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

Optimal. Leaf size=400 \[ -\frac{b^2 e^6 n^2 \text{PolyLog}\left (2,\frac{d}{d+\frac{e}{\sqrt [3]{x}}}\right )}{d^6}-\frac{b e^4 n x^{2/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{2 d^4}-\frac{b e^2 n x^{4/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{4 d^2}+\frac{b e^6 n \log \left (1-\frac{d}{d+\frac{e}{\sqrt [3]{x}}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^6}+\frac{b e^5 n \sqrt [3]{x} \left (d+\frac{e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^6}+\frac{b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^3}+\frac{b e n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{5 d}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2+\frac{47 b^2 e^4 n^2 x^{2/3}}{120 d^4}+\frac{b^2 e^2 n^2 x^{4/3}}{20 d^2}-\frac{77 b^2 e^5 n^2 \sqrt [3]{x}}{60 d^5}-\frac{3 b^2 e^3 n^2 x}{20 d^3}+\frac{77 b^2 e^6 n^2 \log \left (d+\frac{e}{\sqrt [3]{x}}\right )}{60 d^6}+\frac{137 b^2 e^6 n^2 \log (x)}{180 d^6} \]

[Out]

(-77*b^2*e^5*n^2*x^(1/3))/(60*d^5) + (47*b^2*e^4*n^2*x^(2/3))/(120*d^4) - (3*b^2*e^3*n^2*x)/(20*d^3) + (b^2*e^
2*n^2*x^(4/3))/(20*d^2) + (77*b^2*e^6*n^2*Log[d + e/x^(1/3)])/(60*d^6) + (b*e^5*n*(d + e/x^(1/3))*x^(1/3)*(a +
b*Log[c*(d + e/x^(1/3))^n]))/d^6 - (b*e^4*n*x^(2/3)*(a + b*Log[c*(d + e/x^(1/3))^n]))/(2*d^4) + (b*e^3*n*x*(a
+ b*Log[c*(d + e/x^(1/3))^n]))/(3*d^3) - (b*e^2*n*x^(4/3)*(a + b*Log[c*(d + e/x^(1/3))^n]))/(4*d^2) + (b*e*n*
x^(5/3)*(a + b*Log[c*(d + e/x^(1/3))^n]))/(5*d) + (b*e^6*n*Log[1 - d/(d + e/x^(1/3))]*(a + b*Log[c*(d + e/x^(1
/3))^n]))/d^6 + (x^2*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/2 + (137*b^2*e^6*n^2*Log[x])/(180*d^6) - (b^2*e^6*n^2
*PolyLog[2, d/(d + e/x^(1/3))])/d^6

________________________________________________________________________________________

Rubi [A]  time = 1.02412, antiderivative size = 423, normalized size of antiderivative = 1.06, number of steps used = 26, number of rules used = 12, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546, Rules used = {2454, 2398, 2411, 2347, 2344, 2301, 2317, 2391, 2314, 31, 2319, 44} \[ \frac{b^2 e^6 n^2 \text{PolyLog}\left (2,\frac{e}{d \sqrt [3]{x}}+1\right )}{d^6}-\frac{b e^4 n x^{2/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{2 d^4}-\frac{b e^2 n x^{4/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{4 d^2}-\frac{e^6 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{2 d^6}+\frac{b e^6 n \log \left (-\frac{e}{d \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^6}+\frac{b e^5 n \sqrt [3]{x} \left (d+\frac{e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^6}+\frac{b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^3}+\frac{b e n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{5 d}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2+\frac{47 b^2 e^4 n^2 x^{2/3}}{120 d^4}+\frac{b^2 e^2 n^2 x^{4/3}}{20 d^2}-\frac{77 b^2 e^5 n^2 \sqrt [3]{x}}{60 d^5}-\frac{3 b^2 e^3 n^2 x}{20 d^3}+\frac{77 b^2 e^6 n^2 \log \left (d+\frac{e}{\sqrt [3]{x}}\right )}{60 d^6}+\frac{137 b^2 e^6 n^2 \log (x)}{180 d^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-77*b^2*e^5*n^2*x^(1/3))/(60*d^5) + (47*b^2*e^4*n^2*x^(2/3))/(120*d^4) - (3*b^2*e^3*n^2*x)/(20*d^3) + (b^2*e^
2*n^2*x^(4/3))/(20*d^2) + (77*b^2*e^6*n^2*Log[d + e/x^(1/3)])/(60*d^6) + (b*e^5*n*(d + e/x^(1/3))*x^(1/3)*(a +
b*Log[c*(d + e/x^(1/3))^n]))/d^6 - (b*e^4*n*x^(2/3)*(a + b*Log[c*(d + e/x^(1/3))^n]))/(2*d^4) + (b*e^3*n*x*(a
+ b*Log[c*(d + e/x^(1/3))^n]))/(3*d^3) - (b*e^2*n*x^(4/3)*(a + b*Log[c*(d + e/x^(1/3))^n]))/(4*d^2) + (b*e*n*
x^(5/3)*(a + b*Log[c*(d + e/x^(1/3))^n]))/(5*d) - (e^6*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/(2*d^6) + (x^2*(a +
b*Log[c*(d + e/x^(1/3))^n])^2)/2 + (b*e^6*n*(a + b*Log[c*(d + e/x^(1/3))^n])*Log[-(e/(d*x^(1/3)))])/d^6 + (13
7*b^2*e^6*n^2*Log[x])/(180*d^6) + (b^2*e^6*n^2*PolyLog[2, 1 + e/(d*x^(1/3))])/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 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 \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^7} \, dx,x,\frac{1}{\sqrt [3]{x}}\right )\right )\\ &=\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2-(b e n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c (d+e x)^n\right )}{x^6 (d+e x)} \, dx,x,\frac{1}{\sqrt [3]{x}}\right )\\ &=\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^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 )^6} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )\\ &=\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac{(b n) \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 )}{d}+\frac{(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 )^5} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{d}\\ &=\frac{b e n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{5 d}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2+\frac{(b e n) \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 )}{d^2}-\frac{\left (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 )^4} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{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 )^5} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{5 d}\\ &=-\frac{b e^2 n x^{4/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{4 d^2}+\frac{b e n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{5 d}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac{\left (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 )^4} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{d^3}+\frac{\left (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 )^3} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{d^3}-\frac{\left (b^2 e 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 )}{5 d}+\frac{\left (b^2 e^2 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 )}{4 d^2}\\ &=-\frac{b^2 e^5 n^2 \sqrt [3]{x}}{5 d^5}+\frac{b^2 e^4 n^2 x^{2/3}}{10 d^4}-\frac{b^2 e^3 n^2 x}{15 d^3}+\frac{b^2 e^2 n^2 x^{4/3}}{20 d^2}+\frac{b^2 e^6 n^2 \log \left (d+\frac{e}{\sqrt [3]{x}}\right )}{5 d^6}+\frac{b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^3}-\frac{b e^2 n x^{4/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{4 d^2}+\frac{b e n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{5 d}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2+\frac{b^2 e^6 n^2 \log (x)}{15 d^6}+\frac{\left (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 )^3} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{d^4}-\frac{\left (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 )^2} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{d^4}+\frac{\left (b^2 e^2 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 )}{4 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 )^3} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{3 d^3}\\ &=-\frac{9 b^2 e^5 n^2 \sqrt [3]{x}}{20 d^5}+\frac{9 b^2 e^4 n^2 x^{2/3}}{40 d^4}-\frac{3 b^2 e^3 n^2 x}{20 d^3}+\frac{b^2 e^2 n^2 x^{4/3}}{20 d^2}+\frac{9 b^2 e^6 n^2 \log \left (d+\frac{e}{\sqrt [3]{x}}\right )}{20 d^6}-\frac{b e^4 n x^{2/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{2 d^4}+\frac{b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^3}-\frac{b e^2 n x^{4/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{4 d^2}+\frac{b e n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{5 d}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2+\frac{3 b^2 e^6 n^2 \log (x)}{20 d^6}-\frac{\left (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 )^2} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{d^5}+\frac{\left (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 )} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{d^5}-\frac{\left (b^2 e^3 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 )}{3 d^3}+\frac{\left (b^2 e^4 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 )}{2 d^4}\\ &=-\frac{47 b^2 e^5 n^2 \sqrt [3]{x}}{60 d^5}+\frac{47 b^2 e^4 n^2 x^{2/3}}{120 d^4}-\frac{3 b^2 e^3 n^2 x}{20 d^3}+\frac{b^2 e^2 n^2 x^{4/3}}{20 d^2}+\frac{47 b^2 e^6 n^2 \log \left (d+\frac{e}{\sqrt [3]{x}}\right )}{60 d^6}+\frac{b e^5 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 )}{d^6}-\frac{b e^4 n x^{2/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{2 d^4}+\frac{b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^3}-\frac{b e^2 n x^{4/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{4 d^2}+\frac{b e n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{5 d}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2+\frac{47 b^2 e^6 n^2 \log (x)}{180 d^6}+\frac{\left (b e^5 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 )}{d^6}-\frac{\left (b e^6 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 )}{d^6}+\frac{\left (b^2 e^4 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 )}{2 d^4}-\frac{\left (b^2 e^5 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+\frac{e}{\sqrt [3]{x}}\right )}{d^6}\\ &=-\frac{77 b^2 e^5 n^2 \sqrt [3]{x}}{60 d^5}+\frac{47 b^2 e^4 n^2 x^{2/3}}{120 d^4}-\frac{3 b^2 e^3 n^2 x}{20 d^3}+\frac{b^2 e^2 n^2 x^{4/3}}{20 d^2}+\frac{77 b^2 e^6 n^2 \log \left (d+\frac{e}{\sqrt [3]{x}}\right )}{60 d^6}+\frac{b e^5 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 )}{d^6}-\frac{b e^4 n x^{2/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{2 d^4}+\frac{b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^3}-\frac{b e^2 n x^{4/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{4 d^2}+\frac{b e n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{5 d}-\frac{e^6 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{2 d^6}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2+\frac{b e^6 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 )}{d^6}+\frac{137 b^2 e^6 n^2 \log (x)}{180 d^6}-\frac{\left (b^2 e^6 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 )}{d^6}\\ &=-\frac{77 b^2 e^5 n^2 \sqrt [3]{x}}{60 d^5}+\frac{47 b^2 e^4 n^2 x^{2/3}}{120 d^4}-\frac{3 b^2 e^3 n^2 x}{20 d^3}+\frac{b^2 e^2 n^2 x^{4/3}}{20 d^2}+\frac{77 b^2 e^6 n^2 \log \left (d+\frac{e}{\sqrt [3]{x}}\right )}{60 d^6}+\frac{b e^5 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 )}{d^6}-\frac{b e^4 n x^{2/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{2 d^4}+\frac{b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^3}-\frac{b e^2 n x^{4/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{4 d^2}+\frac{b e n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{5 d}-\frac{e^6 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{2 d^6}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2+\frac{b e^6 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 )}{d^6}+\frac{137 b^2 e^6 n^2 \log (x)}{180 d^6}+\frac{b^2 e^6 n^2 \text{Li}_2\left (1+\frac{e}{d \sqrt [3]{x}}\right )}{d^6}\\ \end{align*}

Mathematica [A]  time = 0.238926, size = 546, normalized size = 1.36 \[ \frac{-360 b^2 e^6 n^2 \text{PolyLog}\left (2,\frac{d \sqrt [3]{x}}{e}+1\right )+180 a^2 d^6 x^2+360 a b d^6 x^2 \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )-90 a b d^4 e^2 n x^{4/3}-180 a b d^2 e^4 n x^{2/3}+120 a b d^3 e^3 n x+72 a b d^5 e n x^{5/3}+360 a b d e^5 n \sqrt [3]{x}-360 a b e^6 n \log \left (d \sqrt [3]{x}+e\right )-90 b^2 d^4 e^2 n x^{4/3} \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )-180 b^2 d^2 e^4 n x^{2/3} \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )+120 b^2 d^3 e^3 n x \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )+180 b^2 d^6 x^2 \log ^2\left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )+72 b^2 d^5 e n x^{5/3} \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )+360 b^2 d e^5 n \sqrt [3]{x} \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )-360 b^2 e^6 n \log \left (d \sqrt [3]{x}+e\right ) \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )+18 b^2 d^4 e^2 n^2 x^{4/3}+141 b^2 d^2 e^4 n^2 x^{2/3}-54 b^2 d^3 e^3 n^2 x-462 b^2 d e^5 n^2 \sqrt [3]{x}+180 b^2 e^6 n^2 \log ^2\left (d \sqrt [3]{x}+e\right )+642 b^2 e^6 n^2 \log \left (d+\frac{e}{\sqrt [3]{x}}\right )+180 b^2 e^6 n^2 \log \left (d \sqrt [3]{x}+e\right )-360 b^2 e^6 n^2 \log \left (d \sqrt [3]{x}+e\right ) \log \left (-\frac{d \sqrt [3]{x}}{e}\right )+214 b^2 e^6 n^2 \log (x)}{360 d^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(360*a*b*d*e^5*n*x^(1/3) - 462*b^2*d*e^5*n^2*x^(1/3) - 180*a*b*d^2*e^4*n*x^(2/3) + 141*b^2*d^2*e^4*n^2*x^(2/3)
+ 120*a*b*d^3*e^3*n*x - 54*b^2*d^3*e^3*n^2*x - 90*a*b*d^4*e^2*n*x^(4/3) + 18*b^2*d^4*e^2*n^2*x^(4/3) + 72*a*b
*d^5*e*n*x^(5/3) + 180*a^2*d^6*x^2 + 642*b^2*e^6*n^2*Log[d + e/x^(1/3)] + 360*b^2*d*e^5*n*x^(1/3)*Log[c*(d + e
/x^(1/3))^n] - 180*b^2*d^2*e^4*n*x^(2/3)*Log[c*(d + e/x^(1/3))^n] + 120*b^2*d^3*e^3*n*x*Log[c*(d + e/x^(1/3))^
n] - 90*b^2*d^4*e^2*n*x^(4/3)*Log[c*(d + e/x^(1/3))^n] + 72*b^2*d^5*e*n*x^(5/3)*Log[c*(d + e/x^(1/3))^n] + 360
*a*b*d^6*x^2*Log[c*(d + e/x^(1/3))^n] + 180*b^2*d^6*x^2*Log[c*(d + e/x^(1/3))^n]^2 - 360*a*b*e^6*n*Log[e + d*x
^(1/3)] + 180*b^2*e^6*n^2*Log[e + d*x^(1/3)] - 360*b^2*e^6*n*Log[c*(d + e/x^(1/3))^n]*Log[e + d*x^(1/3)] + 180
*b^2*e^6*n^2*Log[e + d*x^(1/3)]^2 - 360*b^2*e^6*n^2*Log[e + d*x^(1/3)]*Log[-((d*x^(1/3))/e)] + 214*b^2*e^6*n^2
*Log[x] - 360*b^2*e^6*n^2*PolyLog[2, 1 + (d*x^(1/3))/e])/(360*d^6)

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Maple [F]  time = 0.339, size = 0, normalized size = 0. \begin{align*} \int x \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*(a+b*ln(c*(d+e/x^(1/3))^n))^2,x)

[Out]

int(x*(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}{2} \, b^{2} x^{2} \log \left ({\left (d x^{\frac{1}{3}} + e\right )}^{n}\right )^{2} - \int -\frac{3 \,{\left (b^{2} d \log \left (c\right )^{2} + 2 \, a b d \log \left (c\right ) + a^{2} d\right )} x^{2} + 3 \,{\left (b^{2} d x^{2} + b^{2} e x^{\frac{5}{3}}\right )} \log \left (x^{\frac{1}{3} \, n}\right )^{2} + 3 \,{\left (b^{2} e \log \left (c\right )^{2} + 2 \, a b e \log \left (c\right ) + a^{2} e\right )} x^{\frac{5}{3}} -{\left (b^{2} d n x^{2} - 6 \,{\left (b^{2} d \log \left (c\right ) + a b d\right )} x^{2} - 6 \,{\left (b^{2} e \log \left (c\right ) + a b e\right )} x^{\frac{5}{3}} + 6 \,{\left (b^{2} d x^{2} + b^{2} e x^{\frac{5}{3}}\right )} \log \left (x^{\frac{1}{3} \, n}\right )\right )} \log \left ({\left (d x^{\frac{1}{3}} + e\right )}^{n}\right ) - 6 \,{\left ({\left (b^{2} d \log \left (c\right ) + a b d\right )} x^{2} +{\left (b^{2} e \log \left (c\right ) + a b e\right )} x^{\frac{5}{3}}\right )} \log \left (x^{\frac{1}{3} \, n}\right )}{3 \,{\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*(a+b*log(c*(d+e/x^(1/3))^n))^2,x, algorithm="maxima")

[Out]

1/2*b^2*x^2*log((d*x^(1/3) + e)^n)^2 - integrate(-1/3*(3*(b^2*d*log(c)^2 + 2*a*b*d*log(c) + a^2*d)*x^2 + 3*(b^
2*d*x^2 + b^2*e*x^(5/3))*log(x^(1/3*n))^2 + 3*(b^2*e*log(c)^2 + 2*a*b*e*log(c) + a^2*e)*x^(5/3) - (b^2*d*n*x^2
- 6*(b^2*d*log(c) + a*b*d)*x^2 - 6*(b^2*e*log(c) + a*b*e)*x^(5/3) + 6*(b^2*d*x^2 + b^2*e*x^(5/3))*log(x^(1/3*
n)))*log((d*x^(1/3) + e)^n) - 6*((b^2*d*log(c) + a*b*d)*x^2 + (b^2*e*log(c) + a*b*e)*x^(5/3))*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 \log \left (c \left (\frac{d x + e x^{\frac{2}{3}}}{x}\right )^{n}\right )^{2} + 2 \, a b x \log \left (c \left (\frac{d x + e x^{\frac{2}{3}}}{x}\right )^{n}\right ) + a^{2} 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^(1/3))^n))^2,x, algorithm="fricas")

[Out]

integral(b^2*x*log(c*((d*x + e*x^(2/3))/x)^n)^2 + 2*a*b*x*log(c*((d*x + e*x^(2/3))/x)^n) + a^2*x, 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*(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\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(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, x)