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

Optimal. Leaf size=451 \[ -\frac {3 b^2 e^3 n^2 \text {Li}_2\left (\frac {d}{d+e x^{2/3}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d^3}-\frac {3 b^2 e^3 n^2 \log \left (1-\frac {d}{d+e x^{2/3}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 d^3}-\frac {3 b^2 e^3 n^2 \log \left (-\frac {e x^{2/3}}{d}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d^3}-\frac {3 b^2 e^2 n^2 \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 d^3 x^{2/3}}+\frac {3 b e^3 n \log \left (1-\frac {d}{d+e x^{2/3}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 d^3}+\frac {3 b e^2 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 d^3 x^{2/3}}-\frac {3 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 d x^{4/3}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 x^2}+\frac {3 b^3 e^3 n^3 \text {Li}_2\left (\frac {d}{d+e x^{2/3}}\right )}{2 d^3}-\frac {3 b^3 e^3 n^3 \text {Li}_2\left (\frac {x^{2/3} e}{d}+1\right )}{d^3}-\frac {3 b^3 e^3 n^3 \text {Li}_3\left (\frac {d}{d+e x^{2/3}}\right )}{d^3}+\frac {b^3 e^3 n^3 \log (x)}{d^3} \]

[Out]

-3/2*b^2*e^2*n^2*(d+e*x^(2/3))*(a+b*ln(c*(d+e*x^(2/3))^n))/d^3/x^(2/3)-3/2*b^2*e^3*n^2*ln(1-d/(d+e*x^(2/3)))*(
a+b*ln(c*(d+e*x^(2/3))^n))/d^3-3/4*b*e*n*(a+b*ln(c*(d+e*x^(2/3))^n))^2/d/x^(4/3)+3/2*b*e^2*n*(d+e*x^(2/3))*(a+
b*ln(c*(d+e*x^(2/3))^n))^2/d^3/x^(2/3)+3/2*b*e^3*n*ln(1-d/(d+e*x^(2/3)))*(a+b*ln(c*(d+e*x^(2/3))^n))^2/d^3-1/2
*(a+b*ln(c*(d+e*x^(2/3))^n))^3/x^2-3*b^2*e^3*n^2*(a+b*ln(c*(d+e*x^(2/3))^n))*ln(-e*x^(2/3)/d)/d^3+b^3*e^3*n^3*
ln(x)/d^3+3/2*b^3*e^3*n^3*polylog(2,d/(d+e*x^(2/3)))/d^3-3*b^2*e^3*n^2*(a+b*ln(c*(d+e*x^(2/3))^n))*polylog(2,d
/(d+e*x^(2/3)))/d^3-3*b^3*e^3*n^3*polylog(2,1+e*x^(2/3)/d)/d^3-3*b^3*e^3*n^3*polylog(3,d/(d+e*x^(2/3)))/d^3

________________________________________________________________________________________

Rubi [A]  time = 1.01, antiderivative size = 428, normalized size of antiderivative = 0.95, 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} \[ \frac {3 b^2 e^3 n^2 \text {PolyLog}\left (2,\frac {e x^{2/3}}{d}+1\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d^3}-\frac {9 b^3 e^3 n^3 \text {PolyLog}\left (2,\frac {e x^{2/3}}{d}+1\right )}{2 d^3}-\frac {3 b^3 e^3 n^3 \text {PolyLog}\left (3,\frac {e x^{2/3}}{d}+1\right )}{d^3}-\frac {9 b^2 e^3 n^2 \log \left (-\frac {e x^{2/3}}{d}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 d^3}-\frac {3 b^2 e^2 n^2 \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 d^3 x^{2/3}}-\frac {e^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 d^3}+\frac {3 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 d^3}+\frac {3 b e^3 n \log \left (-\frac {e x^{2/3}}{d}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 d^3}+\frac {3 b e^2 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 d^3 x^{2/3}}-\frac {3 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 d x^{4/3}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 x^2}+\frac {b^3 e^3 n^3 \log (x)}{d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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 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 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 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 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 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 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 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 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 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 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 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 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]

Rubi steps

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

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Mathematica [A]  time = 0.87, size = 764, normalized size = 1.69 \[ \frac {-6 b^2 n^2 \left (\left (d^3+e^3 x^2\right ) \log ^2\left (d+e x^{2/3}\right )+\log \left (d+e x^{2/3}\right ) \left (d^2 e x^{2/3}-2 e^3 x^2 \log \left (-\frac {e x^{2/3}}{d}\right )-2 d e^2 x^{4/3}-3 e^3 x^2\right )-2 e^3 x^2 \text {Li}_2\left (\frac {x^{2/3} e}{d}+1\right )+e^2 x^{4/3} \left (3 e x^{2/3} \log \left (-\frac {e x^{2/3}}{d}\right )+d\right )\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )-b n \log \left (d+e x^{2/3}\right )\right )-2 d^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )-b n \log \left (d+e x^{2/3}\right )\right )^3-6 b d^3 n \log \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )-b n \log \left (d+e x^{2/3}\right )\right )^2-3 b d^2 e n x^{2/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )-b n \log \left (d+e x^{2/3}\right )\right )^2-6 b e^3 n x^2 \log \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )-b n \log \left (d+e x^{2/3}\right )\right )^2+4 b e^3 n x^2 \log (x) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )-b n \log \left (d+e x^{2/3}\right )\right )^2+6 b d e^2 n x^{4/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )-b n \log \left (d+e x^{2/3}\right )\right )^2+b^3 n^3 \left (-2 d^3 \log ^3\left (d+e x^{2/3}\right )-3 d^2 e x^{2/3} \log ^2\left (d+e x^{2/3}\right )-12 e^3 x^2 \text {Li}_3\left (\frac {x^{2/3} e}{d}+1\right )+6 e^3 x^2 \text {Li}_2\left (\frac {x^{2/3} e}{d}+1\right ) \left (2 \log \left (d+e x^{2/3}\right )-3\right )-2 e^3 x^2 \log ^3\left (d+e x^{2/3}\right )+9 e^3 x^2 \log ^2\left (d+e x^{2/3}\right )+6 e^3 x^2 \log ^2\left (d+e x^{2/3}\right ) \log \left (-\frac {e x^{2/3}}{d}\right )-6 e^3 x^2 \log \left (d+e x^{2/3}\right )+6 e^3 x^2 \log \left (-\frac {e x^{2/3}}{d}\right )-18 e^3 x^2 \log \left (d+e x^{2/3}\right ) \log \left (-\frac {e x^{2/3}}{d}\right )+6 d e^2 x^{4/3} \log ^2\left (d+e x^{2/3}\right )-6 d e^2 x^{4/3} \log \left (d+e x^{2/3}\right )\right )}{4 d^3 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right )^{3} + 3 \, a b^{2} \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right )^{2} + 3 \, a^{2} b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a^{3}}{x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a\right )}^{3}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \left (e \,x^{\frac {2}{3}}+d \right )^{n}\right )+a \right )^{3}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.01, size = 740, normalized size = 1.64 \[ \frac {3 \, {\left (\log \left (e x^{\frac {2}{3}} + d\right )^{2} \log \left (-\frac {e x^{\frac {2}{3}} + d}{d} + 1\right ) + 2 \, {\rm Li}_2\left (\frac {e x^{\frac {2}{3}} + d}{d}\right ) \log \left (e x^{\frac {2}{3}} + d\right ) - 2 \, {\rm Li}_{3}(\frac {e x^{\frac {2}{3}} + d}{d})\right )} b^{3} e^{3} n^{3}}{2 \, d^{3}} - \frac {3}{4} \, a^{2} b e n {\left (\frac {2 \, e^{2} \log \left (e x^{\frac {2}{3}} + d\right )}{d^{3}} - \frac {2 \, e^{2} \log \left (x^{\frac {2}{3}}\right )}{d^{3}} - \frac {2 \, e x^{\frac {2}{3}} - d}{d^{2} x^{\frac {4}{3}}}\right )} - \frac {3 \, a^{2} b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right )}{2 \, x^{2}} - \frac {a^{3}}{2 \, x^{2}} + \frac {3 \, {\left (2 \, a b^{2} e^{3} n^{2} - {\left (3 \, e^{3} n^{3} - 2 \, e^{3} n^{2} \log \relax (c)\right )} b^{3}\right )} {\left (\log \left (e x^{\frac {2}{3}} + d\right ) \log \left (-\frac {e x^{\frac {2}{3}} + d}{d} + 1\right ) + {\rm Li}_2\left (\frac {e x^{\frac {2}{3}} + d}{d}\right )\right )}}{2 \, d^{3}} - \frac {{\left ({\left (3 \, e^{3} n^{2} - 2 \, e^{3} n \log \relax (c)\right )} a b^{2} - {\left (e^{3} n^{3} - 3 \, e^{3} n^{2} \log \relax (c) + e^{3} n \log \relax (c)^{2}\right )} b^{3}\right )} \log \relax (x)}{d^{3}} - \frac {2 \, b^{3} d^{3} \log \relax (c)^{3} + 6 \, a b^{2} d^{3} \log \relax (c)^{2} + 2 \, {\left (b^{3} e^{3} n^{3} x^{2} + b^{3} d^{3} n^{3}\right )} \log \left (e x^{\frac {2}{3}} + d\right )^{3} - 3 \, {\left (2 \, b^{3} d e^{2} n^{3} x^{\frac {4}{3}} - b^{3} d^{2} e n^{3} x^{\frac {2}{3}} - 2 \, b^{3} d^{3} n^{2} \log \relax (c) - 2 \, a b^{2} d^{3} n^{2} - {\left (2 \, a b^{2} e^{3} n^{2} - {\left (3 \, e^{3} n^{3} - 2 \, e^{3} n^{2} \log \relax (c)\right )} b^{3}\right )} x^{2}\right )} \log \left (e x^{\frac {2}{3}} + d\right )^{2} + 6 \, {\left ({\left (d e^{2} n^{2} - 2 \, d e^{2} n \log \relax (c)\right )} a b^{2} + {\left (d e^{2} n^{2} \log \relax (c) - d e^{2} n \log \relax (c)^{2}\right )} b^{3}\right )} x^{\frac {4}{3}} + 6 \, {\left (b^{3} d^{3} n \log \relax (c)^{2} + 2 \, a b^{2} d^{3} n \log \relax (c) - {\left ({\left (3 \, e^{3} n^{2} - 2 \, e^{3} n \log \relax (c)\right )} a b^{2} - {\left (e^{3} n^{3} - 3 \, e^{3} n^{2} \log \relax (c) + e^{3} n \log \relax (c)^{2}\right )} b^{3}\right )} x^{2} - {\left (2 \, a b^{2} d e^{2} n^{2} - {\left (d e^{2} n^{3} - 2 \, d e^{2} n^{2} \log \relax (c)\right )} b^{3}\right )} x^{\frac {4}{3}} + {\left (b^{3} d^{2} e n^{2} \log \relax (c) + a b^{2} d^{2} e n^{2}\right )} x^{\frac {2}{3}}\right )} \log \left (e x^{\frac {2}{3}} + d\right ) + 3 \, {\left (b^{3} d^{2} e n \log \relax (c)^{2} + 2 \, a b^{2} d^{2} e n \log \relax (c)\right )} x^{\frac {2}{3}}}{4 \, d^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )\right )}^3}{x^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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