\(\int \frac {(a+b \log (c (d+\frac {e}{x^{2/3}})^n))^2}{x^5} \, dx\) [520]

Optimal result

Integrand size = 24, antiderivative size = 482 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^5} \, dx=-\frac {15 b^2 d^4 n^2 \left (d+\frac {e}{x^{2/3}}\right )^2}{8 e^6}+\frac {10 b^2 d^3 n^2 \left (d+\frac {e}{x^{2/3}}\right )^3}{9 e^6}-\frac {15 b^2 d^2 n^2 \left (d+\frac {e}{x^{2/3}}\right )^4}{32 e^6}+\frac {3 b^2 d n^2 \left (d+\frac {e}{x^{2/3}}\right )^5}{25 e^6}-\frac {b^2 n^2 \left (d+\frac {e}{x^{2/3}}\right )^6}{72 e^6}+\frac {3 b^2 d^5 n^2}{e^5 x^{2/3}}-\frac {b^2 d^6 n^2 \log ^2\left (d+\frac {e}{x^{2/3}}\right )}{4 e^6}-\frac {3 b d^5 n \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^6}+\frac {15 b d^4 n \left (d+\frac {e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{4 e^6}-\frac {10 b d^3 n \left (d+\frac {e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 e^6}+\frac {15 b d^2 n \left (d+\frac {e}{x^{2/3}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{8 e^6}-\frac {3 b d n \left (d+\frac {e}{x^{2/3}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{5 e^6}+\frac {b n \left (d+\frac {e}{x^{2/3}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{12 e^6}+\frac {b d^6 n \log \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{2 e^6}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{4 x^4} \]

[Out]

-15/8*b^2*d^4*n^2*(d+e/x^(2/3))^2/e^6+10/9*b^2*d^3*n^2*(d+e/x^(2/3))^3/e^6-15/32*b^2*d^2*n^2*(d+e/x^(2/3))^4/e
^6+3/25*b^2*d*n^2*(d+e/x^(2/3))^5/e^6-1/72*b^2*n^2*(d+e/x^(2/3))^6/e^6+3*b^2*d^5*n^2/e^5/x^(2/3)-1/4*b^2*d^6*n
^2*ln(d+e/x^(2/3))^2/e^6-3*b*d^5*n*(d+e/x^(2/3))*(a+b*ln(c*(d+e/x^(2/3))^n))/e^6+15/4*b*d^4*n*(d+e/x^(2/3))^2*
(a+b*ln(c*(d+e/x^(2/3))^n))/e^6-10/3*b*d^3*n*(d+e/x^(2/3))^3*(a+b*ln(c*(d+e/x^(2/3))^n))/e^6+15/8*b*d^2*n*(d+e
/x^(2/3))^4*(a+b*ln(c*(d+e/x^(2/3))^n))/e^6-3/5*b*d*n*(d+e/x^(2/3))^5*(a+b*ln(c*(d+e/x^(2/3))^n))/e^6+1/12*b*n
*(d+e/x^(2/3))^6*(a+b*ln(c*(d+e/x^(2/3))^n))/e^6+1/2*b*d^6*n*ln(d+e/x^(2/3))*(a+b*ln(c*(d+e/x^(2/3))^n))/e^6-1
/4*(a+b*ln(c*(d+e/x^(2/3))^n))^2/x^4

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 482, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2504, 2445, 2458, 45, 2372, 12, 14, 2338} \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^5} \, dx=\frac {b d^6 n \log \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{2 e^6}-\frac {3 b d^5 n \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^6}+\frac {15 b d^4 n \left (d+\frac {e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{4 e^6}-\frac {10 b d^3 n \left (d+\frac {e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 e^6}+\frac {15 b d^2 n \left (d+\frac {e}{x^{2/3}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{8 e^6}-\frac {3 b d n \left (d+\frac {e}{x^{2/3}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{5 e^6}+\frac {b n \left (d+\frac {e}{x^{2/3}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{12 e^6}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{4 x^4}-\frac {b^2 d^6 n^2 \log ^2\left (d+\frac {e}{x^{2/3}}\right )}{4 e^6}+\frac {3 b^2 d^5 n^2}{e^5 x^{2/3}}-\frac {15 b^2 d^4 n^2 \left (d+\frac {e}{x^{2/3}}\right )^2}{8 e^6}+\frac {10 b^2 d^3 n^2 \left (d+\frac {e}{x^{2/3}}\right )^3}{9 e^6}-\frac {15 b^2 d^2 n^2 \left (d+\frac {e}{x^{2/3}}\right )^4}{32 e^6}+\frac {3 b^2 d n^2 \left (d+\frac {e}{x^{2/3}}\right )^5}{25 e^6}-\frac {b^2 n^2 \left (d+\frac {e}{x^{2/3}}\right )^6}{72 e^6} \]

[In]

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

[Out]

(-15*b^2*d^4*n^2*(d + e/x^(2/3))^2)/(8*e^6) + (10*b^2*d^3*n^2*(d + e/x^(2/3))^3)/(9*e^6) - (15*b^2*d^2*n^2*(d
+ e/x^(2/3))^4)/(32*e^6) + (3*b^2*d*n^2*(d + e/x^(2/3))^5)/(25*e^6) - (b^2*n^2*(d + e/x^(2/3))^6)/(72*e^6) + (
3*b^2*d^5*n^2)/(e^5*x^(2/3)) - (b^2*d^6*n^2*Log[d + e/x^(2/3)]^2)/(4*e^6) - (3*b*d^5*n*(d + e/x^(2/3))*(a + b*
Log[c*(d + e/x^(2/3))^n]))/e^6 + (15*b*d^4*n*(d + e/x^(2/3))^2*(a + b*Log[c*(d + e/x^(2/3))^n]))/(4*e^6) - (10
*b*d^3*n*(d + e/x^(2/3))^3*(a + b*Log[c*(d + e/x^(2/3))^n]))/(3*e^6) + (15*b*d^2*n*(d + e/x^(2/3))^4*(a + b*Lo
g[c*(d + e/x^(2/3))^n]))/(8*e^6) - (3*b*d*n*(d + e/x^(2/3))^5*(a + b*Log[c*(d + e/x^(2/3))^n]))/(5*e^6) + (b*n
*(d + e/x^(2/3))^6*(a + b*Log[c*(d + e/x^(2/3))^n]))/(12*e^6) + (b*d^6*n*Log[d + e/x^(2/3)]*(a + b*Log[c*(d +
e/x^(2/3))^n]))/(2*e^6) - (a + b*Log[c*(d + e/x^(2/3))^n])^2/(4*x^4)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2372

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]]
/; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -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}& = -\left (\frac {3}{2} \text {Subst}\left (\int x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx,x,\frac {1}{x^{2/3}}\right )\right ) \\ & = -\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{4 x^4}+\frac {1}{2} (b e n) \text {Subst}\left (\int \frac {x^6 \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx,x,\frac {1}{x^{2/3}}\right ) \\ & = -\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{4 x^4}+\frac {1}{2} (b n) \text {Subst}\left (\int \frac {\left (-\frac {d}{e}+\frac {x}{e}\right )^6 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+\frac {e}{x^{2/3}}\right ) \\ & = -\frac {3 b d^5 n \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^6}+\frac {15 b d^4 n \left (d+\frac {e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{4 e^6}-\frac {10 b d^3 n \left (d+\frac {e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 e^6}+\frac {15 b d^2 n \left (d+\frac {e}{x^{2/3}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{8 e^6}-\frac {3 b d n \left (d+\frac {e}{x^{2/3}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{5 e^6}+\frac {b n \left (d+\frac {e}{x^{2/3}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{12 e^6}+\frac {b d^6 n \log \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{2 e^6}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{4 x^4}-\frac {1}{2} \left (b^2 n^2\right ) \text {Subst}\left (\int \frac {x \left (-360 d^5+450 d^4 x-400 d^3 x^2+225 d^2 x^3-72 d x^4+10 x^5\right )+60 d^6 \log (x)}{60 e^6 x} \, dx,x,d+\frac {e}{x^{2/3}}\right ) \\ & = -\frac {3 b d^5 n \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^6}+\frac {15 b d^4 n \left (d+\frac {e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{4 e^6}-\frac {10 b d^3 n \left (d+\frac {e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 e^6}+\frac {15 b d^2 n \left (d+\frac {e}{x^{2/3}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{8 e^6}-\frac {3 b d n \left (d+\frac {e}{x^{2/3}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{5 e^6}+\frac {b n \left (d+\frac {e}{x^{2/3}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{12 e^6}+\frac {b d^6 n \log \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{2 e^6}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{4 x^4}-\frac {\left (b^2 n^2\right ) \text {Subst}\left (\int \frac {x \left (-360 d^5+450 d^4 x-400 d^3 x^2+225 d^2 x^3-72 d x^4+10 x^5\right )+60 d^6 \log (x)}{x} \, dx,x,d+\frac {e}{x^{2/3}}\right )}{120 e^6} \\ & = -\frac {3 b d^5 n \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^6}+\frac {15 b d^4 n \left (d+\frac {e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{4 e^6}-\frac {10 b d^3 n \left (d+\frac {e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 e^6}+\frac {15 b d^2 n \left (d+\frac {e}{x^{2/3}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{8 e^6}-\frac {3 b d n \left (d+\frac {e}{x^{2/3}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{5 e^6}+\frac {b n \left (d+\frac {e}{x^{2/3}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{12 e^6}+\frac {b d^6 n \log \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{2 e^6}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{4 x^4}-\frac {\left (b^2 n^2\right ) \text {Subst}\left (\int \left (-360 d^5+450 d^4 x-400 d^3 x^2+225 d^2 x^3-72 d x^4+10 x^5+\frac {60 d^6 \log (x)}{x}\right ) \, dx,x,d+\frac {e}{x^{2/3}}\right )}{120 e^6} \\ & = -\frac {15 b^2 d^4 n^2 \left (d+\frac {e}{x^{2/3}}\right )^2}{8 e^6}+\frac {10 b^2 d^3 n^2 \left (d+\frac {e}{x^{2/3}}\right )^3}{9 e^6}-\frac {15 b^2 d^2 n^2 \left (d+\frac {e}{x^{2/3}}\right )^4}{32 e^6}+\frac {3 b^2 d n^2 \left (d+\frac {e}{x^{2/3}}\right )^5}{25 e^6}-\frac {b^2 n^2 \left (d+\frac {e}{x^{2/3}}\right )^6}{72 e^6}+\frac {3 b^2 d^5 n^2}{e^5 x^{2/3}}-\frac {3 b d^5 n \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^6}+\frac {15 b d^4 n \left (d+\frac {e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{4 e^6}-\frac {10 b d^3 n \left (d+\frac {e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 e^6}+\frac {15 b d^2 n \left (d+\frac {e}{x^{2/3}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{8 e^6}-\frac {3 b d n \left (d+\frac {e}{x^{2/3}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{5 e^6}+\frac {b n \left (d+\frac {e}{x^{2/3}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{12 e^6}+\frac {b d^6 n \log \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{2 e^6}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{4 x^4}-\frac {\left (b^2 d^6 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,d+\frac {e}{x^{2/3}}\right )}{2 e^6} \\ & = -\frac {15 b^2 d^4 n^2 \left (d+\frac {e}{x^{2/3}}\right )^2}{8 e^6}+\frac {10 b^2 d^3 n^2 \left (d+\frac {e}{x^{2/3}}\right )^3}{9 e^6}-\frac {15 b^2 d^2 n^2 \left (d+\frac {e}{x^{2/3}}\right )^4}{32 e^6}+\frac {3 b^2 d n^2 \left (d+\frac {e}{x^{2/3}}\right )^5}{25 e^6}-\frac {b^2 n^2 \left (d+\frac {e}{x^{2/3}}\right )^6}{72 e^6}+\frac {3 b^2 d^5 n^2}{e^5 x^{2/3}}-\frac {b^2 d^6 n^2 \log ^2\left (d+\frac {e}{x^{2/3}}\right )}{4 e^6}-\frac {3 b d^5 n \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{e^6}+\frac {15 b d^4 n \left (d+\frac {e}{x^{2/3}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{4 e^6}-\frac {10 b d^3 n \left (d+\frac {e}{x^{2/3}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{3 e^6}+\frac {15 b d^2 n \left (d+\frac {e}{x^{2/3}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{8 e^6}-\frac {3 b d n \left (d+\frac {e}{x^{2/3}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{5 e^6}+\frac {b n \left (d+\frac {e}{x^{2/3}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{12 e^6}+\frac {b d^6 n \log \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{2 e^6}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{4 x^4} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.54 (sec) , antiderivative size = 988, normalized size of antiderivative = 2.05 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^5} \, dx=-\frac {1800 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2+\frac {b n \left (-600 a e^6+100 b e^6 n+720 a d e^5 x^{2/3}-264 b d e^5 n x^{2/3}-900 a d^2 e^4 x^{4/3}+555 b d^2 e^4 n x^{4/3}+1200 a d^3 e^3 x^2-1140 b d^3 e^3 n x^2-1800 a d^4 e^2 x^{8/3}+2610 b d^4 e^2 n x^{8/3}+3600 a d^5 e x^{10/3}-8820 b d^5 e n x^{10/3}+8820 b d^6 n x^4 \log \left (d+\frac {e}{x^{2/3}}\right )-600 b e^6 \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )+720 b d e^5 x^{2/3} \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )-900 b d^2 e^4 x^{4/3} \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )+1200 b d^3 e^3 x^2 \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )-1800 b d^4 e^2 x^{8/3} \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )+3600 b d^5 e x^{10/3} \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )-3600 a d^6 x^4 \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )-3600 b d^6 x^4 \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right ) \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )+1800 b d^6 n x^4 \log ^2\left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right )-3600 a d^6 x^4 \log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )-3600 b d^6 x^4 \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right ) \log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )+1800 b d^6 n x^4 \log ^2\left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )+3600 b d^6 n x^4 \log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right ) \log \left (\frac {1}{2}-\frac {\sqrt {-d} \sqrt [3]{x}}{2 \sqrt {e}}\right )+3600 b d^6 n x^4 \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right ) \log \left (\frac {1}{2} \left (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )-3600 b d^6 x^4 \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right ) \log \left (-\frac {e}{d x^{2/3}}\right )-7200 b d^6 n x^4 \log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right ) \log \left (-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )-7200 b d^6 n x^4 \log \left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\right ) \log \left (\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )+2400 a d^6 x^4 \log (x)-3600 b d^6 n x^4 \operatorname {PolyLog}\left (2,1+\frac {e}{d x^{2/3}}\right )-7200 b d^6 n x^4 \operatorname {PolyLog}\left (2,1-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )+3600 b d^6 n x^4 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {\sqrt {-d} \sqrt [3]{x}}{2 \sqrt {e}}\right )+3600 b d^6 n x^4 \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )-7200 b d^6 n x^4 \operatorname {PolyLog}\left (2,1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )}{e^6}}{7200 x^4} \]

[In]

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

[Out]

-1/7200*(1800*(a + b*Log[c*(d + e/x^(2/3))^n])^2 + (b*n*(-600*a*e^6 + 100*b*e^6*n + 720*a*d*e^5*x^(2/3) - 264*
b*d*e^5*n*x^(2/3) - 900*a*d^2*e^4*x^(4/3) + 555*b*d^2*e^4*n*x^(4/3) + 1200*a*d^3*e^3*x^2 - 1140*b*d^3*e^3*n*x^
2 - 1800*a*d^4*e^2*x^(8/3) + 2610*b*d^4*e^2*n*x^(8/3) + 3600*a*d^5*e*x^(10/3) - 8820*b*d^5*e*n*x^(10/3) + 8820
*b*d^6*n*x^4*Log[d + e/x^(2/3)] - 600*b*e^6*Log[c*(d + e/x^(2/3))^n] + 720*b*d*e^5*x^(2/3)*Log[c*(d + e/x^(2/3
))^n] - 900*b*d^2*e^4*x^(4/3)*Log[c*(d + e/x^(2/3))^n] + 1200*b*d^3*e^3*x^2*Log[c*(d + e/x^(2/3))^n] - 1800*b*
d^4*e^2*x^(8/3)*Log[c*(d + e/x^(2/3))^n] + 3600*b*d^5*e*x^(10/3)*Log[c*(d + e/x^(2/3))^n] - 3600*a*d^6*x^4*Log
[Sqrt[e] - Sqrt[-d]*x^(1/3)] - 3600*b*d^6*x^4*Log[c*(d + e/x^(2/3))^n]*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)] + 1800*
b*d^6*n*x^4*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)]^2 - 3600*a*d^6*x^4*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)] - 3600*b*d^6*x^
4*Log[c*(d + e/x^(2/3))^n]*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)] + 1800*b*d^6*n*x^4*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)]^
2 + 3600*b*d^6*n*x^4*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)]*Log[1/2 - (Sqrt[-d]*x^(1/3))/(2*Sqrt[e])] + 3600*b*d^6*n*
x^4*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)]*Log[(1 + (Sqrt[-d]*x^(1/3))/Sqrt[e])/2] - 3600*b*d^6*x^4*Log[c*(d + e/x^(2
/3))^n]*Log[-(e/(d*x^(2/3)))] - 7200*b*d^6*n*x^4*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)]*Log[-((Sqrt[-d]*x^(1/3))/Sqrt
[e])] - 7200*b*d^6*n*x^4*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)]*Log[(Sqrt[-d]*x^(1/3))/Sqrt[e]] + 2400*a*d^6*x^4*Log[
x] - 3600*b*d^6*n*x^4*PolyLog[2, 1 + e/(d*x^(2/3))] - 7200*b*d^6*n*x^4*PolyLog[2, 1 - (Sqrt[-d]*x^(1/3))/Sqrt[
e]] + 3600*b*d^6*n*x^4*PolyLog[2, 1/2 - (Sqrt[-d]*x^(1/3))/(2*Sqrt[e])] + 3600*b*d^6*n*x^4*PolyLog[2, (1 + (Sq
rt[-d]*x^(1/3))/Sqrt[e])/2] - 7200*b*d^6*n*x^4*PolyLog[2, 1 + (Sqrt[-d]*x^(1/3))/Sqrt[e]]))/e^6)/x^4

Maple [F]

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

[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 [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 513, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^5} \, dx=-\frac {100 \, b^{2} e^{6} n^{2} + 1800 \, b^{2} e^{6} \log \left (c\right )^{2} - 600 \, a b e^{6} n + 1800 \, a^{2} e^{6} - 60 \, {\left (19 \, b^{2} d^{3} e^{3} n^{2} - 20 \, a b d^{3} e^{3} n\right )} x^{2} - 1800 \, {\left (b^{2} d^{6} n^{2} x^{4} - b^{2} e^{6} n^{2}\right )} \log \left (\frac {d x + e x^{\frac {1}{3}}}{x}\right )^{2} + 600 \, {\left (2 \, b^{2} d^{3} e^{3} n x^{2} - b^{2} e^{6} n + 6 \, a b e^{6}\right )} \log \left (c\right ) + 60 \, {\left (20 \, b^{2} d^{3} e^{3} n^{2} x^{2} - 10 \, b^{2} e^{6} n^{2} + 60 \, a b e^{6} n + 3 \, {\left (49 \, b^{2} d^{6} n^{2} - 20 \, a b d^{6} n\right )} x^{4} - 60 \, {\left (b^{2} d^{6} n x^{4} - b^{2} e^{6} n\right )} \log \left (c\right ) - 6 \, {\left (5 \, b^{2} d^{4} e^{2} n^{2} x^{2} - 2 \, b^{2} d e^{5} n^{2}\right )} x^{\frac {2}{3}} + 15 \, {\left (4 \, b^{2} d^{5} e n^{2} x^{3} - b^{2} d^{2} e^{4} n^{2} x\right )} x^{\frac {1}{3}}\right )} \log \left (\frac {d x + e x^{\frac {1}{3}}}{x}\right ) - 6 \, {\left (44 \, b^{2} d e^{5} n^{2} - 120 \, a b d e^{5} n - 15 \, {\left (29 \, b^{2} d^{4} e^{2} n^{2} - 20 \, a b d^{4} e^{2} n\right )} x^{2} + 60 \, {\left (5 \, b^{2} d^{4} e^{2} n x^{2} - 2 \, b^{2} d e^{5} n\right )} \log \left (c\right )\right )} x^{\frac {2}{3}} - 15 \, {\left (12 \, {\left (49 \, b^{2} d^{5} e n^{2} - 20 \, a b d^{5} e n\right )} x^{3} - {\left (37 \, b^{2} d^{2} e^{4} n^{2} - 60 \, a b d^{2} e^{4} n\right )} x - 60 \, {\left (4 \, b^{2} d^{5} e n x^{3} - b^{2} d^{2} e^{4} n x\right )} \log \left (c\right )\right )} x^{\frac {1}{3}}}{7200 \, e^{6} x^{4}} \]

[In]

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

[Out]

-1/7200*(100*b^2*e^6*n^2 + 1800*b^2*e^6*log(c)^2 - 600*a*b*e^6*n + 1800*a^2*e^6 - 60*(19*b^2*d^3*e^3*n^2 - 20*
a*b*d^3*e^3*n)*x^2 - 1800*(b^2*d^6*n^2*x^4 - b^2*e^6*n^2)*log((d*x + e*x^(1/3))/x)^2 + 600*(2*b^2*d^3*e^3*n*x^
2 - b^2*e^6*n + 6*a*b*e^6)*log(c) + 60*(20*b^2*d^3*e^3*n^2*x^2 - 10*b^2*e^6*n^2 + 60*a*b*e^6*n + 3*(49*b^2*d^6
*n^2 - 20*a*b*d^6*n)*x^4 - 60*(b^2*d^6*n*x^4 - b^2*e^6*n)*log(c) - 6*(5*b^2*d^4*e^2*n^2*x^2 - 2*b^2*d*e^5*n^2)
*x^(2/3) + 15*(4*b^2*d^5*e*n^2*x^3 - b^2*d^2*e^4*n^2*x)*x^(1/3))*log((d*x + e*x^(1/3))/x) - 6*(44*b^2*d*e^5*n^
2 - 120*a*b*d*e^5*n - 15*(29*b^2*d^4*e^2*n^2 - 20*a*b*d^4*e^2*n)*x^2 + 60*(5*b^2*d^4*e^2*n*x^2 - 2*b^2*d*e^5*n
)*log(c))*x^(2/3) - 15*(12*(49*b^2*d^5*e*n^2 - 20*a*b*d^5*e*n)*x^3 - (37*b^2*d^2*e^4*n^2 - 60*a*b*d^2*e^4*n)*x
- 60*(4*b^2*d^5*e*n*x^3 - b^2*d^2*e^4*n*x)*log(c))*x^(1/3))/(e^6*x^4)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 397, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^5} \, dx=\frac {1}{120} \, a b e n {\left (\frac {60 \, d^{6} \log \left (d x^{\frac {2}{3}} + e\right )}{e^{7}} - \frac {60 \, d^{6} \log \left (x^{\frac {2}{3}}\right )}{e^{7}} - \frac {60 \, d^{5} x^{\frac {10}{3}} - 30 \, d^{4} e x^{\frac {8}{3}} + 20 \, d^{3} e^{2} x^{2} - 15 \, d^{2} e^{3} x^{\frac {4}{3}} + 12 \, d e^{4} x^{\frac {2}{3}} - 10 \, e^{5}}{e^{6} x^{4}}\right )} + \frac {1}{7200} \, {\left (60 \, e n {\left (\frac {60 \, d^{6} \log \left (d x^{\frac {2}{3}} + e\right )}{e^{7}} - \frac {60 \, d^{6} \log \left (x^{\frac {2}{3}}\right )}{e^{7}} - \frac {60 \, d^{5} x^{\frac {10}{3}} - 30 \, d^{4} e x^{\frac {8}{3}} + 20 \, d^{3} e^{2} x^{2} - 15 \, d^{2} e^{3} x^{\frac {4}{3}} + 12 \, d e^{4} x^{\frac {2}{3}} - 10 \, e^{5}}{e^{6} x^{4}}\right )} \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right ) - \frac {{\left (1800 \, d^{6} x^{4} \log \left (d x^{\frac {2}{3}} + e\right )^{2} + 800 \, d^{6} x^{4} \log \left (x\right )^{2} - 5880 \, d^{6} x^{4} \log \left (x\right ) - 8820 \, d^{5} e x^{\frac {10}{3}} + 2610 \, d^{4} e^{2} x^{\frac {8}{3}} - 1140 \, d^{3} e^{3} x^{2} + 555 \, d^{2} e^{4} x^{\frac {4}{3}} - 264 \, d e^{5} x^{\frac {2}{3}} + 100 \, e^{6} - 60 \, {\left (40 \, d^{6} x^{4} \log \left (x\right ) - 147 \, d^{6} x^{4}\right )} \log \left (d x^{\frac {2}{3}} + e\right )\right )} n^{2}}{e^{6} x^{4}}\right )} b^{2} - \frac {b^{2} \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right )^{2}}{4 \, x^{4}} - \frac {a b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right )}{2 \, x^{4}} - \frac {a^{2}}{4 \, x^{4}} \]

[In]

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

[Out]

1/120*a*b*e*n*(60*d^6*log(d*x^(2/3) + e)/e^7 - 60*d^6*log(x^(2/3))/e^7 - (60*d^5*x^(10/3) - 30*d^4*e*x^(8/3) +
20*d^3*e^2*x^2 - 15*d^2*e^3*x^(4/3) + 12*d*e^4*x^(2/3) - 10*e^5)/(e^6*x^4)) + 1/7200*(60*e*n*(60*d^6*log(d*x^
(2/3) + e)/e^7 - 60*d^6*log(x^(2/3))/e^7 - (60*d^5*x^(10/3) - 30*d^4*e*x^(8/3) + 20*d^3*e^2*x^2 - 15*d^2*e^3*x
^(4/3) + 12*d*e^4*x^(2/3) - 10*e^5)/(e^6*x^4))*log(c*(d + e/x^(2/3))^n) - (1800*d^6*x^4*log(d*x^(2/3) + e)^2 +
800*d^6*x^4*log(x)^2 - 5880*d^6*x^4*log(x) - 8820*d^5*e*x^(10/3) + 2610*d^4*e^2*x^(8/3) - 1140*d^3*e^3*x^2 +
555*d^2*e^4*x^(4/3) - 264*d*e^5*x^(2/3) + 100*e^6 - 60*(40*d^6*x^4*log(x) - 147*d^6*x^4)*log(d*x^(2/3) + e))*n
^2/(e^6*x^4))*b^2 - 1/4*b^2*log(c*(d + e/x^(2/3))^n)^2/x^4 - 1/2*a*b*log(c*(d + e/x^(2/3))^n)/x^4 - 1/4*a^2/x^
4

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 2.81 (sec) , antiderivative size = 440, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{x^5} \, dx=\frac {b^2\,d^6\,{\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )}^2}{4\,e^6}-\frac {b^2\,{\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )}^2}{4\,x^4}-\frac {b^2\,n^2}{72\,x^4}-\frac {a\,b\,\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )}{2\,x^4}-\frac {a^2}{4\,x^4}+\frac {a\,b\,n}{12\,x^4}+\frac {b^2\,n\,\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )}{12\,x^4}-\frac {49\,b^2\,d^6\,n^2\,\ln \left (d+\frac {e}{x^{2/3}}\right )}{40\,e^6}+\frac {19\,b^2\,d^3\,n^2}{120\,e^3\,x^2}-\frac {37\,b^2\,d^2\,n^2}{480\,e^2\,x^{8/3}}-\frac {29\,b^2\,d^4\,n^2}{80\,e^4\,x^{4/3}}+\frac {49\,b^2\,d^5\,n^2}{40\,e^5\,x^{2/3}}+\frac {11\,b^2\,d\,n^2}{300\,e\,x^{10/3}}-\frac {b^2\,d^3\,n\,\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )}{6\,e^3\,x^2}+\frac {b^2\,d^2\,n\,\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )}{8\,e^2\,x^{8/3}}+\frac {b^2\,d^4\,n\,\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )}{4\,e^4\,x^{4/3}}-\frac {b^2\,d^5\,n\,\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )}{2\,e^5\,x^{2/3}}-\frac {a\,b\,d\,n}{10\,e\,x^{10/3}}+\frac {a\,b\,d^6\,n\,\ln \left (d+\frac {e}{x^{2/3}}\right )}{2\,e^6}-\frac {b^2\,d\,n\,\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )}{10\,e\,x^{10/3}}-\frac {a\,b\,d^3\,n}{6\,e^3\,x^2}+\frac {a\,b\,d^2\,n}{8\,e^2\,x^{8/3}}+\frac {a\,b\,d^4\,n}{4\,e^4\,x^{4/3}}-\frac {a\,b\,d^5\,n}{2\,e^5\,x^{2/3}} \]

[In]

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

[Out]

(b^2*d^6*log(c*(d + e/x^(2/3))^n)^2)/(4*e^6) - (b^2*log(c*(d + e/x^(2/3))^n)^2)/(4*x^4) - (b^2*n^2)/(72*x^4) -
(a*b*log(c*(d + e/x^(2/3))^n))/(2*x^4) - a^2/(4*x^4) + (a*b*n)/(12*x^4) + (b^2*n*log(c*(d + e/x^(2/3))^n))/(1
2*x^4) - (49*b^2*d^6*n^2*log(d + e/x^(2/3)))/(40*e^6) + (19*b^2*d^3*n^2)/(120*e^3*x^2) - (37*b^2*d^2*n^2)/(480
*e^2*x^(8/3)) - (29*b^2*d^4*n^2)/(80*e^4*x^(4/3)) + (49*b^2*d^5*n^2)/(40*e^5*x^(2/3)) + (11*b^2*d*n^2)/(300*e*
x^(10/3)) - (b^2*d^3*n*log(c*(d + e/x^(2/3))^n))/(6*e^3*x^2) + (b^2*d^2*n*log(c*(d + e/x^(2/3))^n))/(8*e^2*x^(
8/3)) + (b^2*d^4*n*log(c*(d + e/x^(2/3))^n))/(4*e^4*x^(4/3)) - (b^2*d^5*n*log(c*(d + e/x^(2/3))^n))/(2*e^5*x^(
2/3)) - (a*b*d*n)/(10*e*x^(10/3)) + (a*b*d^6*n*log(d + e/x^(2/3)))/(2*e^6) - (b^2*d*n*log(c*(d + e/x^(2/3))^n)
)/(10*e*x^(10/3)) - (a*b*d^3*n)/(6*e^3*x^2) + (a*b*d^2*n)/(8*e^2*x^(8/3)) + (a*b*d^4*n)/(4*e^4*x^(4/3)) - (a*b
*d^5*n)/(2*e^5*x^(2/3))