3.462 \(\int \frac{(a+b \log (c (d+e \sqrt [3]{x})^n))^3}{x^3} \, dx\)

Optimal. Leaf size=765 \[ \text{result too large to display} \]

[Out]

-(b^3*e^3*n^3)/(20*d^3*x) + (3*b^3*e^4*n^3)/(10*d^4*x^(2/3)) - (71*b^3*e^5*n^3)/(40*d^5*x^(1/3)) + (71*b^3*e^6
*n^3*Log[d + e*x^(1/3)])/(40*d^6) - (3*b^2*e^2*n^2*(a + b*Log[c*(d + e*x^(1/3))^n]))/(20*d^2*x^(4/3)) + (9*b^2
*e^3*n^2*(a + b*Log[c*(d + e*x^(1/3))^n]))/(20*d^3*x) - (47*b^2*e^4*n^2*(a + b*Log[c*(d + e*x^(1/3))^n]))/(40*
d^4*x^(2/3)) + (77*b^2*e^5*n^2*(d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n]))/(20*d^6*x^(1/3)) + (77*b^2*e^
6*n^2*Log[1 - d/(d + e*x^(1/3))]*(a + b*Log[c*(d + e*x^(1/3))^n]))/(20*d^6) - (3*b*e*n*(a + b*Log[c*(d + e*x^(
1/3))^n])^2)/(10*d*x^(5/3)) + (3*b*e^2*n*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(8*d^2*x^(4/3)) - (b*e^3*n*(a + b
*Log[c*(d + e*x^(1/3))^n])^2)/(2*d^3*x) + (3*b*e^4*n*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(4*d^4*x^(2/3)) - (3*
b*e^5*n*(d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(2*d^6*x^(1/3)) - (3*b*e^6*n*Log[1 - d/(d + e*x^(1
/3))]*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(2*d^6) - (a + b*Log[c*(d + e*x^(1/3))^n])^3/(2*x^2) + (3*b^2*e^6*n^
2*(a + b*Log[c*(d + e*x^(1/3))^n])*Log[-((e*x^(1/3))/d)])/d^6 - (15*b^3*e^6*n^3*Log[x])/(8*d^6) - (77*b^3*e^6*
n^3*PolyLog[2, d/(d + e*x^(1/3))])/(20*d^6) + (3*b^2*e^6*n^2*(a + b*Log[c*(d + e*x^(1/3))^n])*PolyLog[2, d/(d
+ e*x^(1/3))])/d^6 + (3*b^3*e^6*n^3*PolyLog[2, 1 + (e*x^(1/3))/d])/d^6 + (3*b^3*e^6*n^3*PolyLog[3, d/(d + e*x^
(1/3))])/d^6

________________________________________________________________________________________

Rubi [A]  time = 3.05255, antiderivative size = 742, normalized size of antiderivative = 0.97, number of steps used = 73, number of rules used = 17, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.708, Rules used = {2454, 2398, 2411, 2347, 2344, 2302, 30, 2317, 2374, 6589, 2318, 2391, 2319, 2301, 2314, 31, 44} \[ -\frac{3 b^2 e^6 n^2 \text{PolyLog}\left (2,\frac{e \sqrt [3]{x}}{d}+1\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d^6}+\frac{137 b^3 e^6 n^3 \text{PolyLog}\left (2,\frac{e \sqrt [3]{x}}{d}+1\right )}{20 d^6}+\frac{3 b^3 e^6 n^3 \text{PolyLog}\left (3,\frac{e \sqrt [3]{x}}{d}+1\right )}{d^6}-\frac{47 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{40 d^4 x^{2/3}}-\frac{3 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^2 x^{4/3}}+\frac{137 b^2 e^6 n^2 \log \left (-\frac{e \sqrt [3]{x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^6}+\frac{77 b^2 e^5 n^2 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^6 \sqrt [3]{x}}+\frac{9 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^3 x}+\frac{3 b e^4 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{4 d^4 x^{2/3}}+\frac{3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{8 d^2 x^{4/3}}+\frac{e^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 d^6}-\frac{77 b e^6 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{40 d^6}-\frac{3 b e^6 n \log \left (-\frac{e \sqrt [3]{x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^6}-\frac{3 b e^5 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^6 \sqrt [3]{x}}-\frac{b e^3 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^3 x}-\frac{3 b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{10 d x^{5/3}}-\frac{\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 x^2}+\frac{3 b^3 e^4 n^3}{10 d^4 x^{2/3}}-\frac{71 b^3 e^5 n^3}{40 d^5 \sqrt [3]{x}}-\frac{b^3 e^3 n^3}{20 d^3 x}+\frac{71 b^3 e^6 n^3 \log \left (d+e \sqrt [3]{x}\right )}{40 d^6}-\frac{15 b^3 e^6 n^3 \log (x)}{8 d^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b^3*e^3*n^3)/(20*d^3*x) + (3*b^3*e^4*n^3)/(10*d^4*x^(2/3)) - (71*b^3*e^5*n^3)/(40*d^5*x^(1/3)) + (71*b^3*e^6
*n^3*Log[d + e*x^(1/3)])/(40*d^6) - (3*b^2*e^2*n^2*(a + b*Log[c*(d + e*x^(1/3))^n]))/(20*d^2*x^(4/3)) + (9*b^2
*e^3*n^2*(a + b*Log[c*(d + e*x^(1/3))^n]))/(20*d^3*x) - (47*b^2*e^4*n^2*(a + b*Log[c*(d + e*x^(1/3))^n]))/(40*
d^4*x^(2/3)) + (77*b^2*e^5*n^2*(d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n]))/(20*d^6*x^(1/3)) - (77*b*e^6*
n*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(40*d^6) - (3*b*e*n*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(10*d*x^(5/3)) +
(3*b*e^2*n*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(8*d^2*x^(4/3)) - (b*e^3*n*(a + b*Log[c*(d + e*x^(1/3))^n])^2)
/(2*d^3*x) + (3*b*e^4*n*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(4*d^4*x^(2/3)) - (3*b*e^5*n*(d + e*x^(1/3))*(a +
b*Log[c*(d + e*x^(1/3))^n])^2)/(2*d^6*x^(1/3)) + (e^6*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/(2*d^6) - (a + b*Log
[c*(d + e*x^(1/3))^n])^3/(2*x^2) + (137*b^2*e^6*n^2*(a + b*Log[c*(d + e*x^(1/3))^n])*Log[-((e*x^(1/3))/d)])/(2
0*d^6) - (3*b*e^6*n*(a + b*Log[c*(d + e*x^(1/3))^n])^2*Log[-((e*x^(1/3))/d)])/(2*d^6) - (15*b^3*e^6*n^3*Log[x]
)/(8*d^6) + (137*b^3*e^6*n^3*PolyLog[2, 1 + (e*x^(1/3))/d])/(20*d^6) - (3*b^2*e^6*n^2*(a + b*Log[c*(d + e*x^(1
/3))^n])*PolyLog[2, 1 + (e*x^(1/3))/d])/d^6 + (3*b^3*e^6*n^3*PolyLog[3, 1 + (e*x^(1/3))/d])/d^6

Rule 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rule 2398

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((
f + g*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n])^p)/(g*(q + 1)), x] - Dist[(b*e*n*p)/(g*(q + 1)), Int[((f + g*x)^(q
+ 1)*(a + b*Log[c*(d + e*x)^n])^(p - 1))/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*
f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && IntegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 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 2347

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[((
d + e*x)^(q + 1)*(a + b*Log[c*x^n])^p)/x, x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

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 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 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 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 2318

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_))^2, x_Symbol] :> Simp[(x*(a + b*Log[c*x^n])
^p)/(d*(d + e*x)), x] - Dist[(b*n*p)/d, Int[(a + b*Log[c*x^n])^(p - 1)/(d + e*x), x], x] /; FreeQ[{a, b, c, d,
e, n, p}, x] && GtQ[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 2319

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[((d + e*x)^(q + 1
)*(a + b*Log[c*x^n])^p)/(e*(q + 1)), x] - Dist[(b*n*p)/(e*(q + 1)), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

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 2314

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

Rule 31

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

Rule 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x^3} \, dx &=3 \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{x^7} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 x^2}+\frac{1}{2} (3 b e n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^6 (d+e x)} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 x^2}+\frac{1}{2} (3 b n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^6} \, dx,x,d+e \sqrt [3]{x}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 x^2}+\frac{(3 b n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\left (-\frac{d}{e}+\frac{x}{e}\right )^6} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d}-\frac{(3 b e n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^5} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d}\\ &=-\frac{3 b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{10 d x^{5/3}}-\frac{\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 x^2}-\frac{(3 b e n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\left (-\frac{d}{e}+\frac{x}{e}\right )^5} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d^2}+\frac{\left (3 b e^2 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^4} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d^2}+\frac{\left (3 b^2 e n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^5} \, dx,x,d+e \sqrt [3]{x}\right )}{5 d}\\ &=-\frac{3 b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{10 d x^{5/3}}+\frac{3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{8 d^2 x^{4/3}}-\frac{\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 x^2}+\frac{\left (3 b e^2 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\left (-\frac{d}{e}+\frac{x}{e}\right )^4} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d^3}-\frac{\left (3 b e^3 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^3} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d^3}+\frac{\left (3 b^2 e n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\left (-\frac{d}{e}+\frac{x}{e}\right )^5} \, dx,x,d+e \sqrt [3]{x}\right )}{5 d^2}-\frac{\left (3 b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^4} \, dx,x,d+e \sqrt [3]{x}\right )}{5 d^2}-\frac{\left (3 b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^4} \, dx,x,d+e \sqrt [3]{x}\right )}{4 d^2}\\ &=-\frac{3 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^2 x^{4/3}}-\frac{3 b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{10 d x^{5/3}}+\frac{3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{8 d^2 x^{4/3}}-\frac{b e^3 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^3 x}-\frac{\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 x^2}-\frac{\left (3 b e^3 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\left (-\frac{d}{e}+\frac{x}{e}\right )^3} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d^4}+\frac{\left (3 b e^4 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d^4}-\frac{\left (3 b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\left (-\frac{d}{e}+\frac{x}{e}\right )^4} \, dx,x,d+e \sqrt [3]{x}\right )}{5 d^3}-\frac{\left (3 b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\left (-\frac{d}{e}+\frac{x}{e}\right )^4} \, dx,x,d+e \sqrt [3]{x}\right )}{4 d^3}+\frac{\left (3 b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^3} \, dx,x,d+e \sqrt [3]{x}\right )}{5 d^3}+\frac{\left (3 b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^3} \, dx,x,d+e \sqrt [3]{x}\right )}{4 d^3}+\frac{\left (b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^3} \, dx,x,d+e \sqrt [3]{x}\right )}{d^3}+\frac{\left (3 b^3 e^2 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^4} \, dx,x,d+e \sqrt [3]{x}\right )}{20 d^2}\\ &=-\frac{3 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^2 x^{4/3}}+\frac{9 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^3 x}-\frac{3 b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{10 d x^{5/3}}+\frac{3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{8 d^2 x^{4/3}}-\frac{b e^3 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^3 x}+\frac{3 b e^4 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{4 d^4 x^{2/3}}-\frac{\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 x^2}+\frac{\left (3 b e^4 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d^5}-\frac{\left (3 b e^5 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x \left (-\frac{d}{e}+\frac{x}{e}\right )} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d^5}+\frac{\left (3 b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\left (-\frac{d}{e}+\frac{x}{e}\right )^3} \, dx,x,d+e \sqrt [3]{x}\right )}{5 d^4}+\frac{\left (3 b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\left (-\frac{d}{e}+\frac{x}{e}\right )^3} \, dx,x,d+e \sqrt [3]{x}\right )}{4 d^4}+\frac{\left (b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\left (-\frac{d}{e}+\frac{x}{e}\right )^3} \, dx,x,d+e \sqrt [3]{x}\right )}{d^4}-\frac{\left (3 b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+e \sqrt [3]{x}\right )}{5 d^4}-\frac{\left (3 b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+e \sqrt [3]{x}\right )}{4 d^4}-\frac{\left (b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+e \sqrt [3]{x}\right )}{d^4}-\frac{\left (3 b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d^4}+\frac{\left (3 b^3 e^2 n^3\right ) \operatorname{Subst}\left (\int \left (\frac{e^4}{d (d-x)^4}+\frac{e^4}{d^2 (d-x)^3}+\frac{e^4}{d^3 (d-x)^2}+\frac{e^4}{d^4 (d-x)}+\frac{e^4}{d^4 x}\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{20 d^2}-\frac{\left (b^3 e^3 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^3} \, dx,x,d+e \sqrt [3]{x}\right )}{5 d^3}-\frac{\left (b^3 e^3 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^3} \, dx,x,d+e \sqrt [3]{x}\right )}{4 d^3}\\ &=-\frac{b^3 e^3 n^3}{20 d^3 x}+\frac{3 b^3 e^4 n^3}{40 d^4 x^{2/3}}-\frac{3 b^3 e^5 n^3}{20 d^5 \sqrt [3]{x}}+\frac{3 b^3 e^6 n^3 \log \left (d+e \sqrt [3]{x}\right )}{20 d^6}-\frac{3 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^2 x^{4/3}}+\frac{9 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^3 x}-\frac{47 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{40 d^4 x^{2/3}}-\frac{3 b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{10 d x^{5/3}}+\frac{3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{8 d^2 x^{4/3}}-\frac{b e^3 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^3 x}+\frac{3 b e^4 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{4 d^4 x^{2/3}}-\frac{3 b e^5 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^6 \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 x^2}-\frac{b^3 e^6 n^3 \log (x)}{20 d^6}-\frac{\left (3 b e^5 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d^6}+\frac{\left (3 b e^6 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d^6}-\frac{\left (3 b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+e \sqrt [3]{x}\right )}{5 d^5}-\frac{\left (3 b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+e \sqrt [3]{x}\right )}{4 d^5}-\frac{\left (b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+e \sqrt [3]{x}\right )}{d^5}-\frac{\left (3 b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d^5}+\frac{\left (3 b^2 e^5 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+e \sqrt [3]{x}\right )}{d^6}+\frac{\left (3 b^2 e^5 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )} \, dx,x,d+e \sqrt [3]{x}\right )}{5 d^5}+\frac{\left (3 b^2 e^5 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )} \, dx,x,d+e \sqrt [3]{x}\right )}{4 d^5}+\frac{\left (b^2 e^5 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )} \, dx,x,d+e \sqrt [3]{x}\right )}{d^5}+\frac{\left (3 b^2 e^5 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d^5}-\frac{\left (b^3 e^3 n^3\right ) \operatorname{Subst}\left (\int \left (-\frac{e^3}{d (d-x)^3}-\frac{e^3}{d^2 (d-x)^2}-\frac{e^3}{d^3 (d-x)}-\frac{e^3}{d^3 x}\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{5 d^3}-\frac{\left (b^3 e^3 n^3\right ) \operatorname{Subst}\left (\int \left (-\frac{e^3}{d (d-x)^3}-\frac{e^3}{d^2 (d-x)^2}-\frac{e^3}{d^3 (d-x)}-\frac{e^3}{d^3 x}\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{4 d^3}+\frac{\left (3 b^3 e^4 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+e \sqrt [3]{x}\right )}{10 d^4}+\frac{\left (3 b^3 e^4 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+e \sqrt [3]{x}\right )}{8 d^4}+\frac{\left (b^3 e^4 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d^4}\\ &=-\frac{b^3 e^3 n^3}{20 d^3 x}+\frac{3 b^3 e^4 n^3}{10 d^4 x^{2/3}}-\frac{3 b^3 e^5 n^3}{5 d^5 \sqrt [3]{x}}+\frac{3 b^3 e^6 n^3 \log \left (d+e \sqrt [3]{x}\right )}{5 d^6}-\frac{3 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^2 x^{4/3}}+\frac{9 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^3 x}-\frac{47 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{40 d^4 x^{2/3}}+\frac{77 b^2 e^5 n^2 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^6 \sqrt [3]{x}}-\frac{3 b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{10 d x^{5/3}}+\frac{3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{8 d^2 x^{4/3}}-\frac{b e^3 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^3 x}+\frac{3 b e^4 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{4 d^4 x^{2/3}}-\frac{3 b e^5 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^6 \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 x^2}+\frac{3 b^2 e^6 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \log \left (-\frac{e \sqrt [3]{x}}{d}\right )}{d^6}-\frac{3 b e^6 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \log \left (-\frac{e \sqrt [3]{x}}{d}\right )}{2 d^6}-\frac{b^3 e^6 n^3 \log (x)}{5 d^6}+\frac{\left (3 e^6\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{2 d^6}+\frac{\left (3 b^2 e^5 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+e \sqrt [3]{x}\right )}{5 d^6}+\frac{\left (3 b^2 e^5 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+e \sqrt [3]{x}\right )}{4 d^6}+\frac{\left (b^2 e^5 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+e \sqrt [3]{x}\right )}{d^6}+\frac{\left (3 b^2 e^5 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d^6}-\frac{\left (3 b^2 e^6 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{5 d^6}-\frac{\left (3 b^2 e^6 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{4 d^6}-\frac{\left (b^2 e^6 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{d^6}-\frac{\left (3 b^2 e^6 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d^6}+\frac{\left (3 b^2 e^6 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 \sqrt [3]{x}\right )}{d^6}+\frac{\left (3 b^3 e^4 n^3\right ) \operatorname{Subst}\left (\int \left (\frac{e^2}{d (d-x)^2}+\frac{e^2}{d^2 (d-x)}+\frac{e^2}{d^2 x}\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{10 d^4}+\frac{\left (3 b^3 e^4 n^3\right ) \operatorname{Subst}\left (\int \left (\frac{e^2}{d (d-x)^2}+\frac{e^2}{d^2 (d-x)}+\frac{e^2}{d^2 x}\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{8 d^4}+\frac{\left (b^3 e^4 n^3\right ) \operatorname{Subst}\left (\int \left (\frac{e^2}{d (d-x)^2}+\frac{e^2}{d^2 (d-x)}+\frac{e^2}{d^2 x}\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{2 d^4}-\frac{\left (3 b^3 e^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+e \sqrt [3]{x}\right )}{5 d^6}-\frac{\left (3 b^3 e^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+e \sqrt [3]{x}\right )}{4 d^6}-\frac{\left (b^3 e^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+e \sqrt [3]{x}\right )}{d^6}-\frac{\left (3 b^3 e^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d^6}-\frac{\left (3 b^3 e^6 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{d^6}\\ &=-\frac{b^3 e^3 n^3}{20 d^3 x}+\frac{3 b^3 e^4 n^3}{10 d^4 x^{2/3}}-\frac{71 b^3 e^5 n^3}{40 d^5 \sqrt [3]{x}}+\frac{71 b^3 e^6 n^3 \log \left (d+e \sqrt [3]{x}\right )}{40 d^6}-\frac{3 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^2 x^{4/3}}+\frac{9 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^3 x}-\frac{47 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{40 d^4 x^{2/3}}+\frac{77 b^2 e^5 n^2 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^6 \sqrt [3]{x}}-\frac{77 b e^6 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{40 d^6}-\frac{3 b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{10 d x^{5/3}}+\frac{3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{8 d^2 x^{4/3}}-\frac{b e^3 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^3 x}+\frac{3 b e^4 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{4 d^4 x^{2/3}}-\frac{3 b e^5 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^6 \sqrt [3]{x}}+\frac{e^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 d^6}-\frac{\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 x^2}+\frac{137 b^2 e^6 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \log \left (-\frac{e \sqrt [3]{x}}{d}\right )}{20 d^6}-\frac{3 b e^6 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \log \left (-\frac{e \sqrt [3]{x}}{d}\right )}{2 d^6}-\frac{15 b^3 e^6 n^3 \log (x)}{8 d^6}+\frac{3 b^3 e^6 n^3 \text{Li}_2\left (1+\frac{e \sqrt [3]{x}}{d}\right )}{d^6}-\frac{3 b^2 e^6 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \text{Li}_2\left (1+\frac{e \sqrt [3]{x}}{d}\right )}{d^6}-\frac{\left (3 b^3 e^6 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{5 d^6}-\frac{\left (3 b^3 e^6 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{4 d^6}-\frac{\left (b^3 e^6 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{d^6}-\frac{\left (3 b^3 e^6 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d^6}+\frac{\left (3 b^3 e^6 n^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{d}\right )}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{d^6}\\ &=-\frac{b^3 e^3 n^3}{20 d^3 x}+\frac{3 b^3 e^4 n^3}{10 d^4 x^{2/3}}-\frac{71 b^3 e^5 n^3}{40 d^5 \sqrt [3]{x}}+\frac{71 b^3 e^6 n^3 \log \left (d+e \sqrt [3]{x}\right )}{40 d^6}-\frac{3 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^2 x^{4/3}}+\frac{9 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^3 x}-\frac{47 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{40 d^4 x^{2/3}}+\frac{77 b^2 e^5 n^2 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^6 \sqrt [3]{x}}-\frac{77 b e^6 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{40 d^6}-\frac{3 b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{10 d x^{5/3}}+\frac{3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{8 d^2 x^{4/3}}-\frac{b e^3 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^3 x}+\frac{3 b e^4 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{4 d^4 x^{2/3}}-\frac{3 b e^5 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^6 \sqrt [3]{x}}+\frac{e^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 d^6}-\frac{\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 x^2}+\frac{137 b^2 e^6 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \log \left (-\frac{e \sqrt [3]{x}}{d}\right )}{20 d^6}-\frac{3 b e^6 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \log \left (-\frac{e \sqrt [3]{x}}{d}\right )}{2 d^6}-\frac{15 b^3 e^6 n^3 \log (x)}{8 d^6}+\frac{137 b^3 e^6 n^3 \text{Li}_2\left (1+\frac{e \sqrt [3]{x}}{d}\right )}{20 d^6}-\frac{3 b^2 e^6 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \text{Li}_2\left (1+\frac{e \sqrt [3]{x}}{d}\right )}{d^6}+\frac{3 b^3 e^6 n^3 \text{Li}_3\left (1+\frac{e \sqrt [3]{x}}{d}\right )}{d^6}\\ \end{align*}

Mathematica [A]  time = 1.67626, size = 1074, normalized size = 1.4 \[ -\frac{20 \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 d^6+60 b n \log \left (d+e \sqrt [3]{x}\right ) \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 d^6+12 b e n \sqrt [3]{x} \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 d^5-15 b e^2 n x^{2/3} \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 d^4+20 b e^3 n x \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 d^3-30 b e^4 n x^{4/3} \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 d^2+60 b e^5 n x^{5/3} \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 d-60 b e^6 n x^2 \log \left (d+e \sqrt [3]{x}\right ) \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2+20 b e^6 n x^2 \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \log (x)+b^2 n^2 \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \left (-274 x^2 \log \left (-\frac{e \sqrt [3]{x}}{d}\right ) e^6+120 x^2 \text{PolyLog}\left (2,\frac{\sqrt [3]{x} e}{d}+1\right ) e^6-154 d x^{5/3} e^5+47 d^2 x^{4/3} e^4-18 d^3 x e^3+6 d^4 x^{2/3} e^2+60 \left (d^6-e^6 x^2\right ) \log ^2\left (d+e \sqrt [3]{x}\right )+2 \log \left (d+e \sqrt [3]{x}\right ) \left (137 x^2 e^6+60 x^2 \log \left (-\frac{e \sqrt [3]{x}}{d}\right ) e^6+60 d x^{5/3} e^5-30 d^2 x^{4/3} e^4+20 d^3 x e^3-15 d^4 x^{2/3} e^2+12 d^5 \sqrt [3]{x} e\right )\right )+b^3 n^3 \left (20 \log ^3\left (d+e \sqrt [3]{x}\right ) d^6+12 e \sqrt [3]{x} \log ^2\left (d+e \sqrt [3]{x}\right ) d^5+3 e^2 x^{2/3} \left (2-5 \log \left (d+e \sqrt [3]{x}\right )\right ) \log \left (d+e \sqrt [3]{x}\right ) d^4+2 e^3 x \left (10 \log ^2\left (d+e \sqrt [3]{x}\right )-9 \log \left (d+e \sqrt [3]{x}\right )+1\right ) d^3-e^4 x^{4/3} \left (30 \log ^2\left (d+e \sqrt [3]{x}\right )-47 \log \left (d+e \sqrt [3]{x}\right )+12\right ) d^2+e^5 x^{5/3} \left (60 \log ^2\left (d+e \sqrt [3]{x}\right )-154 \log \left (d+e \sqrt [3]{x}\right )+71\right ) d+225 e^6 x^2 \left (\log \left (-\frac{e \sqrt [3]{x}}{d}\right )-\log \left (d+e \sqrt [3]{x}\right )\right )+137 e^6 x^2 \left (\log \left (d+e \sqrt [3]{x}\right ) \left (\log \left (d+e \sqrt [3]{x}\right )-2 \log \left (-\frac{e \sqrt [3]{x}}{d}\right )\right )-2 \text{PolyLog}\left (2,\frac{\sqrt [3]{x} e}{d}+1\right )\right )-20 e^6 x^2 \left (\left (\log \left (d+e \sqrt [3]{x}\right )-3 \log \left (-\frac{e \sqrt [3]{x}}{d}\right )\right ) \log ^2\left (d+e \sqrt [3]{x}\right )-6 \text{PolyLog}\left (2,\frac{\sqrt [3]{x} e}{d}+1\right ) \log \left (d+e \sqrt [3]{x}\right )+6 \text{PolyLog}\left (3,\frac{\sqrt [3]{x} e}{d}+1\right )\right )\right )}{40 d^6 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(12*b*d^5*e*n*x^(1/3)*(a - b*n*Log[d + e*x^(1/3)] + b*Log[c*(d + e*x^(1/3))^n])^2 - 15*b*d^4*e^2*n*x^(2/3)*(a
- b*n*Log[d + e*x^(1/3)] + b*Log[c*(d + e*x^(1/3))^n])^2 + 20*b*d^3*e^3*n*x*(a - b*n*Log[d + e*x^(1/3)] + b*L
og[c*(d + e*x^(1/3))^n])^2 - 30*b*d^2*e^4*n*x^(4/3)*(a - b*n*Log[d + e*x^(1/3)] + b*Log[c*(d + e*x^(1/3))^n])^
2 + 60*b*d*e^5*n*x^(5/3)*(a - b*n*Log[d + e*x^(1/3)] + b*Log[c*(d + e*x^(1/3))^n])^2 + 60*b*d^6*n*Log[d + e*x^
(1/3)]*(a - b*n*Log[d + e*x^(1/3)] + b*Log[c*(d + e*x^(1/3))^n])^2 - 60*b*e^6*n*x^2*Log[d + e*x^(1/3)]*(a - b*
n*Log[d + e*x^(1/3)] + b*Log[c*(d + e*x^(1/3))^n])^2 + 20*d^6*(a - b*n*Log[d + e*x^(1/3)] + b*Log[c*(d + e*x^(
1/3))^n])^3 + 20*b*e^6*n*x^2*(a - b*n*Log[d + e*x^(1/3)] + b*Log[c*(d + e*x^(1/3))^n])^2*Log[x] + b^2*n^2*(a -
b*n*Log[d + e*x^(1/3)] + b*Log[c*(d + e*x^(1/3))^n])*(6*d^4*e^2*x^(2/3) - 18*d^3*e^3*x + 47*d^2*e^4*x^(4/3) -
154*d*e^5*x^(5/3) + 60*(d^6 - e^6*x^2)*Log[d + e*x^(1/3)]^2 - 274*e^6*x^2*Log[-((e*x^(1/3))/d)] + 2*Log[d + e
*x^(1/3)]*(12*d^5*e*x^(1/3) - 15*d^4*e^2*x^(2/3) + 20*d^3*e^3*x - 30*d^2*e^4*x^(4/3) + 60*d*e^5*x^(5/3) + 137*
e^6*x^2 + 60*e^6*x^2*Log[-((e*x^(1/3))/d)]) + 120*e^6*x^2*PolyLog[2, 1 + (e*x^(1/3))/d]) + b^3*n^3*(3*d^4*e^2*
x^(2/3)*(2 - 5*Log[d + e*x^(1/3)])*Log[d + e*x^(1/3)] + 12*d^5*e*x^(1/3)*Log[d + e*x^(1/3)]^2 + 20*d^6*Log[d +
e*x^(1/3)]^3 + 2*d^3*e^3*x*(1 - 9*Log[d + e*x^(1/3)] + 10*Log[d + e*x^(1/3)]^2) - d^2*e^4*x^(4/3)*(12 - 47*Lo
g[d + e*x^(1/3)] + 30*Log[d + e*x^(1/3)]^2) + d*e^5*x^(5/3)*(71 - 154*Log[d + e*x^(1/3)] + 60*Log[d + e*x^(1/3
)]^2) + 225*e^6*x^2*(-Log[d + e*x^(1/3)] + Log[-((e*x^(1/3))/d)]) + 137*e^6*x^2*(Log[d + e*x^(1/3)]*(Log[d + e
*x^(1/3)] - 2*Log[-((e*x^(1/3))/d)]) - 2*PolyLog[2, 1 + (e*x^(1/3))/d]) - 20*e^6*x^2*(Log[d + e*x^(1/3)]^2*(Lo
g[d + e*x^(1/3)] - 3*Log[-((e*x^(1/3))/d)]) - 6*Log[d + e*x^(1/3)]*PolyLog[2, 1 + (e*x^(1/3))/d] + 6*PolyLog[3
, 1 + (e*x^(1/3))/d])))/(40*d^6*x^2)

________________________________________________________________________________________

Maple [F]  time = 0.099, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \left ( a+b\ln \left ( c \left ( d+e\sqrt [3]{x} \right ) ^{n} \right ) \right ) ^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{b^{3} \log \left ({\left (e x^{\frac{1}{3}} + d\right )}^{n}\right )^{3}}{2 \, x^{2}} + \int \frac{{\left (b^{3} e n x + 6 \,{\left (b^{3} e \log \left (c\right ) + a b^{2} e\right )} x + 6 \,{\left (b^{3} d \log \left (c\right ) + a b^{2} d\right )} x^{\frac{2}{3}}\right )} \log \left ({\left (e x^{\frac{1}{3}} + d\right )}^{n}\right )^{2} + 2 \,{\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 + 6 \,{\left ({\left (b^{3} e \log \left (c\right )^{2} + 2 \, a b^{2} e \log \left (c\right ) + a^{2} b e\right )} x +{\left (b^{3} d \log \left (c\right )^{2} + 2 \, a b^{2} d \log \left (c\right ) + a^{2} b d\right )} x^{\frac{2}{3}}\right )} \log \left ({\left (e x^{\frac{1}{3}} + d\right )}^{n}\right ) + 2 \,{\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\right )} x^{\frac{2}{3}}}{2 \,{\left (e x^{4} + d x^{\frac{11}{3}}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b^3*log((e*x^(1/3) + d)^n)^3/x^2 + integrate(1/2*((b^3*e*n*x + 6*(b^3*e*log(c) + a*b^2*e)*x + 6*(b^3*d*lo
g(c) + a*b^2*d)*x^(2/3))*log((e*x^(1/3) + d)^n)^2 + 2*(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 + 6*((b^3*e*log(c)^2 + 2*a*b^2*e*log(c) + a^2*b*e)*x + (b^3*d*log(c)^2 + 2*a*b^2*d*log(c) + a^2*b*d
)*x^(2/3))*log((e*x^(1/3) + d)^n) + 2*(b^3*d*log(c)^3 + 3*a*b^2*d*log(c)^2 + 3*a^2*b*d*log(c) + a^3*d)*x^(2/3)
)/(e*x^4 + d*x^(11/3)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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*(d+e*x**(1/3))**n))**3/x**3,x)

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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