3.1.19 \(\int x (a+b \log (c x^n))^3 \log (1+e x) \, dx\) [19]

Optimal. Leaf size=530 \[ \frac {9 a b^2 n^2 x}{2 e}-\frac {45 b^3 n^3 x}{8 e}+\frac {3}{4} b^3 n^3 x^2+\frac {9 b^3 n^2 x \log \left (c x^n\right )}{2 e}+\frac {3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{4 e}-\frac {9}{8} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{4 e}+\frac {3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{2 e}-\frac {1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )^3+\frac {3 b^3 n^3 \log (1+e x)}{8 e^2}-\frac {3}{8} b^3 n^3 x^2 \log (1+e x)-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{4 e^2}+\frac {3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{4 e^2}-\frac {3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{2 e^2}+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)-\frac {3 b^3 n^3 \text {Li}_2(-e x)}{4 e^2}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{2 e^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2(-e x)}{2 e^2}-\frac {3 b^3 n^3 \text {Li}_3(-e x)}{2 e^2}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(-e x)}{e^2}-\frac {3 b^3 n^3 \text {Li}_4(-e x)}{e^2} \]

[Out]

9/2*a*b^2*n^2*x/e-45/8*b^3*n^3*x/e+3/4*b^3*n^3*x^2+9/2*b^3*n^2*x*ln(c*x^n)/e+3/4*b^2*n^2*x*(a+b*ln(c*x^n))/e-9
/8*b^2*n^2*x^2*(a+b*ln(c*x^n))-9/4*b*n*x*(a+b*ln(c*x^n))^2/e+3/4*b*n*x^2*(a+b*ln(c*x^n))^2+1/2*x*(a+b*ln(c*x^n
))^3/e-1/4*x^2*(a+b*ln(c*x^n))^3+3/8*b^3*n^3*ln(e*x+1)/e^2-3/8*b^3*n^3*x^2*ln(e*x+1)-3/4*b^2*n^2*(a+b*ln(c*x^n
))*ln(e*x+1)/e^2+3/4*b^2*n^2*x^2*(a+b*ln(c*x^n))*ln(e*x+1)+3/4*b*n*(a+b*ln(c*x^n))^2*ln(e*x+1)/e^2-3/4*b*n*x^2
*(a+b*ln(c*x^n))^2*ln(e*x+1)-1/2*(a+b*ln(c*x^n))^3*ln(e*x+1)/e^2+1/2*x^2*(a+b*ln(c*x^n))^3*ln(e*x+1)-3/4*b^3*n
^3*polylog(2,-e*x)/e^2+3/2*b^2*n^2*(a+b*ln(c*x^n))*polylog(2,-e*x)/e^2-3/2*b*n*(a+b*ln(c*x^n))^2*polylog(2,-e*
x)/e^2-3/2*b^3*n^3*polylog(3,-e*x)/e^2+3*b^2*n^2*(a+b*ln(c*x^n))*polylog(3,-e*x)/e^2-3*b^3*n^3*polylog(4,-e*x)
/e^2

________________________________________________________________________________________

Rubi [A]
time = 0.34, antiderivative size = 530, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 12, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {2442, 45, 2424, 2333, 2332, 2342, 2341, 2421, 2430, 6724, 2423, 2438} \begin {gather*} \frac {3 b^2 n^2 \text {PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )}{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 {x \left (a+b \log \left (c x^n\right )\right )^3}{2 e}-\frac {3}{4} b n x^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2+\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 \end {gather*}

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 45

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

Rule 2332

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

Rule 2333

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2342

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

Rule 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c*x^n])^p/m), x] + Dist[b*n*(p/m), Int[PolyLog[2, (-d)*f*x^m]*((a + b*L
og[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2423

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

Rule 2424

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

Rule 2430

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[PolyLo
g[k + 1, e*x^q]*((a + b*Log[c*x^n])^p/q), x] - Dist[b*n*(p/q), Int[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^(
p - 1)/x), x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]

Rule 2438

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

Rule 2442

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

Rule 6724

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

Rubi steps

\begin {align*} \int x \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*}

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Mathematica [A]
time = 0.14, size = 806, normalized size = 1.52 \begin {gather*} \frac {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 \left (c x^n\right )-36 a b^2 e n x \log \left (c x^n\right )+42 b^3 e n^2 x \log \left (c x^n\right )-6 a^2 b e^2 x^2 \log \left (c x^n\right )+12 a b^2 e^2 n x^2 \log \left (c x^n\right )-9 b^3 e^2 n^2 x^2 \log \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 )-6 a b^2 e^2 x^2 \log ^2\left (c x^n\right )+6 b^3 e^2 n x^2 \log ^2\left (c x^n\right )+4 b^3 e x \log ^3\left (c x^n\right )-2 b^3 e^2 x^2 \log ^3\left (c x^n\right )-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 \left (c x^n\right ) \log (1+e x)+12 a b^2 n \log \left (c x^n\right ) \log (1+e x)-6 b^3 n^2 \log \left (c x^n\right ) \log (1+e x)+12 a^2 b e^2 x^2 \log \left (c x^n\right ) \log (1+e x)-12 a b^2 e^2 n x^2 \log \left (c x^n\right ) \log (1+e x)+6 b^3 e^2 n^2 x^2 \log \left (c x^n\right ) \log (1+e x)-12 a b^2 \log ^2\left (c x^n\right ) \log (1+e x)+6 b^3 n \log ^2\left (c x^n\right ) \log (1+e x)+12 a b^2 e^2 x^2 \log ^2\left (c x^n\right ) \log (1+e x)-6 b^3 e^2 n x^2 \log ^2\left (c x^n\right ) \log (1+e x)-4 b^3 \log ^3\left (c x^n\right ) \log (1+e x)+4 b^3 e^2 x^2 \log ^3\left (c x^n\right ) \log (1+e x)-6 b n \left (2 a^2-2 a b n+b^2 n^2-2 b (-2 a+b n) \log \left (c x^n\right )+2 b^2 \log ^2\left (c x^n\right )\right ) \text {Li}_2(-e x)+12 b^2 n^2 \left (2 a-b n+2 b \log \left (c x^n\right )\right ) \text {Li}_3(-e x)-24 b^3 n^3 \text {Li}_4(-e x)}{8 e^2} \end {gather*}

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(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)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(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(x*e + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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