3.22 \(\int \frac{(a+b \log (c x^n))^3 \log (1+e x)}{x^2} \, dx\)

Optimal. Leaf size=342 \[ 6 b^2 e n^2 \text{PolyLog}\left (2,-\frac{1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )+6 b^2 e n^2 \text{PolyLog}\left (3,-\frac{1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )+3 b e n \text{PolyLog}\left (2,-\frac{1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+6 b^3 e n^3 \text{PolyLog}\left (2,-\frac{1}{e x}\right )+6 b^3 e n^3 \text{PolyLog}\left (3,-\frac{1}{e x}\right )+6 b^3 e n^3 \text{PolyLog}\left (4,-\frac{1}{e x}\right )-6 b^2 e n^2 \log \left (\frac{1}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{6 b^2 n^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{x}-3 b e n \log \left (\frac{1}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{3 b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{x}-e \log \left (\frac{1}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac{\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3}{x}+6 b^3 e n^3 \log (x)-6 b^3 e n^3 \log (e x+1)-\frac{6 b^3 n^3 \log (e x+1)}{x} \]

[Out]

6*b^3*e*n^3*Log[x] - 6*b^2*e*n^2*Log[1 + 1/(e*x)]*(a + b*Log[c*x^n]) - 3*b*e*n*Log[1 + 1/(e*x)]*(a + b*Log[c*x
^n])^2 - e*Log[1 + 1/(e*x)]*(a + b*Log[c*x^n])^3 - 6*b^3*e*n^3*Log[1 + e*x] - (6*b^3*n^3*Log[1 + e*x])/x - (6*
b^2*n^2*(a + b*Log[c*x^n])*Log[1 + e*x])/x - (3*b*n*(a + b*Log[c*x^n])^2*Log[1 + e*x])/x - ((a + b*Log[c*x^n])
^3*Log[1 + e*x])/x + 6*b^3*e*n^3*PolyLog[2, -(1/(e*x))] + 6*b^2*e*n^2*(a + b*Log[c*x^n])*PolyLog[2, -(1/(e*x))
] + 3*b*e*n*(a + b*Log[c*x^n])^2*PolyLog[2, -(1/(e*x))] + 6*b^3*e*n^3*PolyLog[3, -(1/(e*x))] + 6*b^2*e*n^2*(a
+ b*Log[c*x^n])*PolyLog[3, -(1/(e*x))] + 6*b^3*e*n^3*PolyLog[4, -(1/(e*x))]

________________________________________________________________________________________

Rubi [A]  time = 0.601301, antiderivative size = 360, normalized size of antiderivative = 1.05, number of steps used = 22, number of rules used = 15, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.682, Rules used = {2305, 2304, 2378, 36, 29, 31, 2344, 2301, 2317, 2391, 2302, 30, 2374, 6589, 2383} \[ -6 b^2 e n^2 \text{PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )+6 b^2 e n^2 \text{PolyLog}(3,-e x) \left (a+b \log \left (c x^n\right )\right )-3 b e n \text{PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )^2-6 b^3 e n^3 \text{PolyLog}(2,-e x)+6 b^3 e n^3 \text{PolyLog}(3,-e x)-6 b^3 e n^3 \text{PolyLog}(4,-e x)-6 b^2 e n^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )-\frac{6 b^2 n^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{x}+\frac{e \left (a+b \log \left (c x^n\right )\right )^4}{4 b n}+e \left (a+b \log \left (c x^n\right )\right )^3-e \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3-\frac{\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3}{x}+3 b e n \left (a+b \log \left (c x^n\right )\right )^2-3 b e n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2-\frac{3 b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{x}+6 b^3 e n^3 \log (x)-6 b^3 e n^3 \log (e x+1)-\frac{6 b^3 n^3 \log (e x+1)}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

6*b^3*e*n^3*Log[x] + 3*b*e*n*(a + b*Log[c*x^n])^2 + e*(a + b*Log[c*x^n])^3 + (e*(a + b*Log[c*x^n])^4)/(4*b*n)
- 6*b^3*e*n^3*Log[1 + e*x] - (6*b^3*n^3*Log[1 + e*x])/x - 6*b^2*e*n^2*(a + b*Log[c*x^n])*Log[1 + e*x] - (6*b^2
*n^2*(a + b*Log[c*x^n])*Log[1 + e*x])/x - 3*b*e*n*(a + b*Log[c*x^n])^2*Log[1 + e*x] - (3*b*n*(a + b*Log[c*x^n]
)^2*Log[1 + e*x])/x - e*(a + b*Log[c*x^n])^3*Log[1 + e*x] - ((a + b*Log[c*x^n])^3*Log[1 + e*x])/x - 6*b^3*e*n^
3*PolyLog[2, -(e*x)] - 6*b^2*e*n^2*(a + b*Log[c*x^n])*PolyLog[2, -(e*x)] - 3*b*e*n*(a + b*Log[c*x^n])^2*PolyLo
g[2, -(e*x)] + 6*b^3*e*n^3*PolyLog[3, -(e*x)] + 6*b^2*e*n^2*(a + b*Log[c*x^n])*PolyLog[3, -(e*x)] - 6*b^3*e*n^
3*PolyLog[4, -(e*x)]

Rule 2305

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

Rule 2304

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

Rule 2378

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

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2344

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

Rule 2301

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

Rule 2317

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

Rule 2391

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

Rule 2302

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

Rule 30

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

Rule 2374

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

Rule 6589

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

Rule 2383

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

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x^2} \, dx &=-\frac{6 b^3 n^3 \log (1+e x)}{x}-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x}-e \int \left (-\frac{6 b^3 n^3}{x (1+e x)}-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{x (1+e x)}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2}{x (1+e x)}-\frac{\left (a+b \log \left (c x^n\right )\right )^3}{x (1+e x)}\right ) \, dx\\ &=-\frac{6 b^3 n^3 \log (1+e x)}{x}-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x}+e \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{x (1+e x)} \, dx+(3 b e n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x (1+e x)} \, dx+\left (6 b^2 e n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{x (1+e x)} \, dx+\left (6 b^3 e n^3\right ) \int \frac{1}{x (1+e x)} \, dx\\ &=-\frac{6 b^3 n^3 \log (1+e x)}{x}-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x}+e \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{x} \, dx-e^2 \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{1+e x} \, dx+(3 b e n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx-\left (3 b e^2 n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{1+e x} \, dx+\left (6 b^2 e n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{x} \, dx-\left (6 b^2 e^2 n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{1+e x} \, dx+\left (6 b^3 e n^3\right ) \int \frac{1}{x} \, dx-\left (6 b^3 e^2 n^3\right ) \int \frac{1}{1+e x} \, dx\\ &=6 b^3 e n^3 \log (x)+3 b e n \left (a+b \log \left (c x^n\right )\right )^2-6 b^3 e n^3 \log (1+e x)-\frac{6 b^3 n^3 \log (1+e x)}{x}-6 b^2 e n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x}-3 b e n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x}-e \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x}+(3 e) \operatorname{Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )+\frac{e \operatorname{Subst}\left (\int x^3 \, dx,x,a+b \log \left (c x^n\right )\right )}{b n}+(3 b e n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x} \, dx+\left (6 b^2 e n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x} \, dx+\left (6 b^3 e n^3\right ) \int \frac{\log (1+e x)}{x} \, dx\\ &=6 b^3 e n^3 \log (x)+3 b e n \left (a+b \log \left (c x^n\right )\right )^2+e \left (a+b \log \left (c x^n\right )\right )^3+\frac{e \left (a+b \log \left (c x^n\right )\right )^4}{4 b n}-6 b^3 e n^3 \log (1+e x)-\frac{6 b^3 n^3 \log (1+e x)}{x}-6 b^2 e n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x}-3 b e n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x}-e \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x}-6 b^3 e n^3 \text{Li}_2(-e x)-6 b^2 e n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)-3 b e n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2(-e x)+\left (6 b^2 e n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)}{x} \, dx+\left (6 b^3 e n^3\right ) \int \frac{\text{Li}_2(-e x)}{x} \, dx\\ &=6 b^3 e n^3 \log (x)+3 b e n \left (a+b \log \left (c x^n\right )\right )^2+e \left (a+b \log \left (c x^n\right )\right )^3+\frac{e \left (a+b \log \left (c x^n\right )\right )^4}{4 b n}-6 b^3 e n^3 \log (1+e x)-\frac{6 b^3 n^3 \log (1+e x)}{x}-6 b^2 e n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x}-3 b e n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x}-e \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x}-6 b^3 e n^3 \text{Li}_2(-e x)-6 b^2 e n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)-3 b e n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2(-e x)+6 b^3 e n^3 \text{Li}_3(-e x)+6 b^2 e n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(-e x)-\left (6 b^3 e n^3\right ) \int \frac{\text{Li}_3(-e x)}{x} \, dx\\ &=6 b^3 e n^3 \log (x)+3 b e n \left (a+b \log \left (c x^n\right )\right )^2+e \left (a+b \log \left (c x^n\right )\right )^3+\frac{e \left (a+b \log \left (c x^n\right )\right )^4}{4 b n}-6 b^3 e n^3 \log (1+e x)-\frac{6 b^3 n^3 \log (1+e x)}{x}-6 b^2 e n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x}-3 b e n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x}-e \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x}-6 b^3 e n^3 \text{Li}_2(-e x)-6 b^2 e n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)-3 b e n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2(-e x)+6 b^3 e n^3 \text{Li}_3(-e x)+6 b^2 e n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(-e x)-6 b^3 e n^3 \text{Li}_4(-e x)\\ \end{align*}

Mathematica [B]  time = 0.302027, size = 770, normalized size = 2.25 \[ -3 b e n \text{PolyLog}(2,-e x) \left (a^2+2 b (a+b n) \log \left (c x^n\right )+2 a b n+b^2 \log ^2\left (c x^n\right )+2 b^2 n^2\right )+6 b^2 e n^2 \text{PolyLog}(3,-e x) \left (a+b \log \left (c x^n\right )+b n\right )-6 b^3 e n^3 \text{PolyLog}(4,-e x)+3 a^2 b e \log (x) \log \left (c x^n\right )-3 a^2 b e \log (e x+1) \log \left (c x^n\right )-\frac{3 a^2 b \log (e x+1) \log \left (c x^n\right )}{x}-\frac{3}{2} a^2 b e n \log ^2(x)+3 a^2 b e n \log (x)-3 a^2 b e n \log (e x+1)-\frac{3 a^2 b n \log (e x+1)}{x}+a^3 e \log (x)-a^3 e \log (e x+1)-\frac{a^3 \log (e x+1)}{x}-3 a b^2 e n \log ^2(x) \log \left (c x^n\right )+3 a b^2 e \log (x) \log ^2\left (c x^n\right )-3 a b^2 e \log (e x+1) \log ^2\left (c x^n\right )-\frac{3 a b^2 \log (e x+1) \log ^2\left (c x^n\right )}{x}+6 a b^2 e n \log (x) \log \left (c x^n\right )-6 a b^2 e n \log (e x+1) \log \left (c x^n\right )-\frac{6 a b^2 n \log (e x+1) \log \left (c x^n\right )}{x}+a b^2 e n^2 \log ^3(x)-3 a b^2 e n^2 \log ^2(x)+6 a b^2 e n^2 \log (x)-6 a b^2 e n^2 \log (e x+1)-\frac{6 a b^2 n^2 \log (e x+1)}{x}+b^3 e n^2 \log ^3(x) \log \left (c x^n\right )-3 b^3 e n^2 \log ^2(x) \log \left (c x^n\right )+6 b^3 e n^2 \log (x) \log \left (c x^n\right )-6 b^3 e n^2 \log (e x+1) \log \left (c x^n\right )-\frac{6 b^3 n^2 \log (e x+1) \log \left (c x^n\right )}{x}-\frac{3}{2} b^3 e n \log ^2(x) \log ^2\left (c x^n\right )+b^3 e \log (x) \log ^3\left (c x^n\right )+3 b^3 e n \log (x) \log ^2\left (c x^n\right )-b^3 e \log (e x+1) \log ^3\left (c x^n\right )-\frac{b^3 \log (e x+1) \log ^3\left (c x^n\right )}{x}-3 b^3 e n \log (e x+1) \log ^2\left (c x^n\right )-\frac{3 b^3 n \log (e x+1) \log ^2\left (c x^n\right )}{x}-\frac{1}{4} b^3 e n^3 \log ^4(x)+b^3 e n^3 \log ^3(x)-3 b^3 e n^3 \log ^2(x)+6 b^3 e n^3 \log (x)-6 b^3 e n^3 \log (e x+1)-\frac{6 b^3 n^3 \log (e x+1)}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^3*e*Log[x] + 3*a^2*b*e*n*Log[x] + 6*a*b^2*e*n^2*Log[x] + 6*b^3*e*n^3*Log[x] - (3*a^2*b*e*n*Log[x]^2)/2 - 3*a
*b^2*e*n^2*Log[x]^2 - 3*b^3*e*n^3*Log[x]^2 + a*b^2*e*n^2*Log[x]^3 + b^3*e*n^3*Log[x]^3 - (b^3*e*n^3*Log[x]^4)/
4 + 3*a^2*b*e*Log[x]*Log[c*x^n] + 6*a*b^2*e*n*Log[x]*Log[c*x^n] + 6*b^3*e*n^2*Log[x]*Log[c*x^n] - 3*a*b^2*e*n*
Log[x]^2*Log[c*x^n] - 3*b^3*e*n^2*Log[x]^2*Log[c*x^n] + b^3*e*n^2*Log[x]^3*Log[c*x^n] + 3*a*b^2*e*Log[x]*Log[c
*x^n]^2 + 3*b^3*e*n*Log[x]*Log[c*x^n]^2 - (3*b^3*e*n*Log[x]^2*Log[c*x^n]^2)/2 + b^3*e*Log[x]*Log[c*x^n]^3 - a^
3*e*Log[1 + e*x] - 3*a^2*b*e*n*Log[1 + e*x] - 6*a*b^2*e*n^2*Log[1 + e*x] - 6*b^3*e*n^3*Log[1 + e*x] - (a^3*Log
[1 + e*x])/x - (3*a^2*b*n*Log[1 + e*x])/x - (6*a*b^2*n^2*Log[1 + e*x])/x - (6*b^3*n^3*Log[1 + e*x])/x - 3*a^2*
b*e*Log[c*x^n]*Log[1 + e*x] - 6*a*b^2*e*n*Log[c*x^n]*Log[1 + e*x] - 6*b^3*e*n^2*Log[c*x^n]*Log[1 + e*x] - (3*a
^2*b*Log[c*x^n]*Log[1 + e*x])/x - (6*a*b^2*n*Log[c*x^n]*Log[1 + e*x])/x - (6*b^3*n^2*Log[c*x^n]*Log[1 + e*x])/
x - 3*a*b^2*e*Log[c*x^n]^2*Log[1 + e*x] - 3*b^3*e*n*Log[c*x^n]^2*Log[1 + e*x] - (3*a*b^2*Log[c*x^n]^2*Log[1 +
e*x])/x - (3*b^3*n*Log[c*x^n]^2*Log[1 + e*x])/x - b^3*e*Log[c*x^n]^3*Log[1 + e*x] - (b^3*Log[c*x^n]^3*Log[1 +
e*x])/x - 3*b*e*n*(a^2 + 2*a*b*n + 2*b^2*n^2 + 2*b*(a + b*n)*Log[c*x^n] + b^2*Log[c*x^n]^2)*PolyLog[2, -(e*x)]
+ 6*b^2*e*n^2*(a + b*n + b*Log[c*x^n])*PolyLog[3, -(e*x)] - 6*b^3*e*n^3*PolyLog[4, -(e*x)]

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Maple [F]  time = 0.141, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{3}\ln \left ( ex+1 \right ) }{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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