3.5.84 \(\int \frac {(a+b \log (c (d+e x^{2/3})^n))^3}{x^3} \, dx\) [484]

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

[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 = 0.57, antiderivative size = 451, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 13, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {2504, 2445, 2458, 2389, 2379, 2421, 6724, 2355, 2354, 2438, 2356, 2351, 31} \begin {gather*} -\frac {3 b^2 e^3 n^2 \text {PolyLog}\left (2,\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^3 e^3 n^3 \text {PolyLog}\left (2,\frac {d}{d+e x^{2/3}}\right )}{2 d^3}-\frac {3 b^3 e^3 n^3 \text {PolyLog}\left (2,\frac {e x^{2/3}}{d}+1\right )}{d^3}-\frac {3 b^3 e^3 n^3 \text {PolyLog}\left (3,\frac {d}{d+e x^{2/3}}\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 {b^3 e^3 n^3 \log (x)}{d^3} \end {gather*}

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

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*} \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} \text {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) \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,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) \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 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) \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 x^{2/3}\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 )^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) \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 x^{2/3}\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 )} \, dx,x,d+e x^{2/3}\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 )^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 ) \text {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 ) \text {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 ) \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 x^{2/3}\right )}{2 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 )}{-\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 ) \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 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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 = 0.55, size = 764, normalized size = 1.69 \begin {gather*} \frac {-3 b d^2 e n x^{2/3} \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+6 b d e^2 n x^{4/3} \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-6 b d^3 n \log \left (d+e x^{2/3}\right ) \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-6 b e^3 n x^2 \log \left (d+e x^{2/3}\right ) \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-2 d^3 \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3+4 b e^3 n x^2 \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \log (x)-6 b^2 n^2 \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \left (\left (d^3+e^3 x^2\right ) \log ^2\left (d+e x^{2/3}\right )+e^2 x^{4/3} \left (d+3 e x^{2/3} \log \left (-\frac {e x^{2/3}}{d}\right )\right )+\log \left (d+e x^{2/3}\right ) \left (d^2 e x^{2/3}-2 d e^2 x^{4/3}-3 e^3 x^2-2 e^3 x^2 \log \left (-\frac {e x^{2/3}}{d}\right )\right )-2 e^3 x^2 \text {Li}_2\left (1+\frac {e x^{2/3}}{d}\right )\right )+b^3 n^3 \left (-6 d e^2 x^{4/3} \log \left (d+e x^{2/3}\right )-6 e^3 x^2 \log \left (d+e x^{2/3}\right )-3 d^2 e x^{2/3} \log ^2\left (d+e x^{2/3}\right )+6 d e^2 x^{4/3} \log ^2\left (d+e x^{2/3}\right )+9 e^3 x^2 \log ^2\left (d+e x^{2/3}\right )-2 d^3 \log ^3\left (d+e x^{2/3}\right )-2 e^3 x^2 \log ^3\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 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 \left (-3+2 \log \left (d+e x^{2/3}\right )\right ) \text {Li}_2\left (1+\frac {e x^{2/3}}{d}\right )-12 e^3 x^2 \text {Li}_3\left (1+\frac {e x^{2/3}}{d}\right )\right )}{4 d^3 x^2} \end {gather*}

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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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )^{n}\right )\right )^{3}}{x^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.41, size = 690, normalized size = 1.53 \begin {gather*} -\frac {3}{4} \, a^{2} b n {\left (\frac {2 \, e^{2} \log \left (x^{\frac {2}{3}} e + d\right )}{d^{3}} - \frac {2 \, e^{2} \log \left (x^{\frac {2}{3}}\right )}{d^{3}} - \frac {2 \, x^{\frac {2}{3}} e - d}{d^{2} x^{\frac {4}{3}}}\right )} e + \frac {3 \, {\left (\log \left (x^{\frac {2}{3}} e + d\right )^{2} \log \left (-\frac {x^{\frac {2}{3}} e + d}{d} + 1\right ) + 2 \, {\rm Li}_2\left (\frac {x^{\frac {2}{3}} e + d}{d}\right ) \log \left (x^{\frac {2}{3}} e + d\right ) - 2 \, {\rm Li}_{3}(\frac {x^{\frac {2}{3}} e + d}{d})\right )} b^{3} n^{3} e^{3}}{2 \, d^{3}} - \frac {3 \, a^{2} b \log \left ({\left (x^{\frac {2}{3}} e + d\right )}^{n} c\right )}{2 \, x^{2}} - \frac {a^{3}}{2 \, x^{2}} + \frac {3 \, {\left (2 \, a b^{2} n^{2} - {\left (3 \, n^{3} - 2 \, n^{2} \log \left (c\right )\right )} b^{3}\right )} {\left (\log \left (x^{\frac {2}{3}} e + d\right ) \log \left (-\frac {x^{\frac {2}{3}} e + d}{d} + 1\right ) + {\rm Li}_2\left (\frac {x^{\frac {2}{3}} e + d}{d}\right )\right )} e^{3}}{2 \, d^{3}} - \frac {{\left ({\left (3 \, n^{2} - 2 \, n \log \left (c\right )\right )} a b^{2} - {\left (n^{3} - 3 \, n^{2} \log \left (c\right ) + n \log \left (c\right )^{2}\right )} b^{3}\right )} e^{3} \log \left (x\right )}{d^{3}} - \frac {2 \, b^{3} d^{3} \log \left (c\right )^{3} + 6 \, a b^{2} d^{3} \log \left (c\right )^{2} + 2 \, {\left (b^{3} d^{3} n^{3} + b^{3} n^{3} x^{2} e^{3}\right )} \log \left (x^{\frac {2}{3}} e + d\right )^{3} + 6 \, {\left ({\left (d n^{2} - 2 \, d n \log \left (c\right )\right )} a b^{2} + {\left (d n^{2} \log \left (c\right ) - d n \log \left (c\right )^{2}\right )} b^{3}\right )} x^{\frac {4}{3}} e^{2} + 3 \, {\left (b^{3} d^{2} n^{3} x^{\frac {2}{3}} e - 2 \, b^{3} d n^{3} x^{\frac {4}{3}} e^{2} + 2 \, b^{3} d^{3} n^{2} \log \left (c\right ) + 2 \, a b^{2} d^{3} n^{2} + {\left (2 \, a b^{2} n^{2} - {\left (3 \, n^{3} - 2 \, n^{2} \log \left (c\right )\right )} b^{3}\right )} x^{2} e^{3}\right )} \log \left (x^{\frac {2}{3}} e + d\right )^{2} + 3 \, {\left (b^{3} d^{2} n \log \left (c\right )^{2} + 2 \, a b^{2} d^{2} n \log \left (c\right )\right )} x^{\frac {2}{3}} e + 6 \, {\left (b^{3} d^{3} n \log \left (c\right )^{2} + 2 \, a b^{2} d^{3} n \log \left (c\right ) - {\left ({\left (3 \, n^{2} - 2 \, n \log \left (c\right )\right )} a b^{2} - {\left (n^{3} - 3 \, n^{2} \log \left (c\right ) + n \log \left (c\right )^{2}\right )} b^{3}\right )} x^{2} e^{3} - {\left (2 \, a b^{2} d n^{2} - {\left (d n^{3} - 2 \, d n^{2} \log \left (c\right )\right )} b^{3}\right )} x^{\frac {4}{3}} e^{2} + {\left (b^{3} d^{2} n^{2} \log \left (c\right ) + a b^{2} d^{2} n^{2}\right )} x^{\frac {2}{3}} e\right )} \log \left (x^{\frac {2}{3}} e + d\right )}{4 \, d^{3} x^{2}} \end {gather*}

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]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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((x^(2/3)*e + d)^n*c) + a)^3/x^3, x)

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

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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