6 \, e x^{2} - 12 \, x}{e^{4}} + \frac{12 \, \log \left (e x + 1\right )}{e^{5}}\right )}\right )} b^{2} e^{4} \log \left (c\right )^{2} - \frac{3}{2} \, b^{2} e^{2} n^{2} x^{2} +{\left (12 \, x^{4} \log \left (e x + 1\right ) - e{\left (\frac{3 \, e^{3} x^{4} - 4 \, e^{2} x^{3} + 6 \, e x^{2} - 12 \, x}{e^{4}} + \frac{12 \, \log \left (e x + 1\right )}{e^{5}}\right )}\right )} a b e^{4} \log \left (c\right ) + 3 \, b^{2} e^{2} n x^{2} \log \left (x^{n}\right ) + \frac{1}{2} \,{\left (12 \, x^{4} \log \left (e x + 1\right ) - e{\left (\frac{3 \, e^{3} x^{4} - 4 \, e^{2} x^{3} + 6 \, e x^{2} - 12 \, x}{e^{4}} + \frac{12 \, \log \left (e x + 1\right )}{e^{5}}\right )}\right )} a^{2} e^{4} + 12 \, b^{2} e n^{2} x - 12 \, b^{2} e n x \log \left (x^{n}\right ) + \int \frac{12 \,{\left ({\left (4 \, a b e^{4} -{\left (e^{4} n - 4 \, e^{4} \log \left (c\right )\right )} b^{2}\right )} x^{4} + b^{2} n\right )} \log \left (e x + 1\right ) \log \left (x^{n}\right )}{x}\,{d x}}{24 \, e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*(3*b^2*e^4*x^4 - 4*b^2*e^3*x^3 + 6*b^2*e^2*x^2 - 12*b^2*e*x - 12*(b^2*e^4*x^4 - b^2)*log(e*x + 1))*log(x
^n)^2/e^4 + 1/24*integrate((24*(b^2*e^4*log(c)^2 + 2*a*b*e^4*log(c) + a^2*e^4)*x^4*log(e*x + 1) + (3*b^2*e^4*n
*x^4 - 4*b^2*e^3*n*x^3 + 6*b^2*e^2*n*x^2 - 12*b^2*e*n*x + 12*((4*a*b*e^4 - (e^4*n - 4*e^4*log(c))*b^2)*x^4 + b
^2*n)*log(e*x + 1))*log(x^n))/x, x)/e^4

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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