\(\int \frac {(a+b \log (c (d+e \sqrt [3]{x})^n))^3}{x^2} \, dx\) [461]

Optimal result

Integrand size = 24, antiderivative size = 439 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x^2} \, dx=-\frac {3 b^2 e^2 n^2 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d^3 \sqrt [3]{x}}-\frac {3 b^2 e^3 n^2 \log \left (1-\frac {d}{d+e \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d^3}-\frac {3 b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d x^{2/3}}+\frac {3 b e^2 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{d^3 \sqrt [3]{x}}+\frac {3 b e^3 n \log \left (1-\frac {d}{d+e \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{d^3}-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x}-\frac {6 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt [3]{x}}{d}\right )}{d^3}+\frac {b^3 e^3 n^3 \log (x)}{d^3}+\frac {3 b^3 e^3 n^3 \operatorname {PolyLog}\left (2,\frac {d}{d+e \sqrt [3]{x}}\right )}{d^3}-\frac {6 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {d}{d+e \sqrt [3]{x}}\right )}{d^3}-\frac {6 b^3 e^3 n^3 \operatorname {PolyLog}\left (2,1+\frac {e \sqrt [3]{x}}{d}\right )}{d^3}-\frac {6 b^3 e^3 n^3 \operatorname {PolyLog}\left (3,\frac {d}{d+e \sqrt [3]{x}}\right )}{d^3} \]

[Out]

-3*b^2*e^2*n^2*(d+e*x^(1/3))*(a+b*ln(c*(d+e*x^(1/3))^n))/d^3/x^(1/3)-3*b^2*e^3*n^2*ln(1-d/(d+e*x^(1/3)))*(a+b*
ln(c*(d+e*x^(1/3))^n))/d^3-3/2*b*e*n*(a+b*ln(c*(d+e*x^(1/3))^n))^2/d/x^(2/3)+3*b*e^2*n*(d+e*x^(1/3))*(a+b*ln(c
*(d+e*x^(1/3))^n))^2/d^3/x^(1/3)+3*b*e^3*n*ln(1-d/(d+e*x^(1/3)))*(a+b*ln(c*(d+e*x^(1/3))^n))^2/d^3-(a+b*ln(c*(
d+e*x^(1/3))^n))^3/x-6*b^2*e^3*n^2*(a+b*ln(c*(d+e*x^(1/3))^n))*ln(-e*x^(1/3)/d)/d^3+b^3*e^3*n^3*ln(x)/d^3+3*b^
3*e^3*n^3*polylog(2,d/(d+e*x^(1/3)))/d^3-6*b^2*e^3*n^2*(a+b*ln(c*(d+e*x^(1/3))^n))*polylog(2,d/(d+e*x^(1/3)))/
d^3-6*b^3*e^3*n^3*polylog(2,1+e*x^(1/3)/d)/d^3-6*b^3*e^3*n^3*polylog(3,d/(d+e*x^(1/3)))/d^3

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {2504, 2445, 2458, 2389, 2379, 2421, 6724, 2355, 2354, 2438, 2356, 2351, 31} \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x^2} \, dx=-\frac {6 b^2 e^3 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+e \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d^3}-\frac {3 b^2 e^3 n^2 \log \left (1-\frac {d}{d+e \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d^3}-\frac {6 b^2 e^3 n^2 \log \left (-\frac {e \sqrt [3]{x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d^3}-\frac {3 b^2 e^2 n^2 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d^3 \sqrt [3]{x}}+\frac {3 b e^3 n \log \left (1-\frac {d}{d+e \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{d^3}+\frac {3 b e^2 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{d^3 \sqrt [3]{x}}-\frac {3 b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d x^{2/3}}-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x}+\frac {3 b^3 e^3 n^3 \operatorname {PolyLog}\left (2,\frac {d}{d+e \sqrt [3]{x}}\right )}{d^3}-\frac {6 b^3 e^3 n^3 \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{x} e}{d}+1\right )}{d^3}-\frac {6 b^3 e^3 n^3 \operatorname {PolyLog}\left (3,\frac {d}{d+e \sqrt [3]{x}}\right )}{d^3}+\frac {b^3 e^3 n^3 \log (x)}{d^3} \]

[In]

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

[Out]

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

Rule 31

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

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^4} \, dx,x,\sqrt [3]{x}\right ) \\ & = -\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x}+(3 b e n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3 (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}{x}+(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 )^3} \, 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}{x}+\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 )^3} \, dx,x,d+e \sqrt [3]{x}\right )}{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 )^2} \, dx,x,d+e \sqrt [3]{x}\right )}{d} \\ & = -\frac {3 b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d x^{2/3}}-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x}-\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 )^2} \, dx,x,d+e \sqrt [3]{x}\right )}{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 )} \, dx,x,d+e \sqrt [3]{x}\right )}{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 )^2} \, dx,x,d+e \sqrt [3]{x}\right )}{d} \\ & = -\frac {3 b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d x^{2/3}}+\frac {3 b e^2 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}{d^3 \sqrt [3]{x}}+\frac {3 b e^3 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}{d^3}-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x}+\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 )^2} \, dx,x,d+e \sqrt [3]{x}\right )}{d^2}-\frac {\left (6 b^2 e^2 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e \sqrt [3]{x}\right )}{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 )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )} \, dx,x,d+e \sqrt [3]{x}\right )}{d^2}-\frac {\left (6 b^2 e^3 n^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {d}{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{d^3} \\ & = -\frac {3 b^2 e^2 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 )}{d^3 \sqrt [3]{x}}-\frac {3 b^2 e^3 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 )}{d^3}-\frac {3 b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d x^{2/3}}+\frac {3 b e^2 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}{d^3 \sqrt [3]{x}}+\frac {3 b e^3 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}{d^3}-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x}-\frac {6 b^2 e^3 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^3}-\frac {6 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \text {Li}_2\left (\frac {d}{d+e \sqrt [3]{x}}\right )}{d^3}+\frac {\left (3 b^3 e^2 n^3\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e \sqrt [3]{x}\right )}{d^3}+\frac {\left (3 b^3 e^3 n^3\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {d}{x}\right )}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{d^3}+\frac {\left (6 b^3 e^3 n^3\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{d}\right )}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{d^3}+\frac {\left (6 b^3 e^3 n^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {d}{x}\right )}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{d^3} \\ & = -\frac {3 b^2 e^2 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 )}{d^3 \sqrt [3]{x}}-\frac {3 b^2 e^3 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 )}{d^3}-\frac {3 b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d x^{2/3}}+\frac {3 b e^2 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}{d^3 \sqrt [3]{x}}+\frac {3 b e^3 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}{d^3}-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x}-\frac {6 b^2 e^3 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^3}+\frac {b^3 e^3 n^3 \log (x)}{d^3}+\frac {3 b^3 e^3 n^3 \text {Li}_2\left (\frac {d}{d+e \sqrt [3]{x}}\right )}{d^3}-\frac {6 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \text {Li}_2\left (\frac {d}{d+e \sqrt [3]{x}}\right )}{d^3}-\frac {6 b^3 e^3 n^3 \text {Li}_2\left (1+\frac {e \sqrt [3]{x}}{d}\right )}{d^3}-\frac {6 b^3 e^3 n^3 \text {Li}_3\left (\frac {d}{d+e \sqrt [3]{x}}\right )}{d^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.61 (sec) , antiderivative size = 733, normalized size of antiderivative = 1.67 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x^2} \, dx=\frac {-3 b d^2 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+6 b d 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 )^2-6 b d^3 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 )^2-6 b e^3 n x \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 )^2-2 d^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 )^3+2 b 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 \log (x)-6 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 (d e^2 x^{2/3}+\left (d^3+e^3 x\right ) \log ^2\left (d+e \sqrt [3]{x}\right )+3 e^3 x \log \left (-\frac {e \sqrt [3]{x}}{d}\right )+\log \left (d+e \sqrt [3]{x}\right ) \left (d^2 e \sqrt [3]{x}-2 d e^2 x^{2/3}-3 e^3 x-2 e^3 x \log \left (-\frac {e \sqrt [3]{x}}{d}\right )\right )-2 e^3 x \operatorname {PolyLog}\left (2,1+\frac {e \sqrt [3]{x}}{d}\right )\right )+b^3 n^3 \left (-6 d e^2 x^{2/3} \log \left (d+e \sqrt [3]{x}\right )-6 e^3 x \log \left (d+e \sqrt [3]{x}\right )-3 d^2 e \sqrt [3]{x} \log ^2\left (d+e \sqrt [3]{x}\right )+6 d e^2 x^{2/3} \log ^2\left (d+e \sqrt [3]{x}\right )+9 e^3 x \log ^2\left (d+e \sqrt [3]{x}\right )-2 d^3 \log ^3\left (d+e \sqrt [3]{x}\right )-2 e^3 x \log ^3\left (d+e \sqrt [3]{x}\right )+6 e^3 x \log \left (-\frac {e \sqrt [3]{x}}{d}\right )-18 e^3 x \log \left (d+e \sqrt [3]{x}\right ) \log \left (-\frac {e \sqrt [3]{x}}{d}\right )+6 e^3 x \log ^2\left (d+e \sqrt [3]{x}\right ) \log \left (-\frac {e \sqrt [3]{x}}{d}\right )+6 e^3 x \left (-3+2 \log \left (d+e \sqrt [3]{x}\right )\right ) \operatorname {PolyLog}\left (2,1+\frac {e \sqrt [3]{x}}{d}\right )-12 e^3 x \operatorname {PolyLog}\left (3,1+\frac {e \sqrt [3]{x}}{d}\right )\right )}{2 d^3 x} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {{\left (a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )^{n}\right )\right )}^{3}}{x^{2}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right ) + a\right )}^{3}}{x^{2}} \,d x } \]

[In]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right ) + a\right )}^{3}}{x^{2}} \,d x } \]

[In]

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

[Out]

-1/2*(2*b^3*d^3*x^(2/3)*log((e*x^(1/3) + d)^n)^3 + (6*b^3*e^3*n*x^(5/3)*log(e*x^(1/3) + d) - 6*b^3*d*e^2*n*x^(
4/3) + 3*b^3*d^2*e*n*x - 2*(b^3*e^3*n*x*log(x) - 3*b^3*d^3*log(c) - 3*a*b^2*d^3)*x^(2/3))*log((e*x^(1/3) + d)^
n)^2)/(d^3*x^(5/3)) + integrate(1/3*(3*(b^3*d^3*e*log(c)^3 + 3*a*b^2*d^3*e*log(c)^2 + 3*a^2*b*d^3*e*log(c) + a
^3*d^3*e)*x^(5/3) + 3*(b^3*d^4*log(c)^3 + 3*a*b^2*d^4*log(c)^2 + 3*a^2*b*d^4*log(c) + a^3*d^4)*x^(4/3) + (6*b^
3*e^4*n^2*x^(8/3)*log(e*x^(1/3) + d) - 6*b^3*d*e^3*n^2*x^(7/3) + 3*b^3*d^2*e^2*n^2*x^2 + 9*(b^3*d^3*e*log(c)^2
+ 2*a*b^2*d^3*e*log(c) + a^2*b*d^3*e)*x^(5/3) + 9*(b^3*d^4*log(c)^2 + 2*a*b^2*d^4*log(c) + a^2*b*d^4)*x^(4/3)
- 2*(b^3*e^4*n^2*x^2*log(x) - 3*(b^3*d^3*e*n*log(c) + a*b^2*d^3*e*n)*x)*x^(2/3))*log((e*x^(1/3) + d)^n))/(d^3
*e*x^(11/3) + d^4*x^(10/3)), x)

Giac [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right ) + a\right )}^{3}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x^2} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )\right )}^3}{x^2} \,d x \]

[In]

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

[Out]

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