\(\int \frac {(a+b \log (c (d+e \sqrt [3]{x})^n))^2}{x^3} \, dx\) [455]

Optimal result

Integrand size = 24, antiderivative size = 405 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{x^3} \, dx=-\frac {b^2 e^2 n^2}{20 d^2 x^{4/3}}+\frac {3 b^2 e^3 n^2}{20 d^3 x}-\frac {47 b^2 e^4 n^2}{120 d^4 x^{2/3}}+\frac {77 b^2 e^5 n^2}{60 d^5 \sqrt [3]{x}}-\frac {77 b^2 e^6 n^2 \log \left (d+e \sqrt [3]{x}\right )}{60 d^6}-\frac {b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{5 d x^{5/3}}+\frac {b e^2 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{4 d^2 x^{4/3}}-\frac {b e^3 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 d^3 x}+\frac {b e^4 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{2 d^4 x^{2/3}}-\frac {b e^5 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d^6 \sqrt [3]{x}}-\frac {b e^6 n \log \left (1-\frac {d}{d+e \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d^6}-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 x^2}+\frac {137 b^2 e^6 n^2 \log (x)}{180 d^6}+\frac {b^2 e^6 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+e \sqrt [3]{x}}\right )}{d^6} \]

[Out]

-1/20*b^2*e^2*n^2/d^2/x^(4/3)+3/20*b^2*e^3*n^2/d^3/x-47/120*b^2*e^4*n^2/d^4/x^(2/3)+77/60*b^2*e^5*n^2/d^5/x^(1
/3)-77/60*b^2*e^6*n^2*ln(d+e*x^(1/3))/d^6-1/5*b*e*n*(a+b*ln(c*(d+e*x^(1/3))^n))/d/x^(5/3)+1/4*b*e^2*n*(a+b*ln(
c*(d+e*x^(1/3))^n))/d^2/x^(4/3)-1/3*b*e^3*n*(a+b*ln(c*(d+e*x^(1/3))^n))/d^3/x+1/2*b*e^4*n*(a+b*ln(c*(d+e*x^(1/
3))^n))/d^4/x^(2/3)-b*e^5*n*(d+e*x^(1/3))*(a+b*ln(c*(d+e*x^(1/3))^n))/d^6/x^(1/3)-b*e^6*n*ln(1-d/(d+e*x^(1/3))
)*(a+b*ln(c*(d+e*x^(1/3))^n))/d^6-1/2*(a+b*ln(c*(d+e*x^(1/3))^n))^2/x^2+137/180*b^2*e^6*n^2*ln(x)/d^6+b^2*e^6*
n^2*polylog(2,d/(d+e*x^(1/3)))/d^6

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2504, 2445, 2458, 2389, 2379, 2438, 2351, 31, 2356, 46} \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{x^3} \, dx=-\frac {b e^6 n \log \left (1-\frac {d}{d+e \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d^6}-\frac {b e^5 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d^6 \sqrt [3]{x}}+\frac {b e^4 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{2 d^4 x^{2/3}}-\frac {b e^3 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 d^3 x}+\frac {b e^2 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{4 d^2 x^{4/3}}-\frac {b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{5 d x^{5/3}}-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 x^2}+\frac {b^2 e^6 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+e \sqrt [3]{x}}\right )}{d^6}-\frac {77 b^2 e^6 n^2 \log \left (d+e \sqrt [3]{x}\right )}{60 d^6}+\frac {137 b^2 e^6 n^2 \log (x)}{180 d^6}+\frac {77 b^2 e^5 n^2}{60 d^5 \sqrt [3]{x}}-\frac {47 b^2 e^4 n^2}{120 d^4 x^{2/3}}+\frac {3 b^2 e^3 n^2}{20 d^3 x}-\frac {b^2 e^2 n^2}{20 d^2 x^{4/3}} \]

[In]

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

[Out]

-1/20*(b^2*e^2*n^2)/(d^2*x^(4/3)) + (3*b^2*e^3*n^2)/(20*d^3*x) - (47*b^2*e^4*n^2)/(120*d^4*x^(2/3)) + (77*b^2*
e^5*n^2)/(60*d^5*x^(1/3)) - (77*b^2*e^6*n^2*Log[d + e*x^(1/3)])/(60*d^6) - (b*e*n*(a + b*Log[c*(d + e*x^(1/3))
^n]))/(5*d*x^(5/3)) + (b*e^2*n*(a + b*Log[c*(d + e*x^(1/3))^n]))/(4*d^2*x^(4/3)) - (b*e^3*n*(a + b*Log[c*(d +
e*x^(1/3))^n]))/(3*d^3*x) + (b*e^4*n*(a + b*Log[c*(d + e*x^(1/3))^n]))/(2*d^4*x^(2/3)) - (b*e^5*n*(d + e*x^(1/
3))*(a + b*Log[c*(d + e*x^(1/3))^n]))/(d^6*x^(1/3)) - (b*e^6*n*Log[1 - d/(d + e*x^(1/3))]*(a + b*Log[c*(d + e*
x^(1/3))^n]))/d^6 - (a + b*Log[c*(d + e*x^(1/3))^n])^2/(2*x^2) + (137*b^2*e^6*n^2*Log[x])/(180*d^6) + (b^2*e^6
*n^2*PolyLog[2, 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 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 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])

Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{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 )^2}{2 x^2}+(b e n) \text {Subst}\left (\int \frac {a+b \log \left (c (d+e x)^n\right )}{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 )^2}{2 x^2}+(b n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^6} \, dx,x,d+e \sqrt [3]{x}\right ) \\ & = -\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 x^2}+\frac {(b n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^6} \, dx,x,d+e \sqrt [3]{x}\right )}{d}-\frac {(b e n) \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 )}{d} \\ & = -\frac {b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{5 d x^{5/3}}-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 x^2}-\frac {(b e n) \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 )}{d^2}+\frac {\left (b e^2 n\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 )}{d^2}+\frac {\left (b^2 e n^2\right ) \text {Subst}\left (\int \frac {1}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^5} \, dx,x,d+e \sqrt [3]{x}\right )}{5 d} \\ & = -\frac {b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{5 d x^{5/3}}+\frac {b e^2 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{4 d^2 x^{4/3}}-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 x^2}+\frac {\left (b e^2 n\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 )}{d^3}-\frac {\left (b e^3 n\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 (b^2 e n^2\right ) \text {Subst}\left (\int \left (-\frac {e^5}{d (d-x)^5}-\frac {e^5}{d^2 (d-x)^4}-\frac {e^5}{d^3 (d-x)^3}-\frac {e^5}{d^4 (d-x)^2}-\frac {e^5}{d^5 (d-x)}-\frac {e^5}{d^5 x}\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{5 d}-\frac {\left (b^2 e^2 n^2\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 )}{4 d^2} \\ & = -\frac {b^2 e^2 n^2}{20 d^2 x^{4/3}}+\frac {b^2 e^3 n^2}{15 d^3 x}-\frac {b^2 e^4 n^2}{10 d^4 x^{2/3}}+\frac {b^2 e^5 n^2}{5 d^5 \sqrt [3]{x}}-\frac {b^2 e^6 n^2 \log \left (d+e \sqrt [3]{x}\right )}{5 d^6}-\frac {b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{5 d x^{5/3}}+\frac {b e^2 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{4 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 )}{3 d^3 x}-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 x^2}+\frac {b^2 e^6 n^2 \log (x)}{15 d^6}-\frac {\left (b e^3 n\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 (b e^4 n\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 (b^2 e^2 n^2\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 )}{4 d^2}+\frac {\left (b^2 e^3 n^2\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 )}{3 d^3} \\ & = -\frac {b^2 e^2 n^2}{20 d^2 x^{4/3}}+\frac {3 b^2 e^3 n^2}{20 d^3 x}-\frac {9 b^2 e^4 n^2}{40 d^4 x^{2/3}}+\frac {9 b^2 e^5 n^2}{20 d^5 \sqrt [3]{x}}-\frac {9 b^2 e^6 n^2 \log \left (d+e \sqrt [3]{x}\right )}{20 d^6}-\frac {b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{5 d x^{5/3}}+\frac {b e^2 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{4 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 )}{3 d^3 x}+\frac {b e^4 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{2 d^4 x^{2/3}}-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 x^2}+\frac {3 b^2 e^6 n^2 \log (x)}{20 d^6}+\frac {\left (b e^4 n\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^5}-\frac {\left (b e^5 n\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^5}+\frac {\left (b^2 e^3 n^2\right ) \text {Subst}\left (\int \left (-\frac {e^3}{d (d-x)^3}-\frac {e^3}{d^2 (d-x)^2}-\frac {e^3}{d^3 (d-x)}-\frac {e^3}{d^3 x}\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{3 d^3}-\frac {\left (b^2 e^4 n^2\right ) \text {Subst}\left (\int \frac {1}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d^4} \\ & = -\frac {b^2 e^2 n^2}{20 d^2 x^{4/3}}+\frac {3 b^2 e^3 n^2}{20 d^3 x}-\frac {47 b^2 e^4 n^2}{120 d^4 x^{2/3}}+\frac {47 b^2 e^5 n^2}{60 d^5 \sqrt [3]{x}}-\frac {47 b^2 e^6 n^2 \log \left (d+e \sqrt [3]{x}\right )}{60 d^6}-\frac {b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{5 d x^{5/3}}+\frac {b e^2 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{4 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 )}{3 d^3 x}+\frac {b e^4 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{2 d^4 x^{2/3}}-\frac {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 )}{d^6 \sqrt [3]{x}}-\frac {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 )}{d^6}-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 x^2}+\frac {47 b^2 e^6 n^2 \log (x)}{180 d^6}-\frac {\left (b^2 e^4 n^2\right ) \text {Subst}\left (\int \left (\frac {e^2}{d (d-x)^2}+\frac {e^2}{d^2 (d-x)}+\frac {e^2}{d^2 x}\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{2 d^4}+\frac {\left (b^2 e^5 n^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e \sqrt [3]{x}\right )}{d^6}+\frac {\left (b^2 e^6 n^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {d}{x}\right )}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{d^6} \\ & = -\frac {b^2 e^2 n^2}{20 d^2 x^{4/3}}+\frac {3 b^2 e^3 n^2}{20 d^3 x}-\frac {47 b^2 e^4 n^2}{120 d^4 x^{2/3}}+\frac {77 b^2 e^5 n^2}{60 d^5 \sqrt [3]{x}}-\frac {77 b^2 e^6 n^2 \log \left (d+e \sqrt [3]{x}\right )}{60 d^6}-\frac {b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{5 d x^{5/3}}+\frac {b e^2 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{4 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 )}{3 d^3 x}+\frac {b e^4 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{2 d^4 x^{2/3}}-\frac {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 )}{d^6 \sqrt [3]{x}}-\frac {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 )}{d^6}-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 x^2}+\frac {137 b^2 e^6 n^2 \log (x)}{180 d^6}+\frac {b^2 e^6 n^2 \text {Li}_2\left (\frac {d}{d+e \sqrt [3]{x}}\right )}{d^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{x^3} \, dx=-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 x^2}-\frac {b e \left (72 a d^5 n-90 a d^4 e n \sqrt [3]{x}+18 b d^4 e n^2 \sqrt [3]{x}+120 a d^3 e^2 n x^{2/3}-54 b d^3 e^2 n^2 x^{2/3}-180 a d^2 e^3 n x+141 b d^2 e^3 n^2 x+360 a d e^4 n x^{4/3}-462 b d e^4 n^2 x^{4/3}+6 e^5 n (-60 a+137 b n) x^{5/3} \log \left (d+e \sqrt [3]{x}\right )+72 b d^5 n \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )-90 b d^4 e n \sqrt [3]{x} \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )+120 b d^3 e^2 n x^{2/3} \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )-180 b d^2 e^3 n x \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )+360 b d e^4 n x^{4/3} \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )-180 b e^5 x^{5/3} \log ^2\left (c \left (d+e \sqrt [3]{x}\right )^n\right )+360 b e^5 n x^{5/3} \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right ) \log \left (-\frac {e \sqrt [3]{x}}{d}\right )+120 a e^5 n x^{5/3} \log (x)-274 b e^5 n^2 x^{5/3} \log (x)+360 b e^5 n^2 x^{5/3} \operatorname {PolyLog}\left (2,1+\frac {e \sqrt [3]{x}}{d}\right )\right )}{360 d^6 x^{5/3}} \]

[In]

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

[Out]

-1/2*(a + b*Log[c*(d + e*x^(1/3))^n])^2/x^2 - (b*e*(72*a*d^5*n - 90*a*d^4*e*n*x^(1/3) + 18*b*d^4*e*n^2*x^(1/3)
+ 120*a*d^3*e^2*n*x^(2/3) - 54*b*d^3*e^2*n^2*x^(2/3) - 180*a*d^2*e^3*n*x + 141*b*d^2*e^3*n^2*x + 360*a*d*e^4*
n*x^(4/3) - 462*b*d*e^4*n^2*x^(4/3) + 6*e^5*n*(-60*a + 137*b*n)*x^(5/3)*Log[d + e*x^(1/3)] + 72*b*d^5*n*Log[c*
(d + e*x^(1/3))^n] - 90*b*d^4*e*n*x^(1/3)*Log[c*(d + e*x^(1/3))^n] + 120*b*d^3*e^2*n*x^(2/3)*Log[c*(d + e*x^(1
/3))^n] - 180*b*d^2*e^3*n*x*Log[c*(d + e*x^(1/3))^n] + 360*b*d*e^4*n*x^(4/3)*Log[c*(d + e*x^(1/3))^n] - 180*b*
e^5*x^(5/3)*Log[c*(d + e*x^(1/3))^n]^2 + 360*b*e^5*n*x^(5/3)*Log[c*(d + e*x^(1/3))^n]*Log[-((e*x^(1/3))/d)] +
120*a*e^5*n*x^(5/3)*Log[x] - 274*b*e^5*n^2*x^(5/3)*Log[x] + 360*b*e^5*n^2*x^(5/3)*PolyLog[2, 1 + (e*x^(1/3))/d
]))/(360*d^6*x^(5/3))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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