3.373 \(\int \log (f x^m) (a+b \log (c (d+e x)^n))^3 \, dx\)

Optimal. Leaf size=522 \[ \frac{6 b^2 d m n^2 \text{PolyLog}\left (2,\frac{e x}{d}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e}+\frac{6 b^2 d m n^2 \text{PolyLog}\left (3,\frac{e x}{d}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e}-\frac{3 b d m n \text{PolyLog}\left (2,\frac{e x}{d}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{6 b^3 d m n^3 \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{e}-\frac{6 b^3 d m n^3 \text{PolyLog}\left (3,\frac{e x}{d}+1\right )}{e}-\frac{6 b^3 d m n^3 \text{PolyLog}\left (4,\frac{e x}{d}+1\right )}{e}+6 a b^2 n^2 x \log \left (f x^m\right )-12 a b^2 m n^2 x-6 b^2 m n^2 x (a-b n)-\frac{3 b n (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac{6 b m n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{3 b d m n \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac{d m \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac{6 b^3 n^2 (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}-\frac{18 b^3 m n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac{6 b^3 d m n^2 \log \left (-\frac{e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e}-6 b^3 n^3 x \log \left (f x^m\right )+18 b^3 m n^3 x \]

[Out]

-12*a*b^2*m*n^2*x + 18*b^3*m*n^3*x - 6*b^2*m*n^2*(a - b*n)*x + 6*a*b^2*n^2*x*Log[f*x^m] - 6*b^3*n^3*x*Log[f*x^
m] - (18*b^3*m*n^2*(d + e*x)*Log[c*(d + e*x)^n])/e - (6*b^3*d*m*n^2*Log[-((e*x)/d)]*Log[c*(d + e*x)^n])/e + (6
*b^3*n^2*(d + e*x)*Log[f*x^m]*Log[c*(d + e*x)^n])/e + (6*b*m*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/e + (3*
b*d*m*n*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n])^2)/e - (3*b*n*(d + e*x)*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n
])^2)/e - (m*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/e - (d*m*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n])^3)/e
+ ((d + e*x)*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^3)/e - (6*b^3*d*m*n^3*PolyLog[2, 1 + (e*x)/d])/e + (6*b^2*d
*m*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, 1 + (e*x)/d])/e - (3*b*d*m*n*(a + b*Log[c*(d + e*x)^n])^2*PolyLog
[2, 1 + (e*x)/d])/e - (6*b^3*d*m*n^3*PolyLog[3, 1 + (e*x)/d])/e + (6*b^2*d*m*n^2*(a + b*Log[c*(d + e*x)^n])*Po
lyLog[3, 1 + (e*x)/d])/e - (6*b^3*d*m*n^3*PolyLog[4, 1 + (e*x)/d])/e

________________________________________________________________________________________

Rubi [A]  time = 0.857952, antiderivative size = 522, normalized size of antiderivative = 1., number of steps used = 28, number of rules used = 13, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.565, Rules used = {2389, 2296, 2295, 2423, 2411, 43, 2351, 2317, 2391, 2353, 2374, 6589, 2383} \[ \frac{6 b^2 d m n^2 \text{PolyLog}\left (2,\frac{e x}{d}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e}+\frac{6 b^2 d m n^2 \text{PolyLog}\left (3,\frac{e x}{d}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e}-\frac{3 b d m n \text{PolyLog}\left (2,\frac{e x}{d}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{6 b^3 d m n^3 \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{e}-\frac{6 b^3 d m n^3 \text{PolyLog}\left (3,\frac{e x}{d}+1\right )}{e}-\frac{6 b^3 d m n^3 \text{PolyLog}\left (4,\frac{e x}{d}+1\right )}{e}+6 a b^2 n^2 x \log \left (f x^m\right )-12 a b^2 m n^2 x-6 b^2 m n^2 x (a-b n)-\frac{3 b n (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac{6 b m n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{3 b d m n \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac{d m \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac{6 b^3 n^2 (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}-\frac{18 b^3 m n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac{6 b^3 d m n^2 \log \left (-\frac{e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e}-6 b^3 n^3 x \log \left (f x^m\right )+18 b^3 m n^3 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

-12*a*b^2*m*n^2*x + 18*b^3*m*n^3*x - 6*b^2*m*n^2*(a - b*n)*x + 6*a*b^2*n^2*x*Log[f*x^m] - 6*b^3*n^3*x*Log[f*x^
m] - (18*b^3*m*n^2*(d + e*x)*Log[c*(d + e*x)^n])/e - (6*b^3*d*m*n^2*Log[-((e*x)/d)]*Log[c*(d + e*x)^n])/e + (6
*b^3*n^2*(d + e*x)*Log[f*x^m]*Log[c*(d + e*x)^n])/e + (6*b*m*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/e + (3*
b*d*m*n*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n])^2)/e - (3*b*n*(d + e*x)*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n
])^2)/e - (m*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/e - (d*m*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n])^3)/e
+ ((d + e*x)*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^3)/e - (6*b^3*d*m*n^3*PolyLog[2, 1 + (e*x)/d])/e + (6*b^2*d
*m*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, 1 + (e*x)/d])/e - (3*b*d*m*n*(a + b*Log[c*(d + e*x)^n])^2*PolyLog
[2, 1 + (e*x)/d])/e - (6*b^3*d*m*n^3*PolyLog[3, 1 + (e*x)/d])/e + (6*b^2*d*m*n^2*(a + b*Log[c*(d + e*x)^n])*Po
lyLog[3, 1 + (e*x)/d])/e - (6*b^3*d*m*n^3*PolyLog[4, 1 + (e*x)/d])/e

Rule 2389

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

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 2423

Int[Log[(f_.)*(x_)^(m_.)]*((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_), x_Symbol] :> With[{u = In
tHide[(a + b*Log[c*(d + e*x)^n])^p, x]}, Dist[Log[f*x^m], u, x] - Dist[m, Int[Dist[1/x, u, x], x], x]] /; Free
Q[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 1]

Rule 2411

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

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 2351

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c,
d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && IntegerQ[r]))

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 2353

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

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 \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx &=6 a b^2 n^2 x \log \left (f x^m\right )-6 b^3 n^3 x \log \left (f x^m\right )+\frac{6 b^3 n^2 (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}-\frac{3 b n (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-m \int \left (6 a b^2 n^2-6 b^3 n^3+\frac{6 b^3 n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e x}-\frac{3 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e x}+\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e x}\right ) \, dx\\ &=-6 b^2 m n^2 (a-b n) x+6 a b^2 n^2 x \log \left (f x^m\right )-6 b^3 n^3 x \log \left (f x^m\right )+\frac{6 b^3 n^2 (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}-\frac{3 b n (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac{m \int \frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{x} \, dx}{e}+\frac{(3 b m n) \int \frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x} \, dx}{e}-\frac{\left (6 b^3 m n^2\right ) \int \frac{(d+e x) \log \left (c (d+e x)^n\right )}{x} \, dx}{e}\\ &=-6 b^2 m n^2 (a-b n) x+6 a b^2 n^2 x \log \left (f x^m\right )-6 b^3 n^3 x \log \left (f x^m\right )+\frac{6 b^3 n^2 (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}-\frac{3 b n (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac{m \operatorname{Subst}\left (\int \frac{x \left (a+b \log \left (c x^n\right )\right )^3}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+e x\right )}{e^2}+\frac{(3 b m n) \operatorname{Subst}\left (\int \frac{x \left (a+b \log \left (c x^n\right )\right )^2}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+e x\right )}{e^2}-\frac{\left (6 b^3 m n^2\right ) \operatorname{Subst}\left (\int \frac{x \log \left (c x^n\right )}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+e x\right )}{e^2}\\ &=-6 b^2 m n^2 (a-b n) x+6 a b^2 n^2 x \log \left (f x^m\right )-6 b^3 n^3 x \log \left (f x^m\right )+\frac{6 b^3 n^2 (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}-\frac{3 b n (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac{m \operatorname{Subst}\left (\int \left (e \left (a+b \log \left (c x^n\right )\right )^3-\frac{d e \left (a+b \log \left (c x^n\right )\right )^3}{d-x}\right ) \, dx,x,d+e x\right )}{e^2}+\frac{(3 b m n) \operatorname{Subst}\left (\int \left (e \left (a+b \log \left (c x^n\right )\right )^2-\frac{d e \left (a+b \log \left (c x^n\right )\right )^2}{d-x}\right ) \, dx,x,d+e x\right )}{e^2}-\frac{\left (6 b^3 m n^2\right ) \operatorname{Subst}\left (\int \left (e \log \left (c x^n\right )-\frac{d e \log \left (c x^n\right )}{d-x}\right ) \, dx,x,d+e x\right )}{e^2}\\ &=-6 b^2 m n^2 (a-b n) x+6 a b^2 n^2 x \log \left (f x^m\right )-6 b^3 n^3 x \log \left (f x^m\right )+\frac{6 b^3 n^2 (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}-\frac{3 b n (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac{m \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x\right )}{e}+\frac{(d m) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{d-x} \, dx,x,d+e x\right )}{e}+\frac{(3 b m n) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e}-\frac{(3 b d m n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{d-x} \, dx,x,d+e x\right )}{e}-\frac{\left (6 b^3 m n^2\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e}+\frac{\left (6 b^3 d m n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (c x^n\right )}{d-x} \, dx,x,d+e x\right )}{e}\\ &=6 b^3 m n^3 x-6 b^2 m n^2 (a-b n) x+6 a b^2 n^2 x \log \left (f x^m\right )-6 b^3 n^3 x \log \left (f x^m\right )-\frac{6 b^3 m n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac{6 b^3 d m n^2 \log \left (-\frac{e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e}+\frac{6 b^3 n^2 (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}+\frac{3 b m n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{3 b d m n \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{3 b n (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac{d m \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac{(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac{(3 b m n) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e}+\frac{(3 b d m n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+e x\right )}{e}-\frac{\left (6 b^2 m n^2\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e}-\frac{\left (6 b^2 d m n^2\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+e x\right )}{e}+\frac{\left (6 b^3 d m n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+e x\right )}{e}\\ &=-6 a b^2 m n^2 x+6 b^3 m n^3 x-6 b^2 m n^2 (a-b n) x+6 a b^2 n^2 x \log \left (f x^m\right )-6 b^3 n^3 x \log \left (f x^m\right )-\frac{6 b^3 m n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac{6 b^3 d m n^2 \log \left (-\frac{e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e}+\frac{6 b^3 n^2 (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}+\frac{6 b m n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{3 b d m n \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{3 b n (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac{d m \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac{(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac{6 b^3 d m n^3 \text{Li}_2\left (1+\frac{e x}{d}\right )}{e}+\frac{6 b^2 d m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (1+\frac{e x}{d}\right )}{e}-\frac{3 b d m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text{Li}_2\left (1+\frac{e x}{d}\right )}{e}-\frac{\left (6 b^2 m n^2\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e}-\frac{\left (6 b^3 m n^2\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e}+\frac{\left (6 b^2 d m n^2\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (\frac{x}{d}\right )}{x} \, dx,x,d+e x\right )}{e}-\frac{\left (6 b^3 d m n^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{d}\right )}{x} \, dx,x,d+e x\right )}{e}\\ &=-12 a b^2 m n^2 x+12 b^3 m n^3 x-6 b^2 m n^2 (a-b n) x+6 a b^2 n^2 x \log \left (f x^m\right )-6 b^3 n^3 x \log \left (f x^m\right )-\frac{12 b^3 m n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac{6 b^3 d m n^2 \log \left (-\frac{e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e}+\frac{6 b^3 n^2 (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}+\frac{6 b m n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{3 b d m n \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{3 b n (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac{d m \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac{(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac{6 b^3 d m n^3 \text{Li}_2\left (1+\frac{e x}{d}\right )}{e}+\frac{6 b^2 d m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (1+\frac{e x}{d}\right )}{e}-\frac{3 b d m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text{Li}_2\left (1+\frac{e x}{d}\right )}{e}-\frac{6 b^3 d m n^3 \text{Li}_3\left (1+\frac{e x}{d}\right )}{e}+\frac{6 b^2 d m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_3\left (1+\frac{e x}{d}\right )}{e}-\frac{\left (6 b^3 m n^2\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e}-\frac{\left (6 b^3 d m n^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (\frac{x}{d}\right )}{x} \, dx,x,d+e x\right )}{e}\\ &=-12 a b^2 m n^2 x+18 b^3 m n^3 x-6 b^2 m n^2 (a-b n) x+6 a b^2 n^2 x \log \left (f x^m\right )-6 b^3 n^3 x \log \left (f x^m\right )-\frac{18 b^3 m n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac{6 b^3 d m n^2 \log \left (-\frac{e x}{d}\right ) \log \left (c (d+e x)^n\right )}{e}+\frac{6 b^3 n^2 (d+e x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )}{e}+\frac{6 b m n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{3 b d m n \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{3 b n (d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac{m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac{d m \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac{(d+e x) \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac{6 b^3 d m n^3 \text{Li}_2\left (1+\frac{e x}{d}\right )}{e}+\frac{6 b^2 d m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (1+\frac{e x}{d}\right )}{e}-\frac{3 b d m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text{Li}_2\left (1+\frac{e x}{d}\right )}{e}-\frac{6 b^3 d m n^3 \text{Li}_3\left (1+\frac{e x}{d}\right )}{e}+\frac{6 b^2 d m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_3\left (1+\frac{e x}{d}\right )}{e}-\frac{6 b^3 d m n^3 \text{Li}_4\left (1+\frac{e x}{d}\right )}{e}\\ \end{align*}

Mathematica [F]  time = 0.486366, size = 0, normalized size = 0. \[ \int \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^3,x]

[Out]

Integrate[Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^3, x]

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Maple [F]  time = 2.416, size = 0, normalized size = 0. \begin{align*} \int \ln \left ( f{x}^{m} \right ) \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(f*x^m)*(a+b*ln(c*(e*x+d)^n))^3,x)

[Out]

int(ln(f*x^m)*(a+b*ln(c*(e*x+d)^n))^3,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b^3*(m - log(f))*x - b^3*x*log(x^m))*log((e*x + d)^n)^3 + integrate((b^3*d*log(c)^3*log(f) + 3*a*b^2*d*log(c
)^2*log(f) + 3*a^2*b*d*log(c)*log(f) + a^3*d*log(f) + 3*(b^3*d*log(c)*log(f) + a*b^2*d*log(f) + (a*b^2*e*log(f
) + (e*log(c)*log(f) + (m*n - n*log(f))*e)*b^3)*x + (b^3*d*log(c) + a*b^2*d - ((e*n - e*log(c))*b^3 - a*b^2*e)
*x)*log(x^m))*log((e*x + d)^n)^2 + (b^3*e*log(c)^3*log(f) + 3*a*b^2*e*log(c)^2*log(f) + 3*a^2*b*e*log(c)*log(f
) + a^3*e*log(f))*x + 3*(b^3*d*log(c)^2*log(f) + 2*a*b^2*d*log(c)*log(f) + a^2*b*d*log(f) + (b^3*e*log(c)^2*lo
g(f) + 2*a*b^2*e*log(c)*log(f) + a^2*b*e*log(f))*x + (b^3*d*log(c)^2 + 2*a*b^2*d*log(c) + a^2*b*d + (b^3*e*log
(c)^2 + 2*a*b^2*e*log(c) + a^2*b*e)*x)*log(x^m))*log((e*x + d)^n) + (b^3*d*log(c)^3 + 3*a*b^2*d*log(c)^2 + 3*a
^2*b*d*log(c) + a^3*d + (b^3*e*log(c)^3 + 3*a*b^2*e*log(c)^2 + 3*a^2*b*e*log(c) + a^3*e)*x)*log(x^m))/(e*x + d
), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(b^3*log((e*x + d)^n*c)^3*log(f*x^m) + 3*a*b^2*log((e*x + d)^n*c)^2*log(f*x^m) + 3*a^2*b*log((e*x + d)
^n*c)*log(f*x^m) + a^3*log(f*x^m), 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(ln(f*x**m)*(a+b*ln(c*(e*x+d)**n))**3,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*log((e*x + d)^n*c) + a)^3*log(f*x^m), x)