3.461 \(\int \frac{(a+b \log (c (d+e \sqrt [3]{x})^n))^3}{x^2} \, dx\)

Optimal. Leaf size=439 \[ -\frac{6 b^2 e^3 n^2 \text{PolyLog}\left (2,\frac{d}{d+e \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d^3}+\frac{3 b^3 e^3 n^3 \text{PolyLog}\left (2,\frac{d}{d+e \sqrt [3]{x}}\right )}{d^3}-\frac{6 b^3 e^3 n^3 \text{PolyLog}\left (2,\frac{e \sqrt [3]{x}}{d}+1\right )}{d^3}-\frac{6 b^3 e^3 n^3 \text{PolyLog}\left (3,\frac{d}{d+e \sqrt [3]{x}}\right )}{d^3}-\frac{3 b^2 e^3 n^2 \log \left (1-\frac{d}{d+e \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d^3}-\frac{6 b^2 e^3 n^2 \log \left (-\frac{e \sqrt [3]{x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d^3}-\frac{3 b^2 e^2 n^2 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d^3 \sqrt [3]{x}}+\frac{3 b e^3 n \log \left (1-\frac{d}{d+e \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{d^3}+\frac{3 b e^2 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{d^3 \sqrt [3]{x}}-\frac{3 b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d x^{2/3}}-\frac{\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x}+\frac{b^3 e^3 n^3 \log (x)}{d^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.00594, antiderivative size = 414, normalized size of antiderivative = 0.94, number of steps used = 22, number of rules used = 16, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {2454, 2398, 2411, 2347, 2344, 2302, 30, 2317, 2374, 6589, 2318, 2391, 2319, 2301, 2314, 31} \[ \frac{6 b^2 e^3 n^2 \text{PolyLog}\left (2,\frac{e \sqrt [3]{x}}{d}+1\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d^3}-\frac{9 b^3 e^3 n^3 \text{PolyLog}\left (2,\frac{e \sqrt [3]{x}}{d}+1\right )}{d^3}-\frac{6 b^3 e^3 n^3 \text{PolyLog}\left (3,\frac{e \sqrt [3]{x}}{d}+1\right )}{d^3}-\frac{9 b^2 e^3 n^2 \log \left (-\frac{e \sqrt [3]{x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d^3}-\frac{3 b^2 e^2 n^2 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d^3 \sqrt [3]{x}}-\frac{e^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{d^3}+\frac{3 b e^3 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^3}+\frac{3 b e^3 n \log \left (-\frac{e \sqrt [3]{x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{d^3}+\frac{3 b e^2 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{d^3 \sqrt [3]{x}}-\frac{3 b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d x^{2/3}}-\frac{\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x}+\frac{b^3 e^3 n^3 \log (x)}{d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*b^2*e^2*n^2*(d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n]))/(d^3*x^(1/3)) + (3*b*e^3*n*(a + b*Log[c*(d +
e*x^(1/3))^n])^2)/(2*d^3) - (3*b*e*n*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(2*d*x^(2/3)) + (3*b*e^2*n*(d + e*x^
(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(d^3*x^(1/3)) - (e^3*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/d^3 - (a +
b*Log[c*(d + e*x^(1/3))^n])^3/x - (9*b^2*e^3*n^2*(a + b*Log[c*(d + e*x^(1/3))^n])*Log[-((e*x^(1/3))/d)])/d^3
+ (3*b*e^3*n*(a + b*Log[c*(d + e*x^(1/3))^n])^2*Log[-((e*x^(1/3))/d)])/d^3 + (b^3*e^3*n^3*Log[x])/d^3 - (9*b^3
*e^3*n^3*PolyLog[2, 1 + (e*x^(1/3))/d])/d^3 + (6*b^2*e^3*n^2*(a + b*Log[c*(d + e*x^(1/3))^n])*PolyLog[2, 1 + (
e*x^(1/3))/d])/d^3 - (6*b^3*e^3*n^3*PolyLog[3, 1 + (e*x^(1/3))/d])/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 \frac{\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x^2} \, dx &=3 \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{x^4} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x}+(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,\sqrt [3]{x}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x}+(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+e \sqrt [3]{x}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x}+\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+e \sqrt [3]{x}\right )}{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+e \sqrt [3]{x}\right )}{d}\\ &=-\frac{3 b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d x^{2/3}}-\frac{\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x}-\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+e \sqrt [3]{x}\right )}{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+e \sqrt [3]{x}\right )}{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+e \sqrt [3]{x}\right )}{d}\\ &=-\frac{3 b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d x^{2/3}}+\frac{3 b e^2 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}{d^3 \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x}+\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+e \sqrt [3]{x}\right )}{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+e \sqrt [3]{x}\right )}{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+e \sqrt [3]{x}\right )}{d^2}-\frac{\left (6 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+e \sqrt [3]{x}\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+e \sqrt [3]{x}\right )}{d^2}\\ &=-\frac{3 b^2 e^2 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 )}{d^3 \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}{2 d x^{2/3}}+\frac{3 b e^2 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}{d^3 \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x}-\frac{6 b^2 e^3 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^3}+\frac{3 b e^3 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 )}{d^3}-\frac{\left (3 e^3\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\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 )}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+e \sqrt [3]{x}\right )}{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+e \sqrt [3]{x}\right )}{d^3}-\frac{\left (6 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+e \sqrt [3]{x}\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+e \sqrt [3]{x}\right )}{d^3}+\frac{\left (6 b^3 e^3 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^3}\\ &=-\frac{3 b^2 e^2 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 )}{d^3 \sqrt [3]{x}}+\frac{3 b e^3 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^3}-\frac{3 b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d x^{2/3}}+\frac{3 b e^2 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}{d^3 \sqrt [3]{x}}-\frac{e^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{d^3}-\frac{\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^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 ) \log \left (-\frac{e \sqrt [3]{x}}{d}\right )}{d^3}+\frac{3 b e^3 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 )}{d^3}+\frac{b^3 e^3 n^3 \log (x)}{d^3}-\frac{6 b^3 e^3 n^3 \text{Li}_2\left (1+\frac{e \sqrt [3]{x}}{d}\right )}{d^3}+\frac{6 b^2 e^3 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^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+e \sqrt [3]{x}\right )}{d^3}-\frac{\left (6 b^3 e^3 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^3}\\ &=-\frac{3 b^2 e^2 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 )}{d^3 \sqrt [3]{x}}+\frac{3 b e^3 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^3}-\frac{3 b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d x^{2/3}}+\frac{3 b e^2 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}{d^3 \sqrt [3]{x}}-\frac{e^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{d^3}-\frac{\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^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 ) \log \left (-\frac{e \sqrt [3]{x}}{d}\right )}{d^3}+\frac{3 b e^3 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 )}{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 \sqrt [3]{x}}{d}\right )}{d^3}+\frac{6 b^2 e^3 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^3}-\frac{6 b^3 e^3 n^3 \text{Li}_3\left (1+\frac{e \sqrt [3]{x}}{d}\right )}{d^3}\\ \end{align*}

Mathematica [A]  time = 0.708821, size = 733, normalized size = 1.67 \[ \frac{-6 b^2 n^2 \left (-2 e^3 x \text{PolyLog}\left (2,\frac{e \sqrt [3]{x}}{d}+1\right )+\log \left (d+e \sqrt [3]{x}\right ) \left (d^2 e \sqrt [3]{x}-2 d e^2 x^{2/3}-2 e^3 x \log \left (-\frac{e \sqrt [3]{x}}{d}\right )-3 e^3 x\right )+\left (d^3+e^3 x\right ) \log ^2\left (d+e \sqrt [3]{x}\right )+d e^2 x^{2/3}+3 e^3 x \log \left (-\frac{e \sqrt [3]{x}}{d}\right )\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )-b n \log \left (d+e \sqrt [3]{x}\right )\right )+b^3 n^3 \left (-12 e^3 x \text{PolyLog}\left (3,\frac{e \sqrt [3]{x}}{d}+1\right )+6 e^3 x \left (2 \log \left (d+e \sqrt [3]{x}\right )-3\right ) \text{PolyLog}\left (2,\frac{e \sqrt [3]{x}}{d}+1\right )-3 d^2 e \sqrt [3]{x} \log ^2\left (d+e \sqrt [3]{x}\right )-2 d^3 \log ^3\left (d+e \sqrt [3]{x}\right )+6 d e^2 x^{2/3} \log ^2\left (d+e \sqrt [3]{x}\right )-6 d e^2 x^{2/3} \log \left (d+e \sqrt [3]{x}\right )-2 e^3 x \log ^3\left (d+e \sqrt [3]{x}\right )+9 e^3 x \log ^2\left (d+e \sqrt [3]{x}\right )+6 e^3 x \log ^2\left (d+e \sqrt [3]{x}\right ) \log \left (-\frac{e \sqrt [3]{x}}{d}\right )-6 e^3 x \log \left (d+e \sqrt [3]{x}\right )+6 e^3 x \log \left (-\frac{e \sqrt [3]{x}}{d}\right )-18 e^3 x \log \left (d+e \sqrt [3]{x}\right ) \log \left (-\frac{e \sqrt [3]{x}}{d}\right )\right )-2 d^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )-b n \log \left (d+e \sqrt [3]{x}\right )\right )^3-6 b d^3 n \log \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )-b n \log \left (d+e \sqrt [3]{x}\right )\right )^2-3 b d^2 e n \sqrt [3]{x} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )-b n \log \left (d+e \sqrt [3]{x}\right )\right )^2+6 b d e^2 n x^{2/3} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )-b n \log \left (d+e \sqrt [3]{x}\right )\right )^2-6 b e^3 n x \log \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )-b n \log \left (d+e \sqrt [3]{x}\right )\right )^2+2 b e^3 n x \log (x) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )-b n \log \left (d+e \sqrt [3]{x}\right )\right )^2}{2 d^3 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.102, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \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^2,x)

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{2 \, b^{3} d^{3} x^{\frac{2}{3}} \log \left ({\left (e x^{\frac{1}{3}} + d\right )}^{n}\right )^{3} +{\left (6 \, b^{3} e^{3} n x^{\frac{5}{3}} \log \left (e x^{\frac{1}{3}} + d\right ) - 6 \, b^{3} d e^{2} n x^{\frac{4}{3}} + 3 \, b^{3} d^{2} e n x - 2 \,{\left (b^{3} e^{3} n x \log \left (x\right ) - 3 \, b^{3} d^{3} \log \left (c\right ) - 3 \, a b^{2} d^{3}\right )} x^{\frac{2}{3}}\right )} \log \left ({\left (e x^{\frac{1}{3}} + d\right )}^{n}\right )^{2}}{2 \, d^{3} x^{\frac{5}{3}}} + \int \frac{3 \,{\left (b^{3} d^{3} e \log \left (c\right )^{3} + 3 \, a b^{2} d^{3} e \log \left (c\right )^{2} + 3 \, a^{2} b d^{3} e \log \left (c\right ) + a^{3} d^{3} e\right )} x^{\frac{5}{3}} + 3 \,{\left (b^{3} d^{4} \log \left (c\right )^{3} + 3 \, a b^{2} d^{4} \log \left (c\right )^{2} + 3 \, a^{2} b d^{4} \log \left (c\right ) + a^{3} d^{4}\right )} x^{\frac{4}{3}} +{\left (6 \, b^{3} e^{4} n^{2} x^{\frac{8}{3}} \log \left (e x^{\frac{1}{3}} + d\right ) - 6 \, b^{3} d e^{3} n^{2} x^{\frac{7}{3}} + 3 \, b^{3} d^{2} e^{2} n^{2} x^{2} + 9 \,{\left (b^{3} d^{3} e \log \left (c\right )^{2} + 2 \, a b^{2} d^{3} e \log \left (c\right ) + a^{2} b d^{3} e\right )} x^{\frac{5}{3}} + 9 \,{\left (b^{3} d^{4} \log \left (c\right )^{2} + 2 \, a b^{2} d^{4} \log \left (c\right ) + a^{2} b d^{4}\right )} x^{\frac{4}{3}} - 2 \,{\left (b^{3} e^{4} n^{2} x^{2} \log \left (x\right ) - 3 \,{\left (b^{3} d^{3} e n \log \left (c\right ) + a b^{2} d^{3} e n\right )} x\right )} x^{\frac{2}{3}}\right )} \log \left ({\left (e x^{\frac{1}{3}} + d\right )}^{n}\right )}{3 \,{\left (d^{3} e x^{\frac{11}{3}} + d^{4} x^{\frac{10}{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^2,x, algorithm="maxima")

[Out]

-1/2*(2*b^3*d^3*x^(2/3)*log((e*x^(1/3) + d)^n)^3 + (6*b^3*e^3*n*x^(5/3)*log(e*x^(1/3) + d) - 6*b^3*d*e^2*n*x^(
4/3) + 3*b^3*d^2*e*n*x - 2*(b^3*e^3*n*x*log(x) - 3*b^3*d^3*log(c) - 3*a*b^2*d^3)*x^(2/3))*log((e*x^(1/3) + d)^
n)^2)/(d^3*x^(5/3)) + integrate(1/3*(3*(b^3*d^3*e*log(c)^3 + 3*a*b^2*d^3*e*log(c)^2 + 3*a^2*b*d^3*e*log(c) + a
^3*d^3*e)*x^(5/3) + 3*(b^3*d^4*log(c)^3 + 3*a*b^2*d^4*log(c)^2 + 3*a^2*b*d^4*log(c) + a^3*d^4)*x^(4/3) + (6*b^
3*e^4*n^2*x^(8/3)*log(e*x^(1/3) + d) - 6*b^3*d*e^3*n^2*x^(7/3) + 3*b^3*d^2*e^2*n^2*x^2 + 9*(b^3*d^3*e*log(c)^2
+ 2*a*b^2*d^3*e*log(c) + a^2*b*d^3*e)*x^(5/3) + 9*(b^3*d^4*log(c)^2 + 2*a*b^2*d^4*log(c) + a^2*b*d^4)*x^(4/3)
- 2*(b^3*e^4*n^2*x^2*log(x) - 3*(b^3*d^3*e*n*log(c) + a*b^2*d^3*e*n)*x)*x^(2/3))*log((e*x^(1/3) + d)^n))/(d^3
*e*x^(11/3) + d^4*x^(10/3)), 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^{2}}, 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^2,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^2, 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**2,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^{2}}\,{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^2,x, algorithm="giac")

[Out]

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