3.475 \(\int \frac{(a+b \log (c (d+e x^{2/3})^n))^2}{x^5} \, dx\)

Optimal. Leaf size=412 \[ \frac{b^2 e^6 n^2 \text{PolyLog}\left (2,\frac{d}{d+e x^{2/3}}\right )}{2 d^6}-\frac{b e^6 n \log \left (1-\frac{d}{d+e x^{2/3}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 d^6}-\frac{b e^5 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 d^6 x^{2/3}}+\frac{b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{4 d^4 x^{4/3}}-\frac{b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{6 d^3 x^2}+\frac{b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{8 d^2 x^{8/3}}-\frac{b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{10 d x^{10/3}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 x^4}+\frac{77 b^2 e^5 n^2}{120 d^5 x^{2/3}}-\frac{47 b^2 e^4 n^2}{240 d^4 x^{4/3}}+\frac{3 b^2 e^3 n^2}{40 d^3 x^2}-\frac{b^2 e^2 n^2}{40 d^2 x^{8/3}}-\frac{77 b^2 e^6 n^2 \log \left (d+e x^{2/3}\right )}{120 d^6}+\frac{137 b^2 e^6 n^2 \log (x)}{180 d^6} \]

[Out]

-(b^2*e^2*n^2)/(40*d^2*x^(8/3)) + (3*b^2*e^3*n^2)/(40*d^3*x^2) - (47*b^2*e^4*n^2)/(240*d^4*x^(4/3)) + (77*b^2*
e^5*n^2)/(120*d^5*x^(2/3)) - (77*b^2*e^6*n^2*Log[d + e*x^(2/3)])/(120*d^6) - (b*e*n*(a + b*Log[c*(d + e*x^(2/3
))^n]))/(10*d*x^(10/3)) + (b*e^2*n*(a + b*Log[c*(d + e*x^(2/3))^n]))/(8*d^2*x^(8/3)) - (b*e^3*n*(a + b*Log[c*(
d + e*x^(2/3))^n]))/(6*d^3*x^2) + (b*e^4*n*(a + b*Log[c*(d + e*x^(2/3))^n]))/(4*d^4*x^(4/3)) - (b*e^5*n*(d + e
*x^(2/3))*(a + b*Log[c*(d + e*x^(2/3))^n]))/(2*d^6*x^(2/3)) - (b*e^6*n*Log[1 - d/(d + e*x^(2/3))]*(a + b*Log[c
*(d + e*x^(2/3))^n]))/(2*d^6) - (a + b*Log[c*(d + e*x^(2/3))^n])^2/(4*x^4) + (137*b^2*e^6*n^2*Log[x])/(180*d^6
) + (b^2*e^6*n^2*PolyLog[2, d/(d + e*x^(2/3))])/(2*d^6)

________________________________________________________________________________________

Rubi [A]  time = 1.01578, antiderivative size = 436, normalized size of antiderivative = 1.06, number of steps used = 26, 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} \[ -\frac{b^2 e^6 n^2 \text{PolyLog}\left (2,\frac{e x^{2/3}}{d}+1\right )}{2 d^6}+\frac{e^6 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 d^6}-\frac{b e^6 n \log \left (-\frac{e x^{2/3}}{d}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 d^6}-\frac{b e^5 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 d^6 x^{2/3}}+\frac{b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{4 d^4 x^{4/3}}-\frac{b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{6 d^3 x^2}+\frac{b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{8 d^2 x^{8/3}}-\frac{b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{10 d x^{10/3}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 x^4}+\frac{77 b^2 e^5 n^2}{120 d^5 x^{2/3}}-\frac{47 b^2 e^4 n^2}{240 d^4 x^{4/3}}+\frac{3 b^2 e^3 n^2}{40 d^3 x^2}-\frac{b^2 e^2 n^2}{40 d^2 x^{8/3}}-\frac{77 b^2 e^6 n^2 \log \left (d+e x^{2/3}\right )}{120 d^6}+\frac{137 b^2 e^6 n^2 \log (x)}{180 d^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d + e*x^(2/3))^n])^2/x^5,x]

[Out]

-(b^2*e^2*n^2)/(40*d^2*x^(8/3)) + (3*b^2*e^3*n^2)/(40*d^3*x^2) - (47*b^2*e^4*n^2)/(240*d^4*x^(4/3)) + (77*b^2*
e^5*n^2)/(120*d^5*x^(2/3)) - (77*b^2*e^6*n^2*Log[d + e*x^(2/3)])/(120*d^6) - (b*e*n*(a + b*Log[c*(d + e*x^(2/3
))^n]))/(10*d*x^(10/3)) + (b*e^2*n*(a + b*Log[c*(d + e*x^(2/3))^n]))/(8*d^2*x^(8/3)) - (b*e^3*n*(a + b*Log[c*(
d + e*x^(2/3))^n]))/(6*d^3*x^2) + (b*e^4*n*(a + b*Log[c*(d + e*x^(2/3))^n]))/(4*d^4*x^(4/3)) - (b*e^5*n*(d + e
*x^(2/3))*(a + b*Log[c*(d + e*x^(2/3))^n]))/(2*d^6*x^(2/3)) + (e^6*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(4*d^6)
- (a + b*Log[c*(d + e*x^(2/3))^n])^2/(4*x^4) - (b*e^6*n*(a + b*Log[c*(d + e*x^(2/3))^n])*Log[-((e*x^(2/3))/d)
])/(2*d^6) + (137*b^2*e^6*n^2*Log[x])/(180*d^6) - (b^2*e^6*n^2*PolyLog[2, 1 + (e*x^(2/3))/d])/(2*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 \frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^5} \, dx &=\frac{3}{2} \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^7} \, dx,x,x^{2/3}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 x^4}+\frac{1}{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,x^{2/3}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 x^4}+\frac{1}{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+e x^{2/3}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 x^4}+\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+e x^{2/3}\right )}{2 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+e x^{2/3}\right )}{2 d}\\ &=-\frac{b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{10 d x^{10/3}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 x^4}-\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+e x^{2/3}\right )}{2 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+e x^{2/3}\right )}{2 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+e x^{2/3}\right )}{10 d}\\ &=-\frac{b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{10 d x^{10/3}}+\frac{b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{8 d^2 x^{8/3}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 x^4}+\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+e x^{2/3}\right )}{2 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+e x^{2/3}\right )}{2 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+e x^{2/3}\right )}{10 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+e x^{2/3}\right )}{8 d^2}\\ &=-\frac{b^2 e^2 n^2}{40 d^2 x^{8/3}}+\frac{b^2 e^3 n^2}{30 d^3 x^2}-\frac{b^2 e^4 n^2}{20 d^4 x^{4/3}}+\frac{b^2 e^5 n^2}{10 d^5 x^{2/3}}-\frac{b^2 e^6 n^2 \log \left (d+e x^{2/3}\right )}{10 d^6}-\frac{b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{10 d x^{10/3}}+\frac{b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{8 d^2 x^{8/3}}-\frac{b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{6 d^3 x^2}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 x^4}+\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+e x^{2/3}\right )}{2 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+e x^{2/3}\right )}{2 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+e x^{2/3}\right )}{8 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+e x^{2/3}\right )}{6 d^3}\\ &=-\frac{b^2 e^2 n^2}{40 d^2 x^{8/3}}+\frac{3 b^2 e^3 n^2}{40 d^3 x^2}-\frac{9 b^2 e^4 n^2}{80 d^4 x^{4/3}}+\frac{9 b^2 e^5 n^2}{40 d^5 x^{2/3}}-\frac{9 b^2 e^6 n^2 \log \left (d+e x^{2/3}\right )}{40 d^6}-\frac{b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{10 d x^{10/3}}+\frac{b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{8 d^2 x^{8/3}}-\frac{b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{6 d^3 x^2}+\frac{b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{4 d^4 x^{4/3}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 x^4}+\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+e x^{2/3}\right )}{2 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+e x^{2/3}\right )}{2 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+e x^{2/3}\right )}{6 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+e x^{2/3}\right )}{4 d^4}\\ &=-\frac{b^2 e^2 n^2}{40 d^2 x^{8/3}}+\frac{3 b^2 e^3 n^2}{40 d^3 x^2}-\frac{47 b^2 e^4 n^2}{240 d^4 x^{4/3}}+\frac{47 b^2 e^5 n^2}{120 d^5 x^{2/3}}-\frac{47 b^2 e^6 n^2 \log \left (d+e x^{2/3}\right )}{120 d^6}-\frac{b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{10 d x^{10/3}}+\frac{b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{8 d^2 x^{8/3}}-\frac{b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{6 d^3 x^2}+\frac{b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{4 d^4 x^{4/3}}-\frac{b e^5 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 d^6 x^{2/3}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 x^4}+\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+e x^{2/3}\right )}{2 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+e x^{2/3}\right )}{2 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+e x^{2/3}\right )}{4 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+e x^{2/3}\right )}{2 d^6}\\ &=-\frac{b^2 e^2 n^2}{40 d^2 x^{8/3}}+\frac{3 b^2 e^3 n^2}{40 d^3 x^2}-\frac{47 b^2 e^4 n^2}{240 d^4 x^{4/3}}+\frac{77 b^2 e^5 n^2}{120 d^5 x^{2/3}}-\frac{77 b^2 e^6 n^2 \log \left (d+e x^{2/3}\right )}{120 d^6}-\frac{b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{10 d x^{10/3}}+\frac{b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{8 d^2 x^{8/3}}-\frac{b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{6 d^3 x^2}+\frac{b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{4 d^4 x^{4/3}}-\frac{b e^5 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 d^6 x^{2/3}}+\frac{e^6 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 d^6}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 x^4}-\frac{b e^6 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \log \left (-\frac{e x^{2/3}}{d}\right )}{2 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+e x^{2/3}\right )}{2 d^6}\\ &=-\frac{b^2 e^2 n^2}{40 d^2 x^{8/3}}+\frac{3 b^2 e^3 n^2}{40 d^3 x^2}-\frac{47 b^2 e^4 n^2}{240 d^4 x^{4/3}}+\frac{77 b^2 e^5 n^2}{120 d^5 x^{2/3}}-\frac{77 b^2 e^6 n^2 \log \left (d+e x^{2/3}\right )}{120 d^6}-\frac{b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{10 d x^{10/3}}+\frac{b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{8 d^2 x^{8/3}}-\frac{b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{6 d^3 x^2}+\frac{b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{4 d^4 x^{4/3}}-\frac{b e^5 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 d^6 x^{2/3}}+\frac{e^6 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 d^6}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 x^4}-\frac{b e^6 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \log \left (-\frac{e x^{2/3}}{d}\right )}{2 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 x^{2/3}}{d}\right )}{2 d^6}\\ \end{align*}

Mathematica [A]  time = 0.275875, size = 539, normalized size = 1.31 \[ -\frac{360 b^2 e^6 n^2 x^4 \text{PolyLog}\left (2,\frac{e x^{2/3}}{d}+1\right )+180 a^2 d^6+360 a b d^6 \log \left (c \left (d+e x^{2/3}\right )^n\right )-360 a b e^6 x^4 \log \left (c \left (d+e x^{2/3}\right )^n\right )-90 a b d^4 e^2 n x^{4/3}+120 a b d^3 e^3 n x^2-180 a b d^2 e^4 n x^{8/3}+72 a b d^5 e n x^{2/3}+360 a b d e^5 n x^{10/3}+360 a b e^6 n x^4 \log \left (-\frac{e x^{2/3}}{d}\right )-90 b^2 d^4 e^2 n x^{4/3} \log \left (c \left (d+e x^{2/3}\right )^n\right )+120 b^2 d^3 e^3 n x^2 \log \left (c \left (d+e x^{2/3}\right )^n\right )-180 b^2 d^2 e^4 n x^{8/3} \log \left (c \left (d+e x^{2/3}\right )^n\right )+180 b^2 d^6 \log ^2\left (c \left (d+e x^{2/3}\right )^n\right )+72 b^2 d^5 e n x^{2/3} \log \left (c \left (d+e x^{2/3}\right )^n\right )-180 b^2 e^6 x^4 \log ^2\left (c \left (d+e x^{2/3}\right )^n\right )+360 b^2 d e^5 n x^{10/3} \log \left (c \left (d+e x^{2/3}\right )^n\right )+360 b^2 e^6 n x^4 \log \left (-\frac{e x^{2/3}}{d}\right ) \log \left (c \left (d+e x^{2/3}\right )^n\right )+18 b^2 d^4 e^2 n^2 x^{4/3}-54 b^2 d^3 e^3 n^2 x^2+141 b^2 d^2 e^4 n^2 x^{8/3}-462 b^2 d e^5 n^2 x^{10/3}+822 b^2 e^6 n^2 x^4 \log \left (d+e x^{2/3}\right )-548 b^2 e^6 n^2 x^4 \log (x)}{720 d^6 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d + e*x^(2/3))^n])^2/x^5,x]

[Out]

-(180*a^2*d^6 + 72*a*b*d^5*e*n*x^(2/3) - 90*a*b*d^4*e^2*n*x^(4/3) + 18*b^2*d^4*e^2*n^2*x^(4/3) + 120*a*b*d^3*e
^3*n*x^2 - 54*b^2*d^3*e^3*n^2*x^2 - 180*a*b*d^2*e^4*n*x^(8/3) + 141*b^2*d^2*e^4*n^2*x^(8/3) + 360*a*b*d*e^5*n*
x^(10/3) - 462*b^2*d*e^5*n^2*x^(10/3) + 822*b^2*e^6*n^2*x^4*Log[d + e*x^(2/3)] + 360*a*b*d^6*Log[c*(d + e*x^(2
/3))^n] + 72*b^2*d^5*e*n*x^(2/3)*Log[c*(d + e*x^(2/3))^n] - 90*b^2*d^4*e^2*n*x^(4/3)*Log[c*(d + e*x^(2/3))^n]
+ 120*b^2*d^3*e^3*n*x^2*Log[c*(d + e*x^(2/3))^n] - 180*b^2*d^2*e^4*n*x^(8/3)*Log[c*(d + e*x^(2/3))^n] + 360*b^
2*d*e^5*n*x^(10/3)*Log[c*(d + e*x^(2/3))^n] - 360*a*b*e^6*x^4*Log[c*(d + e*x^(2/3))^n] + 180*b^2*d^6*Log[c*(d
+ e*x^(2/3))^n]^2 - 180*b^2*e^6*x^4*Log[c*(d + e*x^(2/3))^n]^2 + 360*a*b*e^6*n*x^4*Log[-((e*x^(2/3))/d)] + 360
*b^2*e^6*n*x^4*Log[c*(d + e*x^(2/3))^n]*Log[-((e*x^(2/3))/d)] - 548*b^2*e^6*n^2*x^4*Log[x] + 360*b^2*e^6*n^2*x
^4*PolyLog[2, 1 + (e*x^(2/3))/d])/(720*d^6*x^4)

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Maple [F]  time = 0.343, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{5}} \left ( a+b\ln \left ( c \left ( d+e{x}^{{\frac{2}{3}}} \right ) ^{n} \right ) \right ) ^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(d+e*x^(2/3))^n))^2/x^5,x)

[Out]

int((a+b*ln(c*(d+e*x^(2/3))^n))^2/x^5,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{b^{2} \log \left ({\left (e x^{\frac{2}{3}} + d\right )}^{n}\right )^{2}}{4 \, x^{4}} + \int \frac{3 \,{\left (b^{2} e \log \left (c\right )^{2} + 2 \, a b e \log \left (c\right ) + a^{2} e\right )} x +{\left (b^{2} e n x + 6 \,{\left (b^{2} e \log \left (c\right ) + a b e\right )} x + 6 \,{\left (b^{2} d \log \left (c\right ) + a b d\right )} x^{\frac{1}{3}}\right )} \log \left ({\left (e x^{\frac{2}{3}} + d\right )}^{n}\right ) + 3 \,{\left (b^{2} d \log \left (c\right )^{2} + 2 \, a b d \log \left (c\right ) + a^{2} d\right )} x^{\frac{1}{3}}}{3 \,{\left (e x^{6} + d x^{\frac{16}{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^(2/3))^n))^2/x^5,x, algorithm="maxima")

[Out]

-1/4*b^2*log((e*x^(2/3) + d)^n)^2/x^4 + integrate(1/3*(3*(b^2*e*log(c)^2 + 2*a*b*e*log(c) + a^2*e)*x + (b^2*e*
n*x + 6*(b^2*e*log(c) + a*b*e)*x + 6*(b^2*d*log(c) + a*b*d)*x^(1/3))*log((e*x^(2/3) + d)^n) + 3*(b^2*d*log(c)^
2 + 2*a*b*d*log(c) + a^2*d)*x^(1/3))/(e*x^6 + d*x^(16/3)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \log \left ({\left (e x^{\frac{2}{3}} + d\right )}^{n} c\right )^{2} + 2 \, a b \log \left ({\left (e x^{\frac{2}{3}} + d\right )}^{n} c\right ) + a^{2}}{x^{5}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e*x^(2/3))^n))^2/x^5,x, algorithm="fricas")

[Out]

integral((b^2*log((e*x^(2/3) + d)^n*c)^2 + 2*a*b*log((e*x^(2/3) + d)^n*c) + a^2)/x^5, 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((a+b*ln(c*(d+e*x**(2/3))**n))**2/x**5,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left ({\left (e x^{\frac{2}{3}} + d\right )}^{n} c\right ) + a\right )}^{2}}{x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e*x^(2/3))^n))^2/x^5,x, algorithm="giac")

[Out]

integrate((b*log((e*x^(2/3) + d)^n*c) + a)^2/x^5, x)