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

Optimal. Leaf size=451 \[ \frac{3 b^2 e^3 n^2 \text{PolyLog}\left (2,\frac{d}{d+\frac{e}{x^{2/3}}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{d^3}-\frac{3 b^3 e^3 n^3 \text{PolyLog}\left (2,\frac{d}{d+\frac{e}{x^{2/3}}}\right )}{2 d^3}+\frac{3 b^3 e^3 n^3 \text{PolyLog}\left (2,\frac{e}{d x^{2/3}}+1\right )}{d^3}+\frac{3 b^3 e^3 n^3 \text{PolyLog}\left (3,\frac{d}{d+\frac{e}{x^{2/3}}}\right )}{d^3}+\frac{3 b^2 e^3 n^2 \log \left (1-\frac{d}{d+\frac{e}{x^{2/3}}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{2 d^3}+\frac{3 b^2 e^3 n^2 \log \left (-\frac{e}{d x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{d^3}+\frac{3 b^2 e^2 n^2 x^{2/3} \left (d+\frac{e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{2 d^3}-\frac{3 b e^3 n \log \left (1-\frac{d}{d+\frac{e}{x^{2/3}}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{2 d^3}-\frac{3 b e^2 n x^{2/3} \left (d+\frac{e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{2 d^3}+\frac{3 b e n x^{4/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{4 d}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3+\frac{b^3 e^3 n^3 \log (x)}{d^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.00322, antiderivative size = 428, normalized size of antiderivative = 0.95, number of steps used = 22, number of rules used = 16, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.727, Rules used = {2454, 2398, 2411, 2347, 2344, 2302, 30, 2317, 2374, 6589, 2318, 2391, 2319, 2301, 2314, 31} \[ -\frac{3 b^2 e^3 n^2 \text{PolyLog}\left (2,\frac{e}{d x^{2/3}}+1\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{d^3}+\frac{9 b^3 e^3 n^3 \text{PolyLog}\left (2,\frac{e}{d x^{2/3}}+1\right )}{2 d^3}+\frac{3 b^3 e^3 n^3 \text{PolyLog}\left (3,\frac{e}{d x^{2/3}}+1\right )}{d^3}+\frac{9 b^2 e^3 n^2 \log \left (-\frac{e}{d x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{2 d^3}+\frac{3 b^2 e^2 n^2 x^{2/3} \left (d+\frac{e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{2 d^3}+\frac{e^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{2 d^3}-\frac{3 b e^3 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{4 d^3}-\frac{3 b e^3 n \log \left (-\frac{e}{d x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{2 d^3}-\frac{3 b e^2 n x^{2/3} \left (d+\frac{e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{2 d^3}+\frac{3 b e n x^{4/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{4 d}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3+\frac{b^3 e^3 n^3 \log (x)}{d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*b^2*e^2*n^2*(d + e/x^(2/3))*x^(2/3)*(a + b*Log[c*(d + e/x^(2/3))^n]))/(2*d^3) - (3*b*e^3*n*(a + b*Log[c*(d
+ e/x^(2/3))^n])^2)/(4*d^3) - (3*b*e^2*n*(d + e/x^(2/3))*x^(2/3)*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(2*d^3) +
(3*b*e*n*x^(4/3)*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(4*d) + (e^3*(a + b*Log[c*(d + e/x^(2/3))^n])^3)/(2*d^3)
+ (x^2*(a + b*Log[c*(d + e/x^(2/3))^n])^3)/2 + (9*b^2*e^3*n^2*(a + b*Log[c*(d + e/x^(2/3))^n])*Log[-(e/(d*x^(
2/3)))])/(2*d^3) - (3*b*e^3*n*(a + b*Log[c*(d + e/x^(2/3))^n])^2*Log[-(e/(d*x^(2/3)))])/(2*d^3) + (b^3*e^3*n^3
*Log[x])/d^3 + (9*b^3*e^3*n^3*PolyLog[2, 1 + e/(d*x^(2/3))])/(2*d^3) - (3*b^2*e^3*n^2*(a + b*Log[c*(d + e/x^(2
/3))^n])*PolyLog[2, 1 + e/(d*x^(2/3))])/d^3 + (3*b^3*e^3*n^3*PolyLog[3, 1 + e/(d*x^(2/3))])/d^3

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]

Rubi steps

\begin{align*} \int x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3 \, dx &=-\left (\frac{3}{2} \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{x^4} \, dx,x,\frac{1}{x^{2/3}}\right )\right )\\ &=\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3-\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^3 (d+e x)} \, dx,x,\frac{1}{x^{2/3}}\right )\\ &=\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3-\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 )^3} \, dx,x,d+\frac{e}{x^{2/3}}\right )\\ &=\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3-\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 )^3} \, dx,x,d+\frac{e}{x^{2/3}}\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 )^2} \, dx,x,d+\frac{e}{x^{2/3}}\right )}{2 d}\\ &=\frac{3 b e n x^{4/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{4 d}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3+\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 )^2} \, dx,x,d+\frac{e}{x^{2/3}}\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 )} \, dx,x,d+\frac{e}{x^{2/3}}\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 )^2} \, dx,x,d+\frac{e}{x^{2/3}}\right )}{2 d}\\ &=-\frac{3 b e^2 n \left (d+\frac{e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{2 d^3}+\frac{3 b e n x^{4/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{4 d}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3-\frac{\left (3 b e^2 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+\frac{e}{x^{2/3}}\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} \, dx,x,d+\frac{e}{x^{2/3}}\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 )^2} \, dx,x,d+\frac{e}{x^{2/3}}\right )}{2 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 )}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+\frac{e}{x^{2/3}}\right )}{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 )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )} \, dx,x,d+\frac{e}{x^{2/3}}\right )}{2 d^2}\\ &=\frac{3 b^2 e^2 n^2 \left (d+\frac{e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{2 d^3}-\frac{3 b e^2 n \left (d+\frac{e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{2 d^3}+\frac{3 b e n x^{4/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{4 d}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3+\frac{3 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (-\frac{e}{d x^{2/3}}\right )}{d^3}-\frac{3 b e^3 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2 \log \left (-\frac{e}{d x^{2/3}}\right )}{2 d^3}+\frac{\left (3 e^3\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{2 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 )}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+\frac{e}{x^{2/3}}\right )}{2 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} \, dx,x,d+\frac{e}{x^{2/3}}\right )}{2 d^3}+\frac{\left (3 b^2 e^3 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+\frac{e}{x^{2/3}}\right )}{d^3}-\frac{\left (3 b^3 e^2 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+\frac{e}{x^{2/3}}\right )}{2 d^3}-\frac{\left (3 b^3 e^3 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+\frac{e}{x^{2/3}}\right )}{d^3}\\ &=\frac{3 b^2 e^2 n^2 \left (d+\frac{e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{2 d^3}-\frac{3 b e^3 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{4 d^3}-\frac{3 b e^2 n \left (d+\frac{e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{2 d^3}+\frac{3 b e n x^{4/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{4 d}+\frac{e^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{2 d^3}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3+\frac{9 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (-\frac{e}{d x^{2/3}}\right )}{2 d^3}-\frac{3 b e^3 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2 \log \left (-\frac{e}{d x^{2/3}}\right )}{2 d^3}+\frac{b^3 e^3 n^3 \log (x)}{d^3}+\frac{3 b^3 e^3 n^3 \text{Li}_2\left (1+\frac{e}{d x^{2/3}}\right )}{d^3}-\frac{3 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \text{Li}_2\left (1+\frac{e}{d x^{2/3}}\right )}{d^3}-\frac{\left (3 b^3 e^3 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+\frac{e}{x^{2/3}}\right )}{2 d^3}+\frac{\left (3 b^3 e^3 n^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{d}\right )}{x} \, dx,x,d+\frac{e}{x^{2/3}}\right )}{d^3}\\ &=\frac{3 b^2 e^2 n^2 \left (d+\frac{e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{2 d^3}-\frac{3 b e^3 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{4 d^3}-\frac{3 b e^2 n \left (d+\frac{e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{2 d^3}+\frac{3 b e n x^{4/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{4 d}+\frac{e^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{2 d^3}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3+\frac{9 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (-\frac{e}{d x^{2/3}}\right )}{2 d^3}-\frac{3 b e^3 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2 \log \left (-\frac{e}{d x^{2/3}}\right )}{2 d^3}+\frac{b^3 e^3 n^3 \log (x)}{d^3}+\frac{9 b^3 e^3 n^3 \text{Li}_2\left (1+\frac{e}{d x^{2/3}}\right )}{2 d^3}-\frac{3 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \text{Li}_2\left (1+\frac{e}{d x^{2/3}}\right )}{d^3}+\frac{3 b^3 e^3 n^3 \text{Li}_3\left (1+\frac{e}{d x^{2/3}}\right )}{d^3}\\ \end{align*}

Mathematica [A]  time = 1.35184, size = 683, normalized size = 1.51 \[ \frac{6 b^2 n^2 \left (-2 e^3 \text{PolyLog}\left (2,\frac{e}{d x^{2/3}}+1\right )+\left (d^3 x^2+e^3\right ) \log ^2\left (d+\frac{e}{x^{2/3}}\right )+e \log \left (d+\frac{e}{x^{2/3}}\right ) \left (d^2 x^{4/3}-2 e^2 \log \left (-\frac{e}{d x^{2/3}}\right )-2 d e x^{2/3}-3 e^2\right )+e^2 \left (3 e \log \left (-\frac{e}{d x^{2/3}}\right )+d x^{2/3}\right )\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )-b n \log \left (d+\frac{e}{x^{2/3}}\right )\right )-b^3 n^3 \left (-12 e^3 \text{PolyLog}\left (3,\frac{e}{d x^{2/3}}+1\right )+6 e^3 \left (2 \log \left (d+\frac{e}{x^{2/3}}\right )-3\right ) \text{PolyLog}\left (2,\frac{e}{d x^{2/3}}+1\right )-3 d^2 e x^{4/3} \log ^2\left (d+\frac{e}{x^{2/3}}\right )-2 d^3 x^2 \log ^3\left (d+\frac{e}{x^{2/3}}\right )-2 e^3 \log ^3\left (d+\frac{e}{x^{2/3}}\right )+9 e^3 \log ^2\left (d+\frac{e}{x^{2/3}}\right )+6 e^3 \log ^2\left (d+\frac{e}{x^{2/3}}\right ) \log \left (-\frac{e}{d x^{2/3}}\right )+6 d e^2 x^{2/3} \log ^2\left (d+\frac{e}{x^{2/3}}\right )-6 e^3 \log \left (d+\frac{e}{x^{2/3}}\right )-18 e^3 \log \left (d+\frac{e}{x^{2/3}}\right ) \log \left (-\frac{e}{d x^{2/3}}\right )+6 e^3 \log \left (-\frac{e}{d x^{2/3}}\right )-6 d e^2 x^{2/3} \log \left (d+\frac{e}{x^{2/3}}\right )\right )+2 d^3 x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )-b n \log \left (d+\frac{e}{x^{2/3}}\right )\right )^3+6 b d^3 n x^2 \log \left (d+\frac{e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )-b n \log \left (d+\frac{e}{x^{2/3}}\right )\right )^2+3 b d^2 e n x^{4/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )-b n \log \left (d+\frac{e}{x^{2/3}}\right )\right )^2-6 b d e^2 n x^{2/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )-b n \log \left (d+\frac{e}{x^{2/3}}\right )\right )^2+6 b e^3 n \log \left (d x^{2/3}+e\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )-b n \log \left (d+\frac{e}{x^{2/3}}\right )\right )^2}{4 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*b*d*e^2*n*x^(2/3)*(a - b*n*Log[d + e/x^(2/3)] + b*Log[c*(d + e/x^(2/3))^n])^2 + 3*b*d^2*e*n*x^(4/3)*(a - b
*n*Log[d + e/x^(2/3)] + b*Log[c*(d + e/x^(2/3))^n])^2 + 6*b*d^3*n*x^2*Log[d + e/x^(2/3)]*(a - b*n*Log[d + e/x^
(2/3)] + b*Log[c*(d + e/x^(2/3))^n])^2 + 2*d^3*x^2*(a - b*n*Log[d + e/x^(2/3)] + b*Log[c*(d + e/x^(2/3))^n])^3
+ 6*b*e^3*n*(a - b*n*Log[d + e/x^(2/3)] + b*Log[c*(d + e/x^(2/3))^n])^2*Log[e + d*x^(2/3)] + 6*b^2*n^2*(a - b
*n*Log[d + e/x^(2/3)] + b*Log[c*(d + e/x^(2/3))^n])*((e^3 + d^3*x^2)*Log[d + e/x^(2/3)]^2 + e^2*(d*x^(2/3) + 3
*e*Log[-(e/(d*x^(2/3)))]) + e*Log[d + e/x^(2/3)]*(-3*e^2 - 2*d*e*x^(2/3) + d^2*x^(4/3) - 2*e^2*Log[-(e/(d*x^(2
/3)))]) - 2*e^3*PolyLog[2, 1 + e/(d*x^(2/3))]) - b^3*n^3*(-6*e^3*Log[d + e/x^(2/3)] - 6*d*e^2*x^(2/3)*Log[d +
e/x^(2/3)] + 9*e^3*Log[d + e/x^(2/3)]^2 + 6*d*e^2*x^(2/3)*Log[d + e/x^(2/3)]^2 - 3*d^2*e*x^(4/3)*Log[d + e/x^(
2/3)]^2 - 2*e^3*Log[d + e/x^(2/3)]^3 - 2*d^3*x^2*Log[d + e/x^(2/3)]^3 + 6*e^3*Log[-(e/(d*x^(2/3)))] - 18*e^3*L
og[d + e/x^(2/3)]*Log[-(e/(d*x^(2/3)))] + 6*e^3*Log[d + e/x^(2/3)]^2*Log[-(e/(d*x^(2/3)))] + 6*e^3*(-3 + 2*Log
[d + e/x^(2/3)])*PolyLog[2, 1 + e/(d*x^(2/3))] - 12*e^3*PolyLog[3, 1 + e/(d*x^(2/3))]))/(4*d^3)

________________________________________________________________________________________

Maple [F]  time = 0.352, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b\ln \left ( c \left ( d+{e{x}^{-{\frac{2}{3}}}} \right ) ^{n} \right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, b^{3} x^{2} \log \left ({\left (d x^{\frac{2}{3}} + e\right )}^{n}\right )^{3} - \int \frac{8 \,{\left (b^{3} d x^{2} + b^{3} e x^{\frac{4}{3}}\right )} \log \left (x^{\frac{1}{3} \, n}\right )^{3} -{\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^{2} +{\left (b^{3} d n x^{2} - 3 \,{\left (b^{3} d \log \left (c\right ) + a b^{2} d\right )} x^{2} - 3 \,{\left (b^{3} e \log \left (c\right ) + a b^{2} e\right )} x^{\frac{4}{3}} + 6 \,{\left (b^{3} d x^{2} + b^{3} e x^{\frac{4}{3}}\right )} \log \left (x^{\frac{1}{3} \, n}\right )\right )} \log \left ({\left (d x^{\frac{2}{3}} + e\right )}^{n}\right )^{2} - 12 \,{\left ({\left (b^{3} d \log \left (c\right ) + a b^{2} d\right )} x^{2} +{\left (b^{3} e \log \left (c\right ) + a b^{2} e\right )} x^{\frac{4}{3}}\right )} \log \left (x^{\frac{1}{3} \, n}\right )^{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^{\frac{4}{3}} - 3 \,{\left ({\left (b^{3} d \log \left (c\right )^{2} + 2 \, a b^{2} d \log \left (c\right ) + a^{2} b d\right )} x^{2} + 4 \,{\left (b^{3} d x^{2} + b^{3} e x^{\frac{4}{3}}\right )} \log \left (x^{\frac{1}{3} \, n}\right )^{2} +{\left (b^{3} e \log \left (c\right )^{2} + 2 \, a b^{2} e \log \left (c\right ) + a^{2} b e\right )} x^{\frac{4}{3}} - 4 \,{\left ({\left (b^{3} d \log \left (c\right ) + a b^{2} d\right )} x^{2} +{\left (b^{3} e \log \left (c\right ) + a b^{2} e\right )} x^{\frac{4}{3}}\right )} \log \left (x^{\frac{1}{3} \, n}\right )\right )} \log \left ({\left (d x^{\frac{2}{3}} + e\right )}^{n}\right ) + 6 \,{\left ({\left (b^{3} d \log \left (c\right )^{2} + 2 \, a b^{2} d \log \left (c\right ) + a^{2} b d\right )} x^{2} +{\left (b^{3} e \log \left (c\right )^{2} + 2 \, a b^{2} e \log \left (c\right ) + a^{2} b e\right )} x^{\frac{4}{3}}\right )} \log \left (x^{\frac{1}{3} \, n}\right )}{d x + e x^{\frac{1}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b^3*x^2*log((d*x^(2/3) + e)^n)^3 - integrate((8*(b^3*d*x^2 + b^3*e*x^(4/3))*log(x^(1/3*n))^3 - (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 + (b^3*d*n*x^2 - 3*(b^3*d*log(c) + a*b^2*d)*x^2 - 3*
(b^3*e*log(c) + a*b^2*e)*x^(4/3) + 6*(b^3*d*x^2 + b^3*e*x^(4/3))*log(x^(1/3*n)))*log((d*x^(2/3) + e)^n)^2 - 12
*((b^3*d*log(c) + a*b^2*d)*x^2 + (b^3*e*log(c) + a*b^2*e)*x^(4/3))*log(x^(1/3*n))^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^(4/3) - 3*((b^3*d*log(c)^2 + 2*a*b^2*d*log(c) + a^2*b*d)*x^2 + 4*(b
^3*d*x^2 + b^3*e*x^(4/3))*log(x^(1/3*n))^2 + (b^3*e*log(c)^2 + 2*a*b^2*e*log(c) + a^2*b*e)*x^(4/3) - 4*((b^3*d
*log(c) + a*b^2*d)*x^2 + (b^3*e*log(c) + a*b^2*e)*x^(4/3))*log(x^(1/3*n)))*log((d*x^(2/3) + e)^n) + 6*((b^3*d*
log(c)^2 + 2*a*b^2*d*log(c) + a^2*b*d)*x^2 + (b^3*e*log(c)^2 + 2*a*b^2*e*log(c) + a^2*b*e)*x^(4/3))*log(x^(1/3
*n)))/(d*x + e*x^(1/3)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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