\(\int \frac {(a+b \log (c (d+e \sqrt [3]{x})^n))^3}{x^3} \, dx\) [462]

Optimal result

Integrand size = 24, antiderivative size = 765 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x^3} \, dx=-\frac {b^3 e^3 n^3}{20 d^3 x}+\frac {3 b^3 e^4 n^3}{10 d^4 x^{2/3}}-\frac {71 b^3 e^5 n^3}{40 d^5 \sqrt [3]{x}}+\frac {71 b^3 e^6 n^3 \log \left (d+e \sqrt [3]{x}\right )}{40 d^6}-\frac {3 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^2 x^{4/3}}+\frac {9 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^3 x}-\frac {47 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{40 d^4 x^{2/3}}+\frac {77 b^2 e^5 n^2 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^6 \sqrt [3]{x}}+\frac {77 b^2 e^6 n^2 \log \left (1-\frac {d}{d+e \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^6}-\frac {3 b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{10 d x^{5/3}}+\frac {3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{8 d^2 x^{4/3}}-\frac {b e^3 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^3 x}+\frac {3 b e^4 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{4 d^4 x^{2/3}}-\frac {3 b e^5 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^6 \sqrt [3]{x}}-\frac {3 b e^6 n \log \left (1-\frac {d}{d+e \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^6}-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 x^2}+\frac {3 b^2 e^6 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt [3]{x}}{d}\right )}{d^6}-\frac {15 b^3 e^6 n^3 \log (x)}{8 d^6}-\frac {77 b^3 e^6 n^3 \operatorname {PolyLog}\left (2,\frac {d}{d+e \sqrt [3]{x}}\right )}{20 d^6}+\frac {3 b^2 e^6 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {d}{d+e \sqrt [3]{x}}\right )}{d^6}+\frac {3 b^3 e^6 n^3 \operatorname {PolyLog}\left (2,1+\frac {e \sqrt [3]{x}}{d}\right )}{d^6}+\frac {3 b^3 e^6 n^3 \operatorname {PolyLog}\left (3,\frac {d}{d+e \sqrt [3]{x}}\right )}{d^6} \]

[Out]

-1/20*b^3*e^3*n^3/d^3/x+3/10*b^3*e^4*n^3/d^4/x^(2/3)-71/40*b^3*e^5*n^3/d^5/x^(1/3)+71/40*b^3*e^6*n^3*ln(d+e*x^
(1/3))/d^6-3/20*b^2*e^2*n^2*(a+b*ln(c*(d+e*x^(1/3))^n))/d^2/x^(4/3)+9/20*b^2*e^3*n^2*(a+b*ln(c*(d+e*x^(1/3))^n
))/d^3/x-47/40*b^2*e^4*n^2*(a+b*ln(c*(d+e*x^(1/3))^n))/d^4/x^(2/3)+77/20*b^2*e^5*n^2*(d+e*x^(1/3))*(a+b*ln(c*(
d+e*x^(1/3))^n))/d^6/x^(1/3)+77/20*b^2*e^6*n^2*ln(1-d/(d+e*x^(1/3)))*(a+b*ln(c*(d+e*x^(1/3))^n))/d^6-3/10*b*e*
n*(a+b*ln(c*(d+e*x^(1/3))^n))^2/d/x^(5/3)+3/8*b*e^2*n*(a+b*ln(c*(d+e*x^(1/3))^n))^2/d^2/x^(4/3)-1/2*b*e^3*n*(a
+b*ln(c*(d+e*x^(1/3))^n))^2/d^3/x+3/4*b*e^4*n*(a+b*ln(c*(d+e*x^(1/3))^n))^2/d^4/x^(2/3)-3/2*b*e^5*n*(d+e*x^(1/
3))*(a+b*ln(c*(d+e*x^(1/3))^n))^2/d^6/x^(1/3)-3/2*b*e^6*n*ln(1-d/(d+e*x^(1/3)))*(a+b*ln(c*(d+e*x^(1/3))^n))^2/
d^6-1/2*(a+b*ln(c*(d+e*x^(1/3))^n))^3/x^2+3*b^2*e^6*n^2*(a+b*ln(c*(d+e*x^(1/3))^n))*ln(-e*x^(1/3)/d)/d^6-15/8*
b^3*e^6*n^3*ln(x)/d^6-77/20*b^3*e^6*n^3*polylog(2,d/(d+e*x^(1/3)))/d^6+3*b^2*e^6*n^2*(a+b*ln(c*(d+e*x^(1/3))^n
))*polylog(2,d/(d+e*x^(1/3)))/d^6+3*b^3*e^6*n^3*polylog(2,1+e*x^(1/3)/d)/d^6+3*b^3*e^6*n^3*polylog(3,d/(d+e*x^
(1/3)))/d^6

Rubi [A] (verified)

Time = 1.82 (sec) , antiderivative size = 765, normalized size of antiderivative = 1.00, number of steps used = 62, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {2504, 2445, 2458, 2389, 2379, 2421, 6724, 2355, 2354, 2438, 2356, 2351, 31, 46} \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x^3} \, dx=\frac {3 b^2 e^6 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+e \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d^6}+\frac {77 b^2 e^6 n^2 \log \left (1-\frac {d}{d+e \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^6}+\frac {3 b^2 e^6 n^2 \log \left (-\frac {e \sqrt [3]{x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d^6}+\frac {77 b^2 e^5 n^2 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^6 \sqrt [3]{x}}-\frac {47 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{40 d^4 x^{2/3}}+\frac {9 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^3 x}-\frac {3 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^2 x^{4/3}}-\frac {3 b e^6 n \log \left (1-\frac {d}{d+e \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^6}-\frac {3 b e^5 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^6 \sqrt [3]{x}}+\frac {3 b e^4 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{4 d^4 x^{2/3}}-\frac {b e^3 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^3 x}+\frac {3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{8 d^2 x^{4/3}}-\frac {3 b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{10 d x^{5/3}}-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 x^2}-\frac {77 b^3 e^6 n^3 \operatorname {PolyLog}\left (2,\frac {d}{d+e \sqrt [3]{x}}\right )}{20 d^6}+\frac {3 b^3 e^6 n^3 \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{x} e}{d}+1\right )}{d^6}+\frac {3 b^3 e^6 n^3 \operatorname {PolyLog}\left (3,\frac {d}{d+e \sqrt [3]{x}}\right )}{d^6}+\frac {71 b^3 e^6 n^3 \log \left (d+e \sqrt [3]{x}\right )}{40 d^6}-\frac {15 b^3 e^6 n^3 \log (x)}{8 d^6}-\frac {71 b^3 e^5 n^3}{40 d^5 \sqrt [3]{x}}+\frac {3 b^3 e^4 n^3}{10 d^4 x^{2/3}}-\frac {b^3 e^3 n^3}{20 d^3 x} \]

[In]

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

[Out]

-1/20*(b^3*e^3*n^3)/(d^3*x) + (3*b^3*e^4*n^3)/(10*d^4*x^(2/3)) - (71*b^3*e^5*n^3)/(40*d^5*x^(1/3)) + (71*b^3*e
^6*n^3*Log[d + e*x^(1/3)])/(40*d^6) - (3*b^2*e^2*n^2*(a + b*Log[c*(d + e*x^(1/3))^n]))/(20*d^2*x^(4/3)) + (9*b
^2*e^3*n^2*(a + b*Log[c*(d + e*x^(1/3))^n]))/(20*d^3*x) - (47*b^2*e^4*n^2*(a + b*Log[c*(d + e*x^(1/3))^n]))/(4
0*d^4*x^(2/3)) + (77*b^2*e^5*n^2*(d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n]))/(20*d^6*x^(1/3)) + (77*b^2*
e^6*n^2*Log[1 - d/(d + e*x^(1/3))]*(a + b*Log[c*(d + e*x^(1/3))^n]))/(20*d^6) - (3*b*e*n*(a + b*Log[c*(d + e*x
^(1/3))^n])^2)/(10*d*x^(5/3)) + (3*b*e^2*n*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(8*d^2*x^(4/3)) - (b*e^3*n*(a +
b*Log[c*(d + e*x^(1/3))^n])^2)/(2*d^3*x) + (3*b*e^4*n*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(4*d^4*x^(2/3)) - (
3*b*e^5*n*(d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(2*d^6*x^(1/3)) - (3*b*e^6*n*Log[1 - d/(d + e*x^
(1/3))]*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(2*d^6) - (a + b*Log[c*(d + e*x^(1/3))^n])^3/(2*x^2) + (3*b^2*e^6*
n^2*(a + b*Log[c*(d + e*x^(1/3))^n])*Log[-((e*x^(1/3))/d)])/d^6 - (15*b^3*e^6*n^3*Log[x])/(8*d^6) - (77*b^3*e^
6*n^3*PolyLog[2, d/(d + e*x^(1/3))])/(20*d^6) + (3*b^2*e^6*n^2*(a + b*Log[c*(d + e*x^(1/3))^n])*PolyLog[2, d/(
d + e*x^(1/3))])/d^6 + (3*b^3*e^6*n^3*PolyLog[2, 1 + (e*x^(1/3))/d])/d^6 + (3*b^3*e^6*n^3*PolyLog[3, d/(d + e*
x^(1/3))])/d^6

Rule 31

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

Rule 46

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] && Lt
Q[m + n + 2, 0])

Rule 2351

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 2354

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 2355

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 2356

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 2379

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

Rule 2389

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 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-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*L
og[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 2438

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 2445

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 2458

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 2504

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 6724

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*} \text {integral}& = 3 \text {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{x^7} \, dx,x,\sqrt [3]{x}\right ) \\ & = -\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 x^2}+\frac {1}{2} (3 b e n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^6 (d+e x)} \, dx,x,\sqrt [3]{x}\right ) \\ & = -\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 x^2}+\frac {1}{2} (3 b n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^6} \, dx,x,d+e \sqrt [3]{x}\right ) \\ & = -\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 x^2}+\frac {(3 b n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\left (-\frac {d}{e}+\frac {x}{e}\right )^6} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d}-\frac {(3 b e n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^5} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d} \\ & = -\frac {3 b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{10 d x^{5/3}}-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 x^2}-\frac {(3 b e n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\left (-\frac {d}{e}+\frac {x}{e}\right )^5} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d^2}+\frac {\left (3 b e^2 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^4} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d^2}+\frac {\left (3 b^2 e n^2\right ) \text {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 \sqrt [3]{x}\right )}{5 d} \\ & = -\frac {3 b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{10 d x^{5/3}}+\frac {3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{8 d^2 x^{4/3}}-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 x^2}+\frac {\left (3 b e^2 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\left (-\frac {d}{e}+\frac {x}{e}\right )^4} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d^3}-\frac {\left (3 b e^3 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d^3}+\frac {\left (3 b^2 e n^2\right ) \text {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 \sqrt [3]{x}\right )}{5 d^2}-\frac {\left (3 b^2 e^2 n^2\right ) \text {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 \sqrt [3]{x}\right )}{5 d^2}-\frac {\left (3 b^2 e^2 n^2\right ) \text {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 \sqrt [3]{x}\right )}{4 d^2} \\ & = -\frac {3 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^2 x^{4/3}}-\frac {3 b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{10 d x^{5/3}}+\frac {3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{8 d^2 x^{4/3}}-\frac {b e^3 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^3 x}-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 x^2}-\frac {\left (3 b e^3 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d^4}+\frac {\left (3 b e^4 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d^4}-\frac {\left (3 b^2 e^2 n^2\right ) \text {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 \sqrt [3]{x}\right )}{5 d^3}-\frac {\left (3 b^2 e^2 n^2\right ) \text {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 \sqrt [3]{x}\right )}{4 d^3}+\frac {\left (3 b^2 e^3 n^2\right ) \text {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 \sqrt [3]{x}\right )}{5 d^3}+\frac {\left (3 b^2 e^3 n^2\right ) \text {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 \sqrt [3]{x}\right )}{4 d^3}+\frac {\left (b^2 e^3 n^2\right ) \text {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 \sqrt [3]{x}\right )}{d^3}+\frac {\left (3 b^3 e^2 n^3\right ) \text {Subst}\left (\int \frac {1}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^4} \, dx,x,d+e \sqrt [3]{x}\right )}{20 d^2} \\ & = -\frac {3 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^2 x^{4/3}}+\frac {9 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^3 x}-\frac {3 b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{10 d x^{5/3}}+\frac {3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{8 d^2 x^{4/3}}-\frac {b e^3 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^3 x}+\frac {3 b e^4 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{4 d^4 x^{2/3}}-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 x^2}+\frac {\left (3 b e^4 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d^5}-\frac {\left (3 b e^5 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (-\frac {d}{e}+\frac {x}{e}\right )} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d^5}+\frac {\left (3 b^2 e^3 n^2\right ) \text {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 \sqrt [3]{x}\right )}{5 d^4}+\frac {\left (3 b^2 e^3 n^2\right ) \text {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 \sqrt [3]{x}\right )}{4 d^4}+\frac {\left (b^2 e^3 n^2\right ) \text {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 \sqrt [3]{x}\right )}{d^4}-\frac {\left (3 b^2 e^4 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e \sqrt [3]{x}\right )}{5 d^4}-\frac {\left (3 b^2 e^4 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e \sqrt [3]{x}\right )}{4 d^4}-\frac {\left (b^2 e^4 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e \sqrt [3]{x}\right )}{d^4}-\frac {\left (3 b^2 e^4 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d^4}+\frac {\left (3 b^3 e^2 n^3\right ) \text {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 \sqrt [3]{x}\right )}{20 d^2}-\frac {\left (b^3 e^3 n^3\right ) \text {Subst}\left (\int \frac {1}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+e \sqrt [3]{x}\right )}{5 d^3}-\frac {\left (b^3 e^3 n^3\right ) \text {Subst}\left (\int \frac {1}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+e \sqrt [3]{x}\right )}{4 d^3} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.27 (sec) , antiderivative size = 1074, normalized size of antiderivative = 1.40 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x^3} \, dx=-\frac {12 b d^5 e n \sqrt [3]{x} \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2-15 b d^4 e^2 n x^{2/3} \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2+20 b d^3 e^3 n x \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2-30 b d^2 e^4 n x^{4/3} \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2+60 b d e^5 n x^{5/3} \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2+60 b d^6 n \log \left (d+e \sqrt [3]{x}\right ) \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2-60 b e^6 n x^2 \log \left (d+e \sqrt [3]{x}\right ) \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2+20 d^6 \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3+20 b e^6 n x^2 \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \log (x)+b^2 n^2 \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \left (6 d^4 e^2 x^{2/3}-18 d^3 e^3 x+47 d^2 e^4 x^{4/3}-154 d e^5 x^{5/3}+60 \left (d^6-e^6 x^2\right ) \log ^2\left (d+e \sqrt [3]{x}\right )-274 e^6 x^2 \log \left (-\frac {e \sqrt [3]{x}}{d}\right )+2 \log \left (d+e \sqrt [3]{x}\right ) \left (12 d^5 e \sqrt [3]{x}-15 d^4 e^2 x^{2/3}+20 d^3 e^3 x-30 d^2 e^4 x^{4/3}+60 d e^5 x^{5/3}+137 e^6 x^2+60 e^6 x^2 \log \left (-\frac {e \sqrt [3]{x}}{d}\right )\right )+120 e^6 x^2 \operatorname {PolyLog}\left (2,1+\frac {e \sqrt [3]{x}}{d}\right )\right )+b^3 n^3 \left (3 d^4 e^2 x^{2/3} \left (2-5 \log \left (d+e \sqrt [3]{x}\right )\right ) \log \left (d+e \sqrt [3]{x}\right )+12 d^5 e \sqrt [3]{x} \log ^2\left (d+e \sqrt [3]{x}\right )+20 d^6 \log ^3\left (d+e \sqrt [3]{x}\right )+2 d^3 e^3 x \left (1-9 \log \left (d+e \sqrt [3]{x}\right )+10 \log ^2\left (d+e \sqrt [3]{x}\right )\right )-d^2 e^4 x^{4/3} \left (12-47 \log \left (d+e \sqrt [3]{x}\right )+30 \log ^2\left (d+e \sqrt [3]{x}\right )\right )+d e^5 x^{5/3} \left (71-154 \log \left (d+e \sqrt [3]{x}\right )+60 \log ^2\left (d+e \sqrt [3]{x}\right )\right )+225 e^6 x^2 \left (-\log \left (d+e \sqrt [3]{x}\right )+\log \left (-\frac {e \sqrt [3]{x}}{d}\right )\right )+137 e^6 x^2 \left (\log \left (d+e \sqrt [3]{x}\right ) \left (\log \left (d+e \sqrt [3]{x}\right )-2 \log \left (-\frac {e \sqrt [3]{x}}{d}\right )\right )-2 \operatorname {PolyLog}\left (2,1+\frac {e \sqrt [3]{x}}{d}\right )\right )-20 e^6 x^2 \left (\log ^2\left (d+e \sqrt [3]{x}\right ) \left (\log \left (d+e \sqrt [3]{x}\right )-3 \log \left (-\frac {e \sqrt [3]{x}}{d}\right )\right )-6 \log \left (d+e \sqrt [3]{x}\right ) \operatorname {PolyLog}\left (2,1+\frac {e \sqrt [3]{x}}{d}\right )+6 \operatorname {PolyLog}\left (3,1+\frac {e \sqrt [3]{x}}{d}\right )\right )\right )}{40 d^6 x^2} \]

[In]

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

[Out]

-1/40*(12*b*d^5*e*n*x^(1/3)*(a - b*n*Log[d + e*x^(1/3)] + b*Log[c*(d + e*x^(1/3))^n])^2 - 15*b*d^4*e^2*n*x^(2/
3)*(a - b*n*Log[d + e*x^(1/3)] + b*Log[c*(d + e*x^(1/3))^n])^2 + 20*b*d^3*e^3*n*x*(a - b*n*Log[d + e*x^(1/3)]
+ b*Log[c*(d + e*x^(1/3))^n])^2 - 30*b*d^2*e^4*n*x^(4/3)*(a - b*n*Log[d + e*x^(1/3)] + b*Log[c*(d + e*x^(1/3))
^n])^2 + 60*b*d*e^5*n*x^(5/3)*(a - b*n*Log[d + e*x^(1/3)] + b*Log[c*(d + e*x^(1/3))^n])^2 + 60*b*d^6*n*Log[d +
e*x^(1/3)]*(a - b*n*Log[d + e*x^(1/3)] + b*Log[c*(d + e*x^(1/3))^n])^2 - 60*b*e^6*n*x^2*Log[d + e*x^(1/3)]*(a
- b*n*Log[d + e*x^(1/3)] + b*Log[c*(d + e*x^(1/3))^n])^2 + 20*d^6*(a - b*n*Log[d + e*x^(1/3)] + b*Log[c*(d +
e*x^(1/3))^n])^3 + 20*b*e^6*n*x^2*(a - b*n*Log[d + e*x^(1/3)] + b*Log[c*(d + e*x^(1/3))^n])^2*Log[x] + b^2*n^2
*(a - b*n*Log[d + e*x^(1/3)] + b*Log[c*(d + e*x^(1/3))^n])*(6*d^4*e^2*x^(2/3) - 18*d^3*e^3*x + 47*d^2*e^4*x^(4
/3) - 154*d*e^5*x^(5/3) + 60*(d^6 - e^6*x^2)*Log[d + e*x^(1/3)]^2 - 274*e^6*x^2*Log[-((e*x^(1/3))/d)] + 2*Log[
d + e*x^(1/3)]*(12*d^5*e*x^(1/3) - 15*d^4*e^2*x^(2/3) + 20*d^3*e^3*x - 30*d^2*e^4*x^(4/3) + 60*d*e^5*x^(5/3) +
137*e^6*x^2 + 60*e^6*x^2*Log[-((e*x^(1/3))/d)]) + 120*e^6*x^2*PolyLog[2, 1 + (e*x^(1/3))/d]) + b^3*n^3*(3*d^4
*e^2*x^(2/3)*(2 - 5*Log[d + e*x^(1/3)])*Log[d + e*x^(1/3)] + 12*d^5*e*x^(1/3)*Log[d + e*x^(1/3)]^2 + 20*d^6*Lo
g[d + e*x^(1/3)]^3 + 2*d^3*e^3*x*(1 - 9*Log[d + e*x^(1/3)] + 10*Log[d + e*x^(1/3)]^2) - d^2*e^4*x^(4/3)*(12 -
47*Log[d + e*x^(1/3)] + 30*Log[d + e*x^(1/3)]^2) + d*e^5*x^(5/3)*(71 - 154*Log[d + e*x^(1/3)] + 60*Log[d + e*x
^(1/3)]^2) + 225*e^6*x^2*(-Log[d + e*x^(1/3)] + Log[-((e*x^(1/3))/d)]) + 137*e^6*x^2*(Log[d + e*x^(1/3)]*(Log[
d + e*x^(1/3)] - 2*Log[-((e*x^(1/3))/d)]) - 2*PolyLog[2, 1 + (e*x^(1/3))/d]) - 20*e^6*x^2*(Log[d + e*x^(1/3)]^
2*(Log[d + e*x^(1/3)] - 3*Log[-((e*x^(1/3))/d)]) - 6*Log[d + e*x^(1/3)]*PolyLog[2, 1 + (e*x^(1/3))/d] + 6*Poly
Log[3, 1 + (e*x^(1/3))/d])))/(d^6*x^2)

Maple [F]

\[\int \frac {{\left (a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )^{n}\right )\right )}^{3}}{x^{3}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right ) + a\right )}^{3}}{x^{3}} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right ) + a\right )}^{3}}{x^{3}} \,d x } \]

[In]

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

[Out]

-1/2*b^3*log((e*x^(1/3) + d)^n)^3/x^2 + integrate(1/2*((b^3*e*n*x + 6*(b^3*e*log(c) + a*b^2*e)*x + 6*(b^3*d*lo
g(c) + a*b^2*d)*x^(2/3))*log((e*x^(1/3) + d)^n)^2 + 2*(b^3*e*log(c)^3 + 3*a*b^2*e*log(c)^2 + 3*a^2*b*e*log(c)
+ a^3*e)*x + 6*((b^3*e*log(c)^2 + 2*a*b^2*e*log(c) + a^2*b*e)*x + (b^3*d*log(c)^2 + 2*a*b^2*d*log(c) + a^2*b*d
)*x^(2/3))*log((e*x^(1/3) + d)^n) + 2*(b^3*d*log(c)^3 + 3*a*b^2*d*log(c)^2 + 3*a^2*b*d*log(c) + a^3*d)*x^(2/3)
)/(e*x^4 + d*x^(11/3)), x)

Giac [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right ) + a\right )}^{3}}{x^{3}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x^3} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )\right )}^3}{x^3} \,d x \]

[In]

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

[Out]

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