\(\int \frac {(a+b \log (c (d+e x^{2/3})^n))^2}{x^5} \, dx\) [475]

Optimal result

Integrand size = 24, antiderivative size = 412 \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^5} \, dx=-\frac {b^2 e^2 n^2}{40 d^2 x^{8/3}}+\frac {3 b^2 e^3 n^2}{40 d^3 x^2}-\frac {47 b^2 e^4 n^2}{240 d^4 x^{4/3}}+\frac {77 b^2 e^5 n^2}{120 d^5 x^{2/3}}-\frac {77 b^2 e^6 n^2 \log \left (d+e x^{2/3}\right )}{120 d^6}-\frac {b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{10 d x^{10/3}}+\frac {b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{8 d^2 x^{8/3}}-\frac {b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{6 d^3 x^2}+\frac {b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{4 d^4 x^{4/3}}-\frac {b e^5 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 d^6 x^{2/3}}-\frac {b e^6 n \log \left (1-\frac {d}{d+e x^{2/3}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 d^6}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 x^4}+\frac {137 b^2 e^6 n^2 \log (x)}{180 d^6}+\frac {b^2 e^6 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+e x^{2/3}}\right )}{2 d^6} \]

[Out]

-1/40*b^2*e^2*n^2/d^2/x^(8/3)+3/40*b^2*e^3*n^2/d^3/x^2-47/240*b^2*e^4*n^2/d^4/x^(4/3)+77/120*b^2*e^5*n^2/d^5/x
^(2/3)-77/120*b^2*e^6*n^2*ln(d+e*x^(2/3))/d^6-1/10*b*e*n*(a+b*ln(c*(d+e*x^(2/3))^n))/d/x^(10/3)+1/8*b*e^2*n*(a
+b*ln(c*(d+e*x^(2/3))^n))/d^2/x^(8/3)-1/6*b*e^3*n*(a+b*ln(c*(d+e*x^(2/3))^n))/d^3/x^2+1/4*b*e^4*n*(a+b*ln(c*(d
+e*x^(2/3))^n))/d^4/x^(4/3)-1/2*b*e^5*n*(d+e*x^(2/3))*(a+b*ln(c*(d+e*x^(2/3))^n))/d^6/x^(2/3)-1/2*b*e^6*n*ln(1
-d/(d+e*x^(2/3)))*(a+b*ln(c*(d+e*x^(2/3))^n))/d^6-1/4*(a+b*ln(c*(d+e*x^(2/3))^n))^2/x^4+137/180*b^2*e^6*n^2*ln
(x)/d^6+1/2*b^2*e^6*n^2*polylog(2,d/(d+e*x^(2/3)))/d^6

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 412, 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} \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^5} \, dx=-\frac {b e^6 n \log \left (1-\frac {d}{d+e x^{2/3}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 d^6}-\frac {b e^5 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 d^6 x^{2/3}}+\frac {b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{4 d^4 x^{4/3}}-\frac {b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{6 d^3 x^2}+\frac {b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{8 d^2 x^{8/3}}-\frac {b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{10 d x^{10/3}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 x^4}+\frac {b^2 e^6 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+e x^{2/3}}\right )}{2 d^6}-\frac {77 b^2 e^6 n^2 \log \left (d+e x^{2/3}\right )}{120 d^6}+\frac {137 b^2 e^6 n^2 \log (x)}{180 d^6}+\frac {77 b^2 e^5 n^2}{120 d^5 x^{2/3}}-\frac {47 b^2 e^4 n^2}{240 d^4 x^{4/3}}+\frac {3 b^2 e^3 n^2}{40 d^3 x^2}-\frac {b^2 e^2 n^2}{40 d^2 x^{8/3}} \]

[In]

Int[(a + b*Log[c*(d + e*x^(2/3))^n])^2/x^5,x]

[Out]

-1/40*(b^2*e^2*n^2)/(d^2*x^(8/3)) + (3*b^2*e^3*n^2)/(40*d^3*x^2) - (47*b^2*e^4*n^2)/(240*d^4*x^(4/3)) + (77*b^
2*e^5*n^2)/(120*d^5*x^(2/3)) - (77*b^2*e^6*n^2*Log[d + e*x^(2/3)])/(120*d^6) - (b*e*n*(a + b*Log[c*(d + e*x^(2
/3))^n]))/(10*d*x^(10/3)) + (b*e^2*n*(a + b*Log[c*(d + e*x^(2/3))^n]))/(8*d^2*x^(8/3)) - (b*e^3*n*(a + b*Log[c
*(d + e*x^(2/3))^n]))/(6*d^3*x^2) + (b*e^4*n*(a + b*Log[c*(d + e*x^(2/3))^n]))/(4*d^4*x^(4/3)) - (b*e^5*n*(d +
e*x^(2/3))*(a + b*Log[c*(d + e*x^(2/3))^n]))/(2*d^6*x^(2/3)) - (b*e^6*n*Log[1 - d/(d + e*x^(2/3))]*(a + b*Log
[c*(d + e*x^(2/3))^n]))/(2*d^6) - (a + b*Log[c*(d + e*x^(2/3))^n])^2/(4*x^4) + (137*b^2*e^6*n^2*Log[x])/(180*d
^6) + (b^2*e^6*n^2*PolyLog[2, d/(d + e*x^(2/3))])/(2*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}& = \frac {3}{2} \text {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^7} \, dx,x,x^{2/3}\right ) \\ & = -\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 x^4}+\frac {1}{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,x^{2/3}\right ) \\ & = -\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 x^4}+\frac {1}{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 x^{2/3}\right ) \\ & = -\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 x^4}+\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 x^{2/3}\right )}{2 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 x^{2/3}\right )}{2 d} \\ & = -\frac {b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{10 d x^{10/3}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 x^4}-\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 x^{2/3}\right )}{2 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 x^{2/3}\right )}{2 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 x^{2/3}\right )}{10 d} \\ & = -\frac {b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{10 d x^{10/3}}+\frac {b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{8 d^2 x^{8/3}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 x^4}+\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 x^{2/3}\right )}{2 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 x^{2/3}\right )}{2 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 x^{2/3}\right )}{10 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 x^{2/3}\right )}{8 d^2} \\ & = -\frac {b^2 e^2 n^2}{40 d^2 x^{8/3}}+\frac {b^2 e^3 n^2}{30 d^3 x^2}-\frac {b^2 e^4 n^2}{20 d^4 x^{4/3}}+\frac {b^2 e^5 n^2}{10 d^5 x^{2/3}}-\frac {b^2 e^6 n^2 \log \left (d+e x^{2/3}\right )}{10 d^6}-\frac {b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{10 d x^{10/3}}+\frac {b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{8 d^2 x^{8/3}}-\frac {b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{6 d^3 x^2}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 x^4}+\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 x^{2/3}\right )}{2 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 x^{2/3}\right )}{2 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 x^{2/3}\right )}{8 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 x^{2/3}\right )}{6 d^3} \\ & = -\frac {b^2 e^2 n^2}{40 d^2 x^{8/3}}+\frac {3 b^2 e^3 n^2}{40 d^3 x^2}-\frac {9 b^2 e^4 n^2}{80 d^4 x^{4/3}}+\frac {9 b^2 e^5 n^2}{40 d^5 x^{2/3}}-\frac {9 b^2 e^6 n^2 \log \left (d+e x^{2/3}\right )}{40 d^6}-\frac {b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{10 d x^{10/3}}+\frac {b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{8 d^2 x^{8/3}}-\frac {b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{6 d^3 x^2}+\frac {b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{4 d^4 x^{4/3}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 x^4}+\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 x^{2/3}\right )}{2 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 x^{2/3}\right )}{2 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 x^{2/3}\right )}{6 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 x^{2/3}\right )}{4 d^4} \\ & = -\frac {b^2 e^2 n^2}{40 d^2 x^{8/3}}+\frac {3 b^2 e^3 n^2}{40 d^3 x^2}-\frac {47 b^2 e^4 n^2}{240 d^4 x^{4/3}}+\frac {47 b^2 e^5 n^2}{120 d^5 x^{2/3}}-\frac {47 b^2 e^6 n^2 \log \left (d+e x^{2/3}\right )}{120 d^6}-\frac {b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{10 d x^{10/3}}+\frac {b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{8 d^2 x^{8/3}}-\frac {b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{6 d^3 x^2}+\frac {b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{4 d^4 x^{4/3}}-\frac {b e^5 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 d^6 x^{2/3}}-\frac {b e^6 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 d^6}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 x^4}+\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 x^{2/3}\right )}{4 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 x^{2/3}\right )}{2 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 x^{2/3}\right )}{2 d^6} \\ & = -\frac {b^2 e^2 n^2}{40 d^2 x^{8/3}}+\frac {3 b^2 e^3 n^2}{40 d^3 x^2}-\frac {47 b^2 e^4 n^2}{240 d^4 x^{4/3}}+\frac {77 b^2 e^5 n^2}{120 d^5 x^{2/3}}-\frac {77 b^2 e^6 n^2 \log \left (d+e x^{2/3}\right )}{120 d^6}-\frac {b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{10 d x^{10/3}}+\frac {b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{8 d^2 x^{8/3}}-\frac {b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{6 d^3 x^2}+\frac {b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{4 d^4 x^{4/3}}-\frac {b e^5 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 d^6 x^{2/3}}-\frac {b e^6 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 d^6}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 x^4}+\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 x^{2/3}}\right )}{2 d^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^5} \, dx=-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 x^4}-\frac {b e \left (72 a d^5 n-90 a d^4 e n x^{2/3}+18 b d^4 e n^2 x^{2/3}+120 a d^3 e^2 n x^{4/3}-54 b d^3 e^2 n^2 x^{4/3}-180 a d^2 e^3 n x^2+141 b d^2 e^3 n^2 x^2+360 a d e^4 n x^{8/3}-462 b d e^4 n^2 x^{8/3}+6 e^5 n (-60 a+137 b n) x^{10/3} \log \left (d+e x^{2/3}\right )+72 b d^5 n \log \left (c \left (d+e x^{2/3}\right )^n\right )-90 b d^4 e n x^{2/3} \log \left (c \left (d+e x^{2/3}\right )^n\right )+120 b d^3 e^2 n x^{4/3} \log \left (c \left (d+e x^{2/3}\right )^n\right )-180 b d^2 e^3 n x^2 \log \left (c \left (d+e x^{2/3}\right )^n\right )+360 b d e^4 n x^{8/3} \log \left (c \left (d+e x^{2/3}\right )^n\right )-180 b e^5 x^{10/3} \log ^2\left (c \left (d+e x^{2/3}\right )^n\right )+360 b e^5 n x^{10/3} \log \left (c \left (d+e x^{2/3}\right )^n\right ) \log \left (-\frac {e x^{2/3}}{d}\right )+240 a e^5 n x^{10/3} \log (x)-548 b e^5 n^2 x^{10/3} \log (x)+360 b e^5 n^2 x^{10/3} \operatorname {PolyLog}\left (2,1+\frac {e x^{2/3}}{d}\right )\right )}{720 d^6 x^{10/3}} \]

[In]

Integrate[(a + b*Log[c*(d + e*x^(2/3))^n])^2/x^5,x]

[Out]

-1/4*(a + b*Log[c*(d + e*x^(2/3))^n])^2/x^4 - (b*e*(72*a*d^5*n - 90*a*d^4*e*n*x^(2/3) + 18*b*d^4*e*n^2*x^(2/3)
+ 120*a*d^3*e^2*n*x^(4/3) - 54*b*d^3*e^2*n^2*x^(4/3) - 180*a*d^2*e^3*n*x^2 + 141*b*d^2*e^3*n^2*x^2 + 360*a*d*
e^4*n*x^(8/3) - 462*b*d*e^4*n^2*x^(8/3) + 6*e^5*n*(-60*a + 137*b*n)*x^(10/3)*Log[d + e*x^(2/3)] + 72*b*d^5*n*L
og[c*(d + e*x^(2/3))^n] - 90*b*d^4*e*n*x^(2/3)*Log[c*(d + e*x^(2/3))^n] + 120*b*d^3*e^2*n*x^(4/3)*Log[c*(d + e
*x^(2/3))^n] - 180*b*d^2*e^3*n*x^2*Log[c*(d + e*x^(2/3))^n] + 360*b*d*e^4*n*x^(8/3)*Log[c*(d + e*x^(2/3))^n] -
180*b*e^5*x^(10/3)*Log[c*(d + e*x^(2/3))^n]^2 + 360*b*e^5*n*x^(10/3)*Log[c*(d + e*x^(2/3))^n]*Log[-((e*x^(2/3
))/d)] + 240*a*e^5*n*x^(10/3)*Log[x] - 548*b*e^5*n^2*x^(10/3)*Log[x] + 360*b*e^5*n^2*x^(10/3)*PolyLog[2, 1 + (
e*x^(2/3))/d]))/(720*d^6*x^(10/3))

Maple [F]

\[\int \frac {{\left (a +b \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )^{n}\right )\right )}^{2}}{x^{5}}d x\]

[In]

int((a+b*ln(c*(d+e*x^(2/3))^n))^2/x^5,x)

[Out]

int((a+b*ln(c*(d+e*x^(2/3))^n))^2/x^5,x)

Fricas [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^5} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a\right )}^{2}}{x^{5}} \,d x } \]

[In]

integrate((a+b*log(c*(d+e*x^(2/3))^n))^2/x^5,x, algorithm="fricas")

[Out]

integral((b^2*log((e*x^(2/3) + d)^n*c)^2 + 2*a*b*log((e*x^(2/3) + d)^n*c) + a^2)/x^5, x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^5} \, dx=\text {Timed out} \]

[In]

integrate((a+b*ln(c*(d+e*x**(2/3))**n))**2/x**5,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^5} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a\right )}^{2}}{x^{5}} \,d x } \]

[In]

integrate((a+b*log(c*(d+e*x^(2/3))^n))^2/x^5,x, algorithm="maxima")

[Out]

-1/4*b^2*log((e*x^(2/3) + d)^n)^2/x^4 + 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^(1/3))*log((e*x^(2/3) + d)^n) + 3*(b^2*d*log(c)^
2 + 2*a*b*d*log(c) + a^2*d)*x^(1/3))/(e*x^6 + d*x^(16/3)), x)

Giac [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^5} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a\right )}^{2}}{x^{5}} \,d x } \]

[In]

integrate((a+b*log(c*(d+e*x^(2/3))^n))^2/x^5,x, algorithm="giac")

[Out]

integrate((b*log((e*x^(2/3) + d)^n*c) + a)^2/x^5, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^5} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )\right )}^2}{x^5} \,d x \]

[In]

int((a + b*log(c*(d + e*x^(2/3))^n))^2/x^5,x)

[Out]

int((a + b*log(c*(d + e*x^(2/3))^n))^2/x^5, x)