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

Optimal. Leaf size=572 \[ \frac{2 b^2 e^9 n^2 \text{PolyLog}\left (2,\frac{d}{d+\frac{e}{\sqrt [3]{x}}}\right )}{3 d^9}+\frac{b e^7 n x^{2/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^7}+\frac{b e^5 n x^{4/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{6 d^5}-\frac{2 b e^4 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{15 d^4}+\frac{b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^3}-\frac{2 b e^2 n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{21 d^2}-\frac{2 b e^9 n \log \left (1-\frac{d}{d+\frac{e}{\sqrt [3]{x}}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^9}-\frac{2 b e^8 n \sqrt [3]{x} \left (d+\frac{e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^9}-\frac{2 b e^6 n x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^6}+\frac{b e n x^{8/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 d}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac{341 b^2 e^7 n^2 x^{2/3}}{840 d^7}-\frac{533 b^2 e^5 n^2 x^{4/3}}{5040 d^5}+\frac{73 b^2 e^4 n^2 x^{5/3}}{1260 d^4}-\frac{5 b^2 e^3 n^2 x^2}{168 d^3}+\frac{b^2 e^2 n^2 x^{7/3}}{84 d^2}+\frac{481 b^2 e^8 n^2 \sqrt [3]{x}}{420 d^8}+\frac{743 b^2 e^6 n^2 x}{3780 d^6}-\frac{481 b^2 e^9 n^2 \log \left (d+\frac{e}{\sqrt [3]{x}}\right )}{420 d^9}-\frac{761 b^2 e^9 n^2 \log (x)}{1260 d^9} \]

[Out]

(481*b^2*e^8*n^2*x^(1/3))/(420*d^8) - (341*b^2*e^7*n^2*x^(2/3))/(840*d^7) + (743*b^2*e^6*n^2*x)/(3780*d^6) - (
533*b^2*e^5*n^2*x^(4/3))/(5040*d^5) + (73*b^2*e^4*n^2*x^(5/3))/(1260*d^4) - (5*b^2*e^3*n^2*x^2)/(168*d^3) + (b
^2*e^2*n^2*x^(7/3))/(84*d^2) - (481*b^2*e^9*n^2*Log[d + e/x^(1/3)])/(420*d^9) - (2*b*e^8*n*(d + e/x^(1/3))*x^(
1/3)*(a + b*Log[c*(d + e/x^(1/3))^n]))/(3*d^9) + (b*e^7*n*x^(2/3)*(a + b*Log[c*(d + e/x^(1/3))^n]))/(3*d^7) -
(2*b*e^6*n*x*(a + b*Log[c*(d + e/x^(1/3))^n]))/(9*d^6) + (b*e^5*n*x^(4/3)*(a + b*Log[c*(d + e/x^(1/3))^n]))/(6
*d^5) - (2*b*e^4*n*x^(5/3)*(a + b*Log[c*(d + e/x^(1/3))^n]))/(15*d^4) + (b*e^3*n*x^2*(a + b*Log[c*(d + e/x^(1/
3))^n]))/(9*d^3) - (2*b*e^2*n*x^(7/3)*(a + b*Log[c*(d + e/x^(1/3))^n]))/(21*d^2) + (b*e*n*x^(8/3)*(a + b*Log[c
*(d + e/x^(1/3))^n]))/(12*d) - (2*b*e^9*n*Log[1 - d/(d + e/x^(1/3))]*(a + b*Log[c*(d + e/x^(1/3))^n]))/(3*d^9)
+ (x^3*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/3 - (761*b^2*e^9*n^2*Log[x])/(1260*d^9) + (2*b^2*e^9*n^2*PolyLog[2
, d/(d + e/x^(1/3))])/(3*d^9)

________________________________________________________________________________________

Rubi [A]  time = 1.71509, antiderivative size = 596, normalized size of antiderivative = 1.04, number of steps used = 38, number of rules used = 12, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2454, 2398, 2411, 2347, 2344, 2301, 2317, 2391, 2314, 31, 2319, 44} \[ -\frac{2 b^2 e^9 n^2 \text{PolyLog}\left (2,\frac{e}{d \sqrt [3]{x}}+1\right )}{3 d^9}+\frac{b e^7 n x^{2/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^7}+\frac{b e^5 n x^{4/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{6 d^5}-\frac{2 b e^4 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{15 d^4}+\frac{b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^3}-\frac{2 b e^2 n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{21 d^2}+\frac{e^9 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{3 d^9}-\frac{2 b e^9 n \log \left (-\frac{e}{d \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^9}-\frac{2 b e^8 n \sqrt [3]{x} \left (d+\frac{e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^9}-\frac{2 b e^6 n x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^6}+\frac{b e n x^{8/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 d}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac{341 b^2 e^7 n^2 x^{2/3}}{840 d^7}-\frac{533 b^2 e^5 n^2 x^{4/3}}{5040 d^5}+\frac{73 b^2 e^4 n^2 x^{5/3}}{1260 d^4}-\frac{5 b^2 e^3 n^2 x^2}{168 d^3}+\frac{b^2 e^2 n^2 x^{7/3}}{84 d^2}+\frac{481 b^2 e^8 n^2 \sqrt [3]{x}}{420 d^8}+\frac{743 b^2 e^6 n^2 x}{3780 d^6}-\frac{481 b^2 e^9 n^2 \log \left (d+\frac{e}{\sqrt [3]{x}}\right )}{420 d^9}-\frac{761 b^2 e^9 n^2 \log (x)}{1260 d^9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(481*b^2*e^8*n^2*x^(1/3))/(420*d^8) - (341*b^2*e^7*n^2*x^(2/3))/(840*d^7) + (743*b^2*e^6*n^2*x)/(3780*d^6) - (
533*b^2*e^5*n^2*x^(4/3))/(5040*d^5) + (73*b^2*e^4*n^2*x^(5/3))/(1260*d^4) - (5*b^2*e^3*n^2*x^2)/(168*d^3) + (b
^2*e^2*n^2*x^(7/3))/(84*d^2) - (481*b^2*e^9*n^2*Log[d + e/x^(1/3)])/(420*d^9) - (2*b*e^8*n*(d + e/x^(1/3))*x^(
1/3)*(a + b*Log[c*(d + e/x^(1/3))^n]))/(3*d^9) + (b*e^7*n*x^(2/3)*(a + b*Log[c*(d + e/x^(1/3))^n]))/(3*d^7) -
(2*b*e^6*n*x*(a + b*Log[c*(d + e/x^(1/3))^n]))/(9*d^6) + (b*e^5*n*x^(4/3)*(a + b*Log[c*(d + e/x^(1/3))^n]))/(6
*d^5) - (2*b*e^4*n*x^(5/3)*(a + b*Log[c*(d + e/x^(1/3))^n]))/(15*d^4) + (b*e^3*n*x^2*(a + b*Log[c*(d + e/x^(1/
3))^n]))/(9*d^3) - (2*b*e^2*n*x^(7/3)*(a + b*Log[c*(d + e/x^(1/3))^n]))/(21*d^2) + (b*e*n*x^(8/3)*(a + b*Log[c
*(d + e/x^(1/3))^n]))/(12*d) + (e^9*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/(3*d^9) + (x^3*(a + b*Log[c*(d + e/x^(
1/3))^n])^2)/3 - (2*b*e^9*n*(a + b*Log[c*(d + e/x^(1/3))^n])*Log[-(e/(d*x^(1/3)))])/(3*d^9) - (761*b^2*e^9*n^2
*Log[x])/(1260*d^9) - (2*b^2*e^9*n^2*PolyLog[2, 1 + e/(d*x^(1/3))])/(3*d^9)

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 2301

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

Rule 2317

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

Rule 2391

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

Rule 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 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 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 x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \, dx &=-\left (3 \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^{10}} \, dx,x,\frac{1}{\sqrt [3]{x}}\right )\right )\\ &=\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac{1}{3} (2 b e n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c (d+e x)^n\right )}{x^9 (d+e x)} \, dx,x,\frac{1}{\sqrt [3]{x}}\right )\\ &=\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac{1}{3} (2 b n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^9} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )\\ &=\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac{(2 b n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\left (-\frac{d}{e}+\frac{x}{e}\right )^9} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{3 d}+\frac{(2 b e n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^8} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{3 d}\\ &=\frac{b e n x^{8/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 d}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2+\frac{(2 b e n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\left (-\frac{d}{e}+\frac{x}{e}\right )^8} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{3 d^2}-\frac{\left (2 b e^2 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^7} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{3 d^2}-\frac{\left (b^2 e n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^8} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{12 d}\\ &=-\frac{2 b e^2 n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{21 d^2}+\frac{b e n x^{8/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 d}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac{\left (2 b e^2 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\left (-\frac{d}{e}+\frac{x}{e}\right )^7} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{3 d^3}+\frac{\left (2 b e^3 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^6} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{3 d^3}-\frac{\left (b^2 e n^2\right ) \operatorname{Subst}\left (\int \left (\frac{e^8}{d (d-x)^8}+\frac{e^8}{d^2 (d-x)^7}+\frac{e^8}{d^3 (d-x)^6}+\frac{e^8}{d^4 (d-x)^5}+\frac{e^8}{d^5 (d-x)^4}+\frac{e^8}{d^6 (d-x)^3}+\frac{e^8}{d^7 (d-x)^2}+\frac{e^8}{d^8 (d-x)}+\frac{e^8}{d^8 x}\right ) \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{12 d}+\frac{\left (2 b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^7} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{21 d^2}\\ &=\frac{b^2 e^8 n^2 \sqrt [3]{x}}{12 d^8}-\frac{b^2 e^7 n^2 x^{2/3}}{24 d^7}+\frac{b^2 e^6 n^2 x}{36 d^6}-\frac{b^2 e^5 n^2 x^{4/3}}{48 d^5}+\frac{b^2 e^4 n^2 x^{5/3}}{60 d^4}-\frac{b^2 e^3 n^2 x^2}{72 d^3}+\frac{b^2 e^2 n^2 x^{7/3}}{84 d^2}-\frac{b^2 e^9 n^2 \log \left (d+\frac{e}{\sqrt [3]{x}}\right )}{12 d^9}+\frac{b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^3}-\frac{2 b e^2 n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{21 d^2}+\frac{b e n x^{8/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 d}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac{b^2 e^9 n^2 \log (x)}{36 d^9}+\frac{\left (2 b e^3 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\left (-\frac{d}{e}+\frac{x}{e}\right )^6} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{3 d^4}-\frac{\left (2 b e^4 n\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+\frac{e}{\sqrt [3]{x}}\right )}{3 d^4}+\frac{\left (2 b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \left (-\frac{e^7}{d (d-x)^7}-\frac{e^7}{d^2 (d-x)^6}-\frac{e^7}{d^3 (d-x)^5}-\frac{e^7}{d^4 (d-x)^4}-\frac{e^7}{d^5 (d-x)^3}-\frac{e^7}{d^6 (d-x)^2}-\frac{e^7}{d^7 (d-x)}-\frac{e^7}{d^7 x}\right ) \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{21 d^2}-\frac{\left (b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^6} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{9 d^3}\\ &=\frac{5 b^2 e^8 n^2 \sqrt [3]{x}}{28 d^8}-\frac{5 b^2 e^7 n^2 x^{2/3}}{56 d^7}+\frac{5 b^2 e^6 n^2 x}{84 d^6}-\frac{5 b^2 e^5 n^2 x^{4/3}}{112 d^5}+\frac{b^2 e^4 n^2 x^{5/3}}{28 d^4}-\frac{5 b^2 e^3 n^2 x^2}{168 d^3}+\frac{b^2 e^2 n^2 x^{7/3}}{84 d^2}-\frac{5 b^2 e^9 n^2 \log \left (d+\frac{e}{\sqrt [3]{x}}\right )}{28 d^9}-\frac{2 b e^4 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{15 d^4}+\frac{b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^3}-\frac{2 b e^2 n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{21 d^2}+\frac{b e n x^{8/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 d}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac{5 b^2 e^9 n^2 \log (x)}{84 d^9}-\frac{\left (2 b e^4 n\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+\frac{e}{\sqrt [3]{x}}\right )}{3 d^5}+\frac{\left (2 b e^5 n\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+\frac{e}{\sqrt [3]{x}}\right )}{3 d^5}-\frac{\left (b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \left (\frac{e^6}{d (d-x)^6}+\frac{e^6}{d^2 (d-x)^5}+\frac{e^6}{d^3 (d-x)^4}+\frac{e^6}{d^4 (d-x)^3}+\frac{e^6}{d^5 (d-x)^2}+\frac{e^6}{d^6 (d-x)}+\frac{e^6}{d^6 x}\right ) \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{9 d^3}+\frac{\left (2 b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^5} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{15 d^4}\\ &=\frac{73 b^2 e^8 n^2 \sqrt [3]{x}}{252 d^8}-\frac{73 b^2 e^7 n^2 x^{2/3}}{504 d^7}+\frac{73 b^2 e^6 n^2 x}{756 d^6}-\frac{73 b^2 e^5 n^2 x^{4/3}}{1008 d^5}+\frac{73 b^2 e^4 n^2 x^{5/3}}{1260 d^4}-\frac{5 b^2 e^3 n^2 x^2}{168 d^3}+\frac{b^2 e^2 n^2 x^{7/3}}{84 d^2}-\frac{73 b^2 e^9 n^2 \log \left (d+\frac{e}{\sqrt [3]{x}}\right )}{252 d^9}+\frac{b e^5 n x^{4/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{6 d^5}-\frac{2 b e^4 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{15 d^4}+\frac{b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^3}-\frac{2 b e^2 n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{21 d^2}+\frac{b e n x^{8/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 d}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac{73 b^2 e^9 n^2 \log (x)}{756 d^9}+\frac{\left (2 b e^5 n\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+\frac{e}{\sqrt [3]{x}}\right )}{3 d^6}-\frac{\left (2 b e^6 n\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+\frac{e}{\sqrt [3]{x}}\right )}{3 d^6}+\frac{\left (2 b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \left (-\frac{e^5}{d (d-x)^5}-\frac{e^5}{d^2 (d-x)^4}-\frac{e^5}{d^3 (d-x)^3}-\frac{e^5}{d^4 (d-x)^2}-\frac{e^5}{d^5 (d-x)}-\frac{e^5}{d^5 x}\right ) \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{15 d^4}-\frac{\left (b^2 e^5 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^4} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{6 d^5}\\ &=\frac{533 b^2 e^8 n^2 \sqrt [3]{x}}{1260 d^8}-\frac{533 b^2 e^7 n^2 x^{2/3}}{2520 d^7}+\frac{533 b^2 e^6 n^2 x}{3780 d^6}-\frac{533 b^2 e^5 n^2 x^{4/3}}{5040 d^5}+\frac{73 b^2 e^4 n^2 x^{5/3}}{1260 d^4}-\frac{5 b^2 e^3 n^2 x^2}{168 d^3}+\frac{b^2 e^2 n^2 x^{7/3}}{84 d^2}-\frac{533 b^2 e^9 n^2 \log \left (d+\frac{e}{\sqrt [3]{x}}\right )}{1260 d^9}-\frac{2 b e^6 n x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^6}+\frac{b e^5 n x^{4/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{6 d^5}-\frac{2 b e^4 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{15 d^4}+\frac{b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^3}-\frac{2 b e^2 n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{21 d^2}+\frac{b e n x^{8/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 d}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac{533 b^2 e^9 n^2 \log (x)}{3780 d^9}-\frac{\left (2 b e^6 n\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+\frac{e}{\sqrt [3]{x}}\right )}{3 d^7}+\frac{\left (2 b e^7 n\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}{\sqrt [3]{x}}\right )}{3 d^7}-\frac{\left (b^2 e^5 n^2\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+\frac{e}{\sqrt [3]{x}}\right )}{6 d^5}+\frac{\left (2 b^2 e^6 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^3} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{9 d^6}\\ &=\frac{743 b^2 e^8 n^2 \sqrt [3]{x}}{1260 d^8}-\frac{743 b^2 e^7 n^2 x^{2/3}}{2520 d^7}+\frac{743 b^2 e^6 n^2 x}{3780 d^6}-\frac{533 b^2 e^5 n^2 x^{4/3}}{5040 d^5}+\frac{73 b^2 e^4 n^2 x^{5/3}}{1260 d^4}-\frac{5 b^2 e^3 n^2 x^2}{168 d^3}+\frac{b^2 e^2 n^2 x^{7/3}}{84 d^2}-\frac{743 b^2 e^9 n^2 \log \left (d+\frac{e}{\sqrt [3]{x}}\right )}{1260 d^9}+\frac{b e^7 n x^{2/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^7}-\frac{2 b e^6 n x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^6}+\frac{b e^5 n x^{4/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{6 d^5}-\frac{2 b e^4 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{15 d^4}+\frac{b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^3}-\frac{2 b e^2 n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{21 d^2}+\frac{b e n x^{8/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 d}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac{743 b^2 e^9 n^2 \log (x)}{3780 d^9}+\frac{\left (2 b e^7 n\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}{\sqrt [3]{x}}\right )}{3 d^8}-\frac{\left (2 b e^8 n\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}{\sqrt [3]{x}}\right )}{3 d^8}+\frac{\left (2 b^2 e^6 n^2\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+\frac{e}{\sqrt [3]{x}}\right )}{9 d^6}-\frac{\left (b^2 e^7 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{3 d^7}\\ &=\frac{341 b^2 e^8 n^2 \sqrt [3]{x}}{420 d^8}-\frac{341 b^2 e^7 n^2 x^{2/3}}{840 d^7}+\frac{743 b^2 e^6 n^2 x}{3780 d^6}-\frac{533 b^2 e^5 n^2 x^{4/3}}{5040 d^5}+\frac{73 b^2 e^4 n^2 x^{5/3}}{1260 d^4}-\frac{5 b^2 e^3 n^2 x^2}{168 d^3}+\frac{b^2 e^2 n^2 x^{7/3}}{84 d^2}-\frac{341 b^2 e^9 n^2 \log \left (d+\frac{e}{\sqrt [3]{x}}\right )}{420 d^9}-\frac{2 b e^8 n \left (d+\frac{e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^9}+\frac{b e^7 n x^{2/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^7}-\frac{2 b e^6 n x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^6}+\frac{b e^5 n x^{4/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{6 d^5}-\frac{2 b e^4 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{15 d^4}+\frac{b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^3}-\frac{2 b e^2 n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{21 d^2}+\frac{b e n x^{8/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 d}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac{341 b^2 e^9 n^2 \log (x)}{1260 d^9}-\frac{\left (2 b e^8 n\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}{\sqrt [3]{x}}\right )}{3 d^9}+\frac{\left (2 b e^9 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{3 d^9}-\frac{\left (b^2 e^7 n^2\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+\frac{e}{\sqrt [3]{x}}\right )}{3 d^7}+\frac{\left (2 b^2 e^8 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{3 d^9}\\ &=\frac{481 b^2 e^8 n^2 \sqrt [3]{x}}{420 d^8}-\frac{341 b^2 e^7 n^2 x^{2/3}}{840 d^7}+\frac{743 b^2 e^6 n^2 x}{3780 d^6}-\frac{533 b^2 e^5 n^2 x^{4/3}}{5040 d^5}+\frac{73 b^2 e^4 n^2 x^{5/3}}{1260 d^4}-\frac{5 b^2 e^3 n^2 x^2}{168 d^3}+\frac{b^2 e^2 n^2 x^{7/3}}{84 d^2}-\frac{481 b^2 e^9 n^2 \log \left (d+\frac{e}{\sqrt [3]{x}}\right )}{420 d^9}-\frac{2 b e^8 n \left (d+\frac{e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^9}+\frac{b e^7 n x^{2/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^7}-\frac{2 b e^6 n x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^6}+\frac{b e^5 n x^{4/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{6 d^5}-\frac{2 b e^4 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{15 d^4}+\frac{b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^3}-\frac{2 b e^2 n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{21 d^2}+\frac{b e n x^{8/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 d}+\frac{e^9 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{3 d^9}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac{2 b e^9 n \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right ) \log \left (-\frac{e}{d \sqrt [3]{x}}\right )}{3 d^9}-\frac{761 b^2 e^9 n^2 \log (x)}{1260 d^9}+\frac{\left (2 b^2 e^9 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{3 d^9}\\ &=\frac{481 b^2 e^8 n^2 \sqrt [3]{x}}{420 d^8}-\frac{341 b^2 e^7 n^2 x^{2/3}}{840 d^7}+\frac{743 b^2 e^6 n^2 x}{3780 d^6}-\frac{533 b^2 e^5 n^2 x^{4/3}}{5040 d^5}+\frac{73 b^2 e^4 n^2 x^{5/3}}{1260 d^4}-\frac{5 b^2 e^3 n^2 x^2}{168 d^3}+\frac{b^2 e^2 n^2 x^{7/3}}{84 d^2}-\frac{481 b^2 e^9 n^2 \log \left (d+\frac{e}{\sqrt [3]{x}}\right )}{420 d^9}-\frac{2 b e^8 n \left (d+\frac{e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^9}+\frac{b e^7 n x^{2/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^7}-\frac{2 b e^6 n x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^6}+\frac{b e^5 n x^{4/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{6 d^5}-\frac{2 b e^4 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{15 d^4}+\frac{b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^3}-\frac{2 b e^2 n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{21 d^2}+\frac{b e n x^{8/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 d}+\frac{e^9 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{3 d^9}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac{2 b e^9 n \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right ) \log \left (-\frac{e}{d \sqrt [3]{x}}\right )}{3 d^9}-\frac{761 b^2 e^9 n^2 \log (x)}{1260 d^9}-\frac{2 b^2 e^9 n^2 \text{Li}_2\left (1+\frac{e}{d \sqrt [3]{x}}\right )}{3 d^9}\\ \end{align*}

Mathematica [A]  time = 0.417994, size = 738, normalized size = 1.29 \[ \frac{10080 b^2 e^9 n^2 \text{PolyLog}\left (2,\frac{d \sqrt [3]{x}}{e}+1\right )+5040 a^2 d^9 x^3+10080 a b d^9 x^3 \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )-1440 a b d^7 e^2 n x^{7/3}+1680 a b d^6 e^3 n x^2-2016 a b d^5 e^4 n x^{5/3}+2520 a b d^4 e^5 n x^{4/3}+5040 a b d^2 e^7 n x^{2/3}-3360 a b d^3 e^6 n x+1260 a b d^8 e n x^{8/3}-10080 a b d e^8 n \sqrt [3]{x}+10080 a b e^9 n \log \left (d \sqrt [3]{x}+e\right )-1440 b^2 d^7 e^2 n x^{7/3} \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )+1680 b^2 d^6 e^3 n x^2 \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )-2016 b^2 d^5 e^4 n x^{5/3} \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )+2520 b^2 d^4 e^5 n x^{4/3} \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )+5040 b^2 d^2 e^7 n x^{2/3} \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )-3360 b^2 d^3 e^6 n x \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )+5040 b^2 d^9 x^3 \log ^2\left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )+1260 b^2 d^8 e n x^{8/3} \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )-10080 b^2 d e^8 n \sqrt [3]{x} \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )+10080 b^2 e^9 n \log \left (d \sqrt [3]{x}+e\right ) \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )+180 b^2 d^7 e^2 n^2 x^{7/3}-450 b^2 d^6 e^3 n^2 x^2+876 b^2 d^5 e^4 n^2 x^{5/3}-1599 b^2 d^4 e^5 n^2 x^{4/3}-6138 b^2 d^2 e^7 n^2 x^{2/3}+2972 b^2 d^3 e^6 n^2 x+17316 b^2 d e^8 n^2 \sqrt [3]{x}-5040 b^2 e^9 n^2 \log ^2\left (d \sqrt [3]{x}+e\right )-22356 b^2 e^9 n^2 \log \left (d+\frac{e}{\sqrt [3]{x}}\right )-5040 b^2 e^9 n^2 \log \left (d \sqrt [3]{x}+e\right )+10080 b^2 e^9 n^2 \log \left (d \sqrt [3]{x}+e\right ) \log \left (-\frac{d \sqrt [3]{x}}{e}\right )-7452 b^2 e^9 n^2 \log (x)}{15120 d^9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10080*a*b*d*e^8*n*x^(1/3) + 17316*b^2*d*e^8*n^2*x^(1/3) + 5040*a*b*d^2*e^7*n*x^(2/3) - 6138*b^2*d^2*e^7*n^2*
x^(2/3) - 3360*a*b*d^3*e^6*n*x + 2972*b^2*d^3*e^6*n^2*x + 2520*a*b*d^4*e^5*n*x^(4/3) - 1599*b^2*d^4*e^5*n^2*x^
(4/3) - 2016*a*b*d^5*e^4*n*x^(5/3) + 876*b^2*d^5*e^4*n^2*x^(5/3) + 1680*a*b*d^6*e^3*n*x^2 - 450*b^2*d^6*e^3*n^
2*x^2 - 1440*a*b*d^7*e^2*n*x^(7/3) + 180*b^2*d^7*e^2*n^2*x^(7/3) + 1260*a*b*d^8*e*n*x^(8/3) + 5040*a^2*d^9*x^3
- 22356*b^2*e^9*n^2*Log[d + e/x^(1/3)] - 10080*b^2*d*e^8*n*x^(1/3)*Log[c*(d + e/x^(1/3))^n] + 5040*b^2*d^2*e^
7*n*x^(2/3)*Log[c*(d + e/x^(1/3))^n] - 3360*b^2*d^3*e^6*n*x*Log[c*(d + e/x^(1/3))^n] + 2520*b^2*d^4*e^5*n*x^(4
/3)*Log[c*(d + e/x^(1/3))^n] - 2016*b^2*d^5*e^4*n*x^(5/3)*Log[c*(d + e/x^(1/3))^n] + 1680*b^2*d^6*e^3*n*x^2*Lo
g[c*(d + e/x^(1/3))^n] - 1440*b^2*d^7*e^2*n*x^(7/3)*Log[c*(d + e/x^(1/3))^n] + 1260*b^2*d^8*e*n*x^(8/3)*Log[c*
(d + e/x^(1/3))^n] + 10080*a*b*d^9*x^3*Log[c*(d + e/x^(1/3))^n] + 5040*b^2*d^9*x^3*Log[c*(d + e/x^(1/3))^n]^2
+ 10080*a*b*e^9*n*Log[e + d*x^(1/3)] - 5040*b^2*e^9*n^2*Log[e + d*x^(1/3)] + 10080*b^2*e^9*n*Log[c*(d + e/x^(1
/3))^n]*Log[e + d*x^(1/3)] - 5040*b^2*e^9*n^2*Log[e + d*x^(1/3)]^2 + 10080*b^2*e^9*n^2*Log[e + d*x^(1/3)]*Log[
-((d*x^(1/3))/e)] - 7452*b^2*e^9*n^2*Log[x] + 10080*b^2*e^9*n^2*PolyLog[2, 1 + (d*x^(1/3))/e])/(15120*d^9)

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Maple [F]  time = 0.346, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( a+b\ln \left ( c \left ( d+{e{\frac{1}{\sqrt [3]{x}}}} \right ) ^{n} \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{3} \, b^{2} x^{3} \log \left ({\left (d x^{\frac{1}{3}} + e\right )}^{n}\right )^{2} - \int -\frac{9 \,{\left (b^{2} d \log \left (c\right )^{2} + 2 \, a b d \log \left (c\right ) + a^{2} d\right )} x^{3} + 9 \,{\left (b^{2} e \log \left (c\right )^{2} + 2 \, a b e \log \left (c\right ) + a^{2} e\right )} x^{\frac{8}{3}} + 9 \,{\left (b^{2} d x^{3} + b^{2} e x^{\frac{8}{3}}\right )} \log \left (x^{\frac{1}{3} \, n}\right )^{2} - 2 \,{\left (b^{2} d n x^{3} - 9 \,{\left (b^{2} d \log \left (c\right ) + a b d\right )} x^{3} - 9 \,{\left (b^{2} e \log \left (c\right ) + a b e\right )} x^{\frac{8}{3}} + 9 \,{\left (b^{2} d x^{3} + b^{2} e x^{\frac{8}{3}}\right )} \log \left (x^{\frac{1}{3} \, n}\right )\right )} \log \left ({\left (d x^{\frac{1}{3}} + e\right )}^{n}\right ) - 18 \,{\left ({\left (b^{2} d \log \left (c\right ) + a b d\right )} x^{3} +{\left (b^{2} e \log \left (c\right ) + a b e\right )} x^{\frac{8}{3}}\right )} \log \left (x^{\frac{1}{3} \, n}\right )}{9 \,{\left (d x + e x^{\frac{2}{3}}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b^2*x^3*log((d*x^(1/3) + e)^n)^2 - integrate(-1/9*(9*(b^2*d*log(c)^2 + 2*a*b*d*log(c) + a^2*d)*x^3 + 9*(b^
2*e*log(c)^2 + 2*a*b*e*log(c) + a^2*e)*x^(8/3) + 9*(b^2*d*x^3 + b^2*e*x^(8/3))*log(x^(1/3*n))^2 - 2*(b^2*d*n*x
^3 - 9*(b^2*d*log(c) + a*b*d)*x^3 - 9*(b^2*e*log(c) + a*b*e)*x^(8/3) + 9*(b^2*d*x^3 + b^2*e*x^(8/3))*log(x^(1/
3*n)))*log((d*x^(1/3) + e)^n) - 18*((b^2*d*log(c) + a*b*d)*x^3 + (b^2*e*log(c) + a*b*e)*x^(8/3))*log(x^(1/3*n)
))/(d*x + e*x^(2/3)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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