\(\int \frac {(a+b \log (c (d+\frac {e}{\sqrt [3]{x}})^n))^2}{x^3} \, dx\) [502]

Optimal result

Integrand size = 24, antiderivative size = 479 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{x^3} \, dx=-\frac {15 b^2 d^4 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}{4 e^6}+\frac {20 b^2 d^3 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3}{9 e^6}-\frac {15 b^2 d^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^4}{16 e^6}+\frac {6 b^2 d n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^5}{25 e^6}-\frac {b^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^6}{36 e^6}+\frac {6 b^2 d^5 n^2}{e^5 \sqrt [3]{x}}-\frac {b^2 d^6 n^2 \log ^2\left (d+\frac {e}{\sqrt [3]{x}}\right )}{2 e^6}-\frac {6 b d^5 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^6}+\frac {15 b d^4 n \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{2 e^6}-\frac {20 b d^3 n \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 e^6}+\frac {15 b d^2 n \left (d+\frac {e}{\sqrt [3]{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{4 e^6}-\frac {6 b d n \left (d+\frac {e}{\sqrt [3]{x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{5 e^6}+\frac {b n \left (d+\frac {e}{\sqrt [3]{x}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{6 e^6}+\frac {b d^6 n \log \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^6}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{2 x^2} \]

[Out]

-15/4*b^2*d^4*n^2*(d+e/x^(1/3))^2/e^6+20/9*b^2*d^3*n^2*(d+e/x^(1/3))^3/e^6-15/16*b^2*d^2*n^2*(d+e/x^(1/3))^4/e
^6+6/25*b^2*d*n^2*(d+e/x^(1/3))^5/e^6-1/36*b^2*n^2*(d+e/x^(1/3))^6/e^6+6*b^2*d^5*n^2/e^5/x^(1/3)-1/2*b^2*d^6*n
^2*ln(d+e/x^(1/3))^2/e^6-6*b*d^5*n*(d+e/x^(1/3))*(a+b*ln(c*(d+e/x^(1/3))^n))/e^6+15/2*b*d^4*n*(d+e/x^(1/3))^2*
(a+b*ln(c*(d+e/x^(1/3))^n))/e^6-20/3*b*d^3*n*(d+e/x^(1/3))^3*(a+b*ln(c*(d+e/x^(1/3))^n))/e^6+15/4*b*d^2*n*(d+e
/x^(1/3))^4*(a+b*ln(c*(d+e/x^(1/3))^n))/e^6-6/5*b*d*n*(d+e/x^(1/3))^5*(a+b*ln(c*(d+e/x^(1/3))^n))/e^6+1/6*b*n*
(d+e/x^(1/3))^6*(a+b*ln(c*(d+e/x^(1/3))^n))/e^6+b*d^6*n*ln(d+e/x^(1/3))*(a+b*ln(c*(d+e/x^(1/3))^n))/e^6-1/2*(a
+b*ln(c*(d+e/x^(1/3))^n))^2/x^2

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 479, 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} \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{x^3} \, dx=\frac {b d^6 n \log \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^6}-\frac {6 b d^5 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^6}+\frac {15 b d^4 n \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{2 e^6}-\frac {20 b d^3 n \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 e^6}+\frac {15 b d^2 n \left (d+\frac {e}{\sqrt [3]{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{4 e^6}-\frac {6 b d n \left (d+\frac {e}{\sqrt [3]{x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{5 e^6}+\frac {b n \left (d+\frac {e}{\sqrt [3]{x}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{6 e^6}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{2 x^2}-\frac {b^2 d^6 n^2 \log ^2\left (d+\frac {e}{\sqrt [3]{x}}\right )}{2 e^6}+\frac {6 b^2 d^5 n^2}{e^5 \sqrt [3]{x}}-\frac {15 b^2 d^4 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}{4 e^6}+\frac {20 b^2 d^3 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3}{9 e^6}-\frac {15 b^2 d^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^4}{16 e^6}+\frac {6 b^2 d n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^5}{25 e^6}-\frac {b^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^6}{36 e^6} \]

[In]

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

[Out]

(-15*b^2*d^4*n^2*(d + e/x^(1/3))^2)/(4*e^6) + (20*b^2*d^3*n^2*(d + e/x^(1/3))^3)/(9*e^6) - (15*b^2*d^2*n^2*(d
+ e/x^(1/3))^4)/(16*e^6) + (6*b^2*d*n^2*(d + e/x^(1/3))^5)/(25*e^6) - (b^2*n^2*(d + e/x^(1/3))^6)/(36*e^6) + (
6*b^2*d^5*n^2)/(e^5*x^(1/3)) - (b^2*d^6*n^2*Log[d + e/x^(1/3)]^2)/(2*e^6) - (6*b*d^5*n*(d + e/x^(1/3))*(a + b*
Log[c*(d + e/x^(1/3))^n]))/e^6 + (15*b*d^4*n*(d + e/x^(1/3))^2*(a + b*Log[c*(d + e/x^(1/3))^n]))/(2*e^6) - (20
*b*d^3*n*(d + e/x^(1/3))^3*(a + b*Log[c*(d + e/x^(1/3))^n]))/(3*e^6) + (15*b*d^2*n*(d + e/x^(1/3))^4*(a + b*Lo
g[c*(d + e/x^(1/3))^n]))/(4*e^6) - (6*b*d*n*(d + e/x^(1/3))^5*(a + b*Log[c*(d + e/x^(1/3))^n]))/(5*e^6) + (b*n
*(d + e/x^(1/3))^6*(a + b*Log[c*(d + e/x^(1/3))^n]))/(6*e^6) + (b*d^6*n*Log[d + e/x^(1/3)]*(a + b*Log[c*(d + e
/x^(1/3))^n]))/e^6 - (a + b*Log[c*(d + e/x^(1/3))^n])^2/(2*x^2)

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 (3 \text {Subst}\left (\int x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\right ) \\ & = -\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{2 x^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}{\sqrt [3]{x}}\right ) \\ & = -\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{2 x^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}{\sqrt [3]{x}}\right ) \\ & = -\frac {6 b d^5 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^6}+\frac {15 b d^4 n \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{2 e^6}-\frac {20 b d^3 n \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 e^6}+\frac {15 b d^2 n \left (d+\frac {e}{\sqrt [3]{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{4 e^6}-\frac {6 b d n \left (d+\frac {e}{\sqrt [3]{x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{5 e^6}+\frac {b n \left (d+\frac {e}{\sqrt [3]{x}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{6 e^6}+\frac {b d^6 n \log \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^6}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{2 x^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}{\sqrt [3]{x}}\right ) \\ & = -\frac {6 b d^5 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^6}+\frac {15 b d^4 n \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{2 e^6}-\frac {20 b d^3 n \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 e^6}+\frac {15 b d^2 n \left (d+\frac {e}{\sqrt [3]{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{4 e^6}-\frac {6 b d n \left (d+\frac {e}{\sqrt [3]{x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{5 e^6}+\frac {b n \left (d+\frac {e}{\sqrt [3]{x}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{6 e^6}+\frac {b d^6 n \log \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^6}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{2 x^2}-\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}{\sqrt [3]{x}}\right )}{60 e^6} \\ & = -\frac {6 b d^5 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^6}+\frac {15 b d^4 n \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{2 e^6}-\frac {20 b d^3 n \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 e^6}+\frac {15 b d^2 n \left (d+\frac {e}{\sqrt [3]{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{4 e^6}-\frac {6 b d n \left (d+\frac {e}{\sqrt [3]{x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{5 e^6}+\frac {b n \left (d+\frac {e}{\sqrt [3]{x}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{6 e^6}+\frac {b d^6 n \log \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^6}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{2 x^2}-\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}{\sqrt [3]{x}}\right )}{60 e^6} \\ & = -\frac {15 b^2 d^4 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}{4 e^6}+\frac {20 b^2 d^3 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3}{9 e^6}-\frac {15 b^2 d^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^4}{16 e^6}+\frac {6 b^2 d n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^5}{25 e^6}-\frac {b^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^6}{36 e^6}+\frac {6 b^2 d^5 n^2}{e^5 \sqrt [3]{x}}-\frac {6 b d^5 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^6}+\frac {15 b d^4 n \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{2 e^6}-\frac {20 b d^3 n \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 e^6}+\frac {15 b d^2 n \left (d+\frac {e}{\sqrt [3]{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{4 e^6}-\frac {6 b d n \left (d+\frac {e}{\sqrt [3]{x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{5 e^6}+\frac {b n \left (d+\frac {e}{\sqrt [3]{x}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{6 e^6}+\frac {b d^6 n \log \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^6}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{2 x^2}-\frac {\left (b^2 d^6 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{e^6} \\ & = -\frac {15 b^2 d^4 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}{4 e^6}+\frac {20 b^2 d^3 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3}{9 e^6}-\frac {15 b^2 d^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^4}{16 e^6}+\frac {6 b^2 d n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^5}{25 e^6}-\frac {b^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^6}{36 e^6}+\frac {6 b^2 d^5 n^2}{e^5 \sqrt [3]{x}}-\frac {b^2 d^6 n^2 \log ^2\left (d+\frac {e}{\sqrt [3]{x}}\right )}{2 e^6}-\frac {6 b d^5 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^6}+\frac {15 b d^4 n \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{2 e^6}-\frac {20 b d^3 n \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 e^6}+\frac {15 b d^2 n \left (d+\frac {e}{\sqrt [3]{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{4 e^6}-\frac {6 b d n \left (d+\frac {e}{\sqrt [3]{x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{5 e^6}+\frac {b n \left (d+\frac {e}{\sqrt [3]{x}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{6 e^6}+\frac {b d^6 n \log \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{e^6}-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{2 x^2} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 0.37 (sec) , antiderivative size = 551, normalized size of antiderivative = 1.15 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{x^3} \, dx=\frac {-1800 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2+\frac {b n \left (600 a e^6-100 b e^6 n-720 a d e^5 \sqrt [3]{x}+264 b d e^5 n \sqrt [3]{x}+900 a d^2 e^4 x^{2/3}-555 b d^2 e^4 n x^{2/3}-1200 a d^3 e^3 x+1140 b d^3 e^3 n x+1800 a d^4 e^2 x^{4/3}-2610 b d^4 e^2 n x^{4/3}-3600 a d^5 e x^{5/3}+8820 b d^5 e n x^{5/3}-8820 b d^6 n x^2 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )+600 b e^6 \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )-720 b d e^5 \sqrt [3]{x} \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )+900 b d^2 e^4 x^{2/3} \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )-1200 b d^3 e^3 x \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )+1800 b d^4 e^2 x^{4/3} \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )-3600 b d^5 e x^{5/3} \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )+3600 a d^6 x^2 \log \left (e+d \sqrt [3]{x}\right )+3600 b d^6 x^2 \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right ) \log \left (e+d \sqrt [3]{x}\right )-1800 b d^6 n x^2 \log ^2\left (e+d \sqrt [3]{x}\right )+3600 b d^6 x^2 \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right ) \log \left (-\frac {e}{d \sqrt [3]{x}}\right )+3600 b d^6 n x^2 \log \left (e+d \sqrt [3]{x}\right ) \log \left (-\frac {d \sqrt [3]{x}}{e}\right )-1200 a d^6 x^2 \log (x)+3600 b d^6 n x^2 \operatorname {PolyLog}\left (2,1+\frac {e}{d \sqrt [3]{x}}\right )+3600 b d^6 n x^2 \operatorname {PolyLog}\left (2,1+\frac {d \sqrt [3]{x}}{e}\right )\right )}{e^6}}{3600 x^2} \]

[In]

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

[Out]

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

Maple [F]

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

[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 [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 597, normalized size of antiderivative = 1.25 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{x^3} \, dx=-\frac {100 \, b^{2} e^{6} n^{2} - 600 \, a b e^{6} n + 1800 \, a^{2} e^{6} - 20 \, {\left (90 \, a^{2} e^{6} - {\left (57 \, b^{2} d^{3} e^{3} - 5 \, b^{2} e^{6}\right )} n^{2} + 30 \, {\left (2 \, a b d^{3} e^{3} - a b e^{6}\right )} n\right )} x^{2} - 1800 \, {\left (b^{2} e^{6} x^{2} - b^{2} e^{6}\right )} \log \left (c\right )^{2} - 1800 \, {\left (b^{2} d^{6} n^{2} x^{2} - b^{2} e^{6} n^{2}\right )} \log \left (\frac {d x + e x^{\frac {2}{3}}}{x}\right )^{2} - 60 \, {\left (19 \, b^{2} d^{3} e^{3} n^{2} - 20 \, a b d^{3} e^{3} n\right )} x + 600 \, {\left (2 \, b^{2} d^{3} e^{3} n x - b^{2} e^{6} n + 6 \, a b e^{6} - {\left (6 \, a b e^{6} + {\left (2 \, b^{2} d^{3} e^{3} - b^{2} e^{6}\right )} n\right )} x^{2}\right )} \log \left (c\right ) + 60 \, {\left (20 \, b^{2} d^{3} e^{3} n^{2} x - 10 \, b^{2} e^{6} n^{2} + 60 \, a b e^{6} n + 3 \, {\left (49 \, b^{2} d^{6} n^{2} - 20 \, a b d^{6} n\right )} x^{2} - 60 \, {\left (b^{2} d^{6} n x^{2} - b^{2} e^{6} n\right )} \log \left (c\right ) + 15 \, {\left (4 \, b^{2} d^{5} e n^{2} x - b^{2} d^{2} e^{4} n^{2}\right )} x^{\frac {2}{3}} - 6 \, {\left (5 \, b^{2} d^{4} e^{2} n^{2} x - 2 \, b^{2} d e^{5} n^{2}\right )} x^{\frac {1}{3}}\right )} \log \left (\frac {d x + e x^{\frac {2}{3}}}{x}\right ) + 15 \, {\left (37 \, b^{2} d^{2} e^{4} n^{2} - 60 \, a b d^{2} e^{4} n - 12 \, {\left (49 \, b^{2} d^{5} e n^{2} - 20 \, a b d^{5} e n\right )} x + 60 \, {\left (4 \, b^{2} d^{5} e n x - b^{2} d^{2} e^{4} n\right )} \log \left (c\right )\right )} x^{\frac {2}{3}} - 6 \, {\left (44 \, b^{2} d e^{5} n^{2} - 120 \, a b d e^{5} n - 15 \, {\left (29 \, b^{2} d^{4} e^{2} n^{2} - 20 \, a b d^{4} e^{2} n\right )} x + 60 \, {\left (5 \, b^{2} d^{4} e^{2} n x - 2 \, b^{2} d e^{5} n\right )} \log \left (c\right )\right )} x^{\frac {1}{3}}}{3600 \, e^{6} x^{2}} \]

[In]

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

[Out]

-1/3600*(100*b^2*e^6*n^2 - 600*a*b*e^6*n + 1800*a^2*e^6 - 20*(90*a^2*e^6 - (57*b^2*d^3*e^3 - 5*b^2*e^6)*n^2 +
30*(2*a*b*d^3*e^3 - a*b*e^6)*n)*x^2 - 1800*(b^2*e^6*x^2 - b^2*e^6)*log(c)^2 - 1800*(b^2*d^6*n^2*x^2 - b^2*e^6*
n^2)*log((d*x + e*x^(2/3))/x)^2 - 60*(19*b^2*d^3*e^3*n^2 - 20*a*b*d^3*e^3*n)*x + 600*(2*b^2*d^3*e^3*n*x - b^2*
e^6*n + 6*a*b*e^6 - (6*a*b*e^6 + (2*b^2*d^3*e^3 - b^2*e^6)*n)*x^2)*log(c) + 60*(20*b^2*d^3*e^3*n^2*x - 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^2 - 60*(b^2*d^6*n*x^2 - b^2*e^6*n)*log(c) + 15*(4
*b^2*d^5*e*n^2*x - b^2*d^2*e^4*n^2)*x^(2/3) - 6*(5*b^2*d^4*e^2*n^2*x - 2*b^2*d*e^5*n^2)*x^(1/3))*log((d*x + e*
x^(2/3))/x) + 15*(37*b^2*d^2*e^4*n^2 - 60*a*b*d^2*e^4*n - 12*(49*b^2*d^5*e*n^2 - 20*a*b*d^5*e*n)*x + 60*(4*b^2
*d^5*e*n*x - b^2*d^2*e^4*n)*log(c))*x^(2/3) - 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 + 60*(5*b^2*d^4*e^2*n*x - 2*b^2*d*e^5*n)*log(c))*x^(1/3))/(e^6*x^2)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 387, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{x^3} \, dx=\frac {1}{60} \, a b e n {\left (\frac {60 \, d^{6} \log \left (d x^{\frac {1}{3}} + e\right )}{e^{7}} - \frac {20 \, d^{6} \log \left (x\right )}{e^{7}} - \frac {60 \, d^{5} x^{\frac {5}{3}} - 30 \, d^{4} e x^{\frac {4}{3}} + 20 \, d^{3} e^{2} x - 15 \, d^{2} e^{3} x^{\frac {2}{3}} + 12 \, d e^{4} x^{\frac {1}{3}} - 10 \, e^{5}}{e^{6} x^{2}}\right )} + \frac {1}{3600} \, {\left (60 \, e n {\left (\frac {60 \, d^{6} \log \left (d x^{\frac {1}{3}} + e\right )}{e^{7}} - \frac {20 \, d^{6} \log \left (x\right )}{e^{7}} - \frac {60 \, d^{5} x^{\frac {5}{3}} - 30 \, d^{4} e x^{\frac {4}{3}} + 20 \, d^{3} e^{2} x - 15 \, d^{2} e^{3} x^{\frac {2}{3}} + 12 \, d e^{4} x^{\frac {1}{3}} - 10 \, e^{5}}{e^{6} x^{2}}\right )} \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right ) - \frac {{\left (1800 \, d^{6} x^{2} \log \left (d x^{\frac {1}{3}} + e\right )^{2} + 200 \, d^{6} x^{2} \log \left (x\right )^{2} - 2940 \, d^{6} x^{2} \log \left (x\right ) - 8820 \, d^{5} e x^{\frac {5}{3}} + 2610 \, d^{4} e^{2} x^{\frac {4}{3}} - 1140 \, d^{3} e^{3} x + 555 \, d^{2} e^{4} x^{\frac {2}{3}} - 264 \, d e^{5} x^{\frac {1}{3}} + 100 \, e^{6} - 60 \, {\left (20 \, d^{6} x^{2} \log \left (x\right ) - 147 \, d^{6} x^{2}\right )} \log \left (d x^{\frac {1}{3}} + e\right )\right )} n^{2}}{e^{6} x^{2}}\right )} b^{2} - \frac {b^{2} \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right )^{2}}{2 \, x^{2}} - \frac {a b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right )}{x^{2}} - \frac {a^{2}}{2 \, x^{2}} \]

[In]

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

[Out]

1/60*a*b*e*n*(60*d^6*log(d*x^(1/3) + e)/e^7 - 20*d^6*log(x)/e^7 - (60*d^5*x^(5/3) - 30*d^4*e*x^(4/3) + 20*d^3*
e^2*x - 15*d^2*e^3*x^(2/3) + 12*d*e^4*x^(1/3) - 10*e^5)/(e^6*x^2)) + 1/3600*(60*e*n*(60*d^6*log(d*x^(1/3) + e)
/e^7 - 20*d^6*log(x)/e^7 - (60*d^5*x^(5/3) - 30*d^4*e*x^(4/3) + 20*d^3*e^2*x - 15*d^2*e^3*x^(2/3) + 12*d*e^4*x
^(1/3) - 10*e^5)/(e^6*x^2))*log(c*(d + e/x^(1/3))^n) - (1800*d^6*x^2*log(d*x^(1/3) + e)^2 + 200*d^6*x^2*log(x)
^2 - 2940*d^6*x^2*log(x) - 8820*d^5*e*x^(5/3) + 2610*d^4*e^2*x^(4/3) - 1140*d^3*e^3*x + 555*d^2*e^4*x^(2/3) -
264*d*e^5*x^(1/3) + 100*e^6 - 60*(20*d^6*x^2*log(x) - 147*d^6*x^2)*log(d*x^(1/3) + e))*n^2/(e^6*x^2))*b^2 - 1/
2*b^2*log(c*(d + e/x^(1/3))^n)^2/x^2 - a*b*log(c*(d + e/x^(1/3))^n)/x^2 - 1/2*a^2/x^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 877 vs. \(2 (411) = 822\).

Time = 0.37 (sec) , antiderivative size = 877, normalized size of antiderivative = 1.83 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{x^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/3600*(1800*(6*(d*x^(1/3) + e)*b^2*d^5*n^2/(e^5*x^(1/3)) - 15*(d*x^(1/3) + e)^2*b^2*d^4*n^2/(e^5*x^(2/3)) + 2
0*(d*x^(1/3) + e)^3*b^2*d^3*n^2/(e^5*x) - 15*(d*x^(1/3) + e)^4*b^2*d^2*n^2/(e^5*x^(4/3)) + 6*(d*x^(1/3) + e)^5
*b^2*d*n^2/(e^5*x^(5/3)) - (d*x^(1/3) + e)^6*b^2*n^2/(e^5*x^2))*log((d*x^(1/3) + e)/x^(1/3))^2 + 60*(10*(b^2*n
^2 - 6*b^2*n*log(c) - 6*a*b*n)*(d*x^(1/3) + e)^6/(e^5*x^2) - 72*(b^2*d*n^2 - 5*b^2*d*n*log(c) - 5*a*b*d*n)*(d*
x^(1/3) + e)^5/(e^5*x^(5/3)) + 225*(b^2*d^2*n^2 - 4*b^2*d^2*n*log(c) - 4*a*b*d^2*n)*(d*x^(1/3) + e)^4/(e^5*x^(
4/3)) - 400*(b^2*d^3*n^2 - 3*b^2*d^3*n*log(c) - 3*a*b*d^3*n)*(d*x^(1/3) + e)^3/(e^5*x) + 450*(b^2*d^4*n^2 - 2*
b^2*d^4*n*log(c) - 2*a*b*d^4*n)*(d*x^(1/3) + e)^2/(e^5*x^(2/3)) - 360*(b^2*d^5*n^2 - b^2*d^5*n*log(c) - a*b*d^
5*n)*(d*x^(1/3) + e)/(e^5*x^(1/3)))*log((d*x^(1/3) + e)/x^(1/3)) - 100*(b^2*n^2 - 6*b^2*n*log(c) + 18*b^2*log(
c)^2 - 6*a*b*n + 36*a*b*log(c) + 18*a^2)*(d*x^(1/3) + e)^6/(e^5*x^2) + 432*(2*b^2*d*n^2 - 10*b^2*d*n*log(c) +
25*b^2*d*log(c)^2 - 10*a*b*d*n + 50*a*b*d*log(c) + 25*a^2*d)*(d*x^(1/3) + e)^5/(e^5*x^(5/3)) - 3375*(b^2*d^2*n
^2 - 4*b^2*d^2*n*log(c) + 8*b^2*d^2*log(c)^2 - 4*a*b*d^2*n + 16*a*b*d^2*log(c) + 8*a^2*d^2)*(d*x^(1/3) + e)^4/
(e^5*x^(4/3)) + 4000*(2*b^2*d^3*n^2 - 6*b^2*d^3*n*log(c) + 9*b^2*d^3*log(c)^2 - 6*a*b*d^3*n + 18*a*b*d^3*log(c
) + 9*a^2*d^3)*(d*x^(1/3) + e)^3/(e^5*x) - 13500*(b^2*d^4*n^2 - 2*b^2*d^4*n*log(c) + 2*b^2*d^4*log(c)^2 - 2*a*
b*d^4*n + 4*a*b*d^4*log(c) + 2*a^2*d^4)*(d*x^(1/3) + e)^2/(e^5*x^(2/3)) + 10800*(2*b^2*d^5*n^2 - 2*b^2*d^5*n*l
og(c) + b^2*d^5*log(c)^2 - 2*a*b*d^5*n + 2*a*b*d^5*log(c) + a^2*d^5)*(d*x^(1/3) + e)/(e^5*x^(1/3)))/e

Mupad [B] (verification not implemented)

Time = 3.00 (sec) , antiderivative size = 439, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{x^3} \, dx=\frac {b^2\,d^6\,{\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}^2}{2\,e^6}-\frac {b^2\,{\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}^2}{2\,x^2}-\frac {b^2\,n^2}{36\,x^2}-\frac {a\,b\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}{x^2}-\frac {a^2}{2\,x^2}+\frac {a\,b\,n}{6\,x^2}+\frac {b^2\,n\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}{6\,x^2}-\frac {49\,b^2\,d^6\,n^2\,\ln \left (d+\frac {e}{x^{1/3}}\right )}{20\,e^6}+\frac {19\,b^2\,d^3\,n^2}{60\,e^3\,x}-\frac {37\,b^2\,d^2\,n^2}{240\,e^2\,x^{4/3}}-\frac {29\,b^2\,d^4\,n^2}{40\,e^4\,x^{2/3}}+\frac {49\,b^2\,d^5\,n^2}{20\,e^5\,x^{1/3}}+\frac {11\,b^2\,d\,n^2}{150\,e\,x^{5/3}}-\frac {b^2\,d^3\,n\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}{3\,e^3\,x}+\frac {b^2\,d^2\,n\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}{4\,e^2\,x^{4/3}}+\frac {b^2\,d^4\,n\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}{2\,e^4\,x^{2/3}}-\frac {b^2\,d^5\,n\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}{e^5\,x^{1/3}}-\frac {a\,b\,d\,n}{5\,e\,x^{5/3}}+\frac {a\,b\,d^6\,n\,\ln \left (d+\frac {e}{x^{1/3}}\right )}{e^6}-\frac {b^2\,d\,n\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}{5\,e\,x^{5/3}}-\frac {a\,b\,d^3\,n}{3\,e^3\,x}+\frac {a\,b\,d^2\,n}{4\,e^2\,x^{4/3}}+\frac {a\,b\,d^4\,n}{2\,e^4\,x^{2/3}}-\frac {a\,b\,d^5\,n}{e^5\,x^{1/3}} \]

[In]

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

[Out]

(b^2*d^6*log(c*(d + e/x^(1/3))^n)^2)/(2*e^6) - (b^2*log(c*(d + e/x^(1/3))^n)^2)/(2*x^2) - (b^2*n^2)/(36*x^2) -
(a*b*log(c*(d + e/x^(1/3))^n))/x^2 - a^2/(2*x^2) + (a*b*n)/(6*x^2) + (b^2*n*log(c*(d + e/x^(1/3))^n))/(6*x^2)
- (49*b^2*d^6*n^2*log(d + e/x^(1/3)))/(20*e^6) + (19*b^2*d^3*n^2)/(60*e^3*x) - (37*b^2*d^2*n^2)/(240*e^2*x^(4
/3)) - (29*b^2*d^4*n^2)/(40*e^4*x^(2/3)) + (49*b^2*d^5*n^2)/(20*e^5*x^(1/3)) + (11*b^2*d*n^2)/(150*e*x^(5/3))
- (b^2*d^3*n*log(c*(d + e/x^(1/3))^n))/(3*e^3*x) + (b^2*d^2*n*log(c*(d + e/x^(1/3))^n))/(4*e^2*x^(4/3)) + (b^2
*d^4*n*log(c*(d + e/x^(1/3))^n))/(2*e^4*x^(2/3)) - (b^2*d^5*n*log(c*(d + e/x^(1/3))^n))/(e^5*x^(1/3)) - (a*b*d
*n)/(5*e*x^(5/3)) + (a*b*d^6*n*log(d + e/x^(1/3)))/e^6 - (b^2*d*n*log(c*(d + e/x^(1/3))^n))/(5*e*x^(5/3)) - (a
*b*d^3*n)/(3*e^3*x) + (a*b*d^2*n)/(4*e^2*x^(4/3)) + (a*b*d^4*n)/(2*e^4*x^(2/3)) - (a*b*d^5*n)/(e^5*x^(1/3))