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

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

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.476294, antiderivative size = 355, normalized size of antiderivative = 0.74, number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2454, 2398, 2411, 43, 2334, 12, 14, 2301} \[ -\frac{1}{120} b n \left (\frac{360 d^5 \left (d+\frac{e}{x^{2/3}}\right )}{e^6}-\frac{450 d^4 \left (d+\frac{e}{x^{2/3}}\right )^2}{e^6}+\frac{400 d^3 \left (d+\frac{e}{x^{2/3}}\right )^3}{e^6}-\frac{225 d^2 \left (d+\frac{e}{x^{2/3}}\right )^4}{e^6}-\frac{60 d^6 \log \left (d+\frac{e}{x^{2/3}}\right )}{e^6}+\frac{72 d \left (d+\frac{e}{x^{2/3}}\right )^5}{e^6}-\frac{10 \left (d+\frac{e}{x^{2/3}}\right )^6}{e^6}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{4 x^4}+\frac{3 b^2 d^5 n^2}{e^5 x^{2/3}}-\frac{15 b^2 d^4 n^2 \left (d+\frac{e}{x^{2/3}}\right )^2}{8 e^6}+\frac{10 b^2 d^3 n^2 \left (d+\frac{e}{x^{2/3}}\right )^3}{9 e^6}-\frac{15 b^2 d^2 n^2 \left (d+\frac{e}{x^{2/3}}\right )^4}{32 e^6}-\frac{b^2 d^6 n^2 \log ^2\left (d+\frac{e}{x^{2/3}}\right )}{4 e^6}+\frac{3 b^2 d n^2 \left (d+\frac{e}{x^{2/3}}\right )^5}{25 e^6}-\frac{b^2 n^2 \left (d+\frac{e}{x^{2/3}}\right )^6}{72 e^6} \]

Antiderivative was successfully verified.

[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) - (b*n*((360*d^5*(d + e/x^(2/3)))/e^
6 - (450*d^4*(d + e/x^(2/3))^2)/e^6 + (400*d^3*(d + e/x^(2/3))^3)/e^6 - (225*d^2*(d + e/x^(2/3))^4)/e^6 + (72*
d*(d + e/x^(2/3))^5)/e^6 - (10*(d + e/x^(2/3))^6)/e^6 - (60*d^6*Log[d + e/x^(2/3)])/e^6)*(a + b*Log[c*(d + e/x
^(2/3))^n]))/120 - (a + b*Log[c*(d + e/x^(2/3))^n])^2/(4*x^4)

Rule 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rule 2398

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 2411

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 43

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 2334

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]}, Simp[u*(a + b*Log[c*x^n]), 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 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 2301

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]

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{x^5} \, dx &=-\left (\frac{3}{2} \operatorname{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) \operatorname{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) \operatorname{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{1}{120} b n \left (\frac{360 d^5 \left (d+\frac{e}{x^{2/3}}\right )}{e^6}-\frac{450 d^4 \left (d+\frac{e}{x^{2/3}}\right )^2}{e^6}+\frac{400 d^3 \left (d+\frac{e}{x^{2/3}}\right )^3}{e^6}-\frac{225 d^2 \left (d+\frac{e}{x^{2/3}}\right )^4}{e^6}+\frac{72 d \left (d+\frac{e}{x^{2/3}}\right )^5}{e^6}-\frac{10 \left (d+\frac{e}{x^{2/3}}\right )^6}{e^6}-\frac{60 d^6 \log \left (d+\frac{e}{x^{2/3}}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\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} \left (b^2 n^2\right ) \operatorname{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{1}{120} b n \left (\frac{360 d^5 \left (d+\frac{e}{x^{2/3}}\right )}{e^6}-\frac{450 d^4 \left (d+\frac{e}{x^{2/3}}\right )^2}{e^6}+\frac{400 d^3 \left (d+\frac{e}{x^{2/3}}\right )^3}{e^6}-\frac{225 d^2 \left (d+\frac{e}{x^{2/3}}\right )^4}{e^6}+\frac{72 d \left (d+\frac{e}{x^{2/3}}\right )^5}{e^6}-\frac{10 \left (d+\frac{e}{x^{2/3}}\right )^6}{e^6}-\frac{60 d^6 \log \left (d+\frac{e}{x^{2/3}}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\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{\left (b^2 n^2\right ) \operatorname{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{1}{120} b n \left (\frac{360 d^5 \left (d+\frac{e}{x^{2/3}}\right )}{e^6}-\frac{450 d^4 \left (d+\frac{e}{x^{2/3}}\right )^2}{e^6}+\frac{400 d^3 \left (d+\frac{e}{x^{2/3}}\right )^3}{e^6}-\frac{225 d^2 \left (d+\frac{e}{x^{2/3}}\right )^4}{e^6}+\frac{72 d \left (d+\frac{e}{x^{2/3}}\right )^5}{e^6}-\frac{10 \left (d+\frac{e}{x^{2/3}}\right )^6}{e^6}-\frac{60 d^6 \log \left (d+\frac{e}{x^{2/3}}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\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{\left (b^2 n^2\right ) \operatorname{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{1}{120} b n \left (\frac{360 d^5 \left (d+\frac{e}{x^{2/3}}\right )}{e^6}-\frac{450 d^4 \left (d+\frac{e}{x^{2/3}}\right )^2}{e^6}+\frac{400 d^3 \left (d+\frac{e}{x^{2/3}}\right )^3}{e^6}-\frac{225 d^2 \left (d+\frac{e}{x^{2/3}}\right )^4}{e^6}+\frac{72 d \left (d+\frac{e}{x^{2/3}}\right )^5}{e^6}-\frac{10 \left (d+\frac{e}{x^{2/3}}\right )^6}{e^6}-\frac{60 d^6 \log \left (d+\frac{e}{x^{2/3}}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\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{\left (b^2 d^6 n^2\right ) \operatorname{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{1}{120} b n \left (\frac{360 d^5 \left (d+\frac{e}{x^{2/3}}\right )}{e^6}-\frac{450 d^4 \left (d+\frac{e}{x^{2/3}}\right )^2}{e^6}+\frac{400 d^3 \left (d+\frac{e}{x^{2/3}}\right )^3}{e^6}-\frac{225 d^2 \left (d+\frac{e}{x^{2/3}}\right )^4}{e^6}+\frac{72 d \left (d+\frac{e}{x^{2/3}}\right )^5}{e^6}-\frac{10 \left (d+\frac{e}{x^{2/3}}\right )^6}{e^6}-\frac{60 d^6 \log \left (d+\frac{e}{x^{2/3}}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )-\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]  time = 0.837764, size = 1021, normalized size = 2.12 \[ \frac{\frac{b n \left (-1800 b n x^4 \log ^2\left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right ) d^6-1800 b n x^4 \log ^2\left (\sqrt [3]{x} \sqrt{-d}+\sqrt{e}\right ) d^6-5220 b n x^4 \log \left (d+\frac{e}{x^{2/3}}\right ) d^6-3600 b x^4 \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right ) d^6+3600 a x^4 \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right ) d^6+3600 b 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 ) d^6+3600 a x^4 \log \left (\sqrt [3]{x} \sqrt{-d}+\sqrt{e}\right ) d^6+3600 b x^4 \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right ) \log \left (\sqrt [3]{x} \sqrt{-d}+\sqrt{e}\right ) d^6-3600 b n x^4 \log \left (\sqrt [3]{x} \sqrt{-d}+\sqrt{e}\right ) \log \left (\frac{1}{2}-\frac{\sqrt{-d} \sqrt [3]{x}}{2 \sqrt{e}}\right ) d^6-3600 b n x^4 \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right ) \log \left (\frac{1}{2} \left (\frac{\sqrt [3]{x} \sqrt{-d}}{\sqrt{e}}+1\right )\right ) d^6+3600 a x^4 \log \left (-\frac{e}{d x^{2/3}}\right ) d^6+3600 b x^4 \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right ) \log \left (-\frac{e}{d x^{2/3}}\right ) d^6+7200 b n x^4 \log \left (\sqrt [3]{x} \sqrt{-d}+\sqrt{e}\right ) \log \left (-\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right ) d^6+7200 b n x^4 \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right ) \log \left (\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right ) d^6+3600 b n x^4 \text{PolyLog}\left (2,\frac{e}{d x^{2/3}}+1\right ) d^6+7200 b n x^4 \text{PolyLog}\left (2,1-\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right ) d^6-3600 b n x^4 \text{PolyLog}\left (2,\frac{1}{2}-\frac{\sqrt{-d} \sqrt [3]{x}}{2 \sqrt{e}}\right ) d^6-3600 b n x^4 \text{PolyLog}\left (2,\frac{1}{2} \left (\frac{\sqrt [3]{x} \sqrt{-d}}{\sqrt{e}}+1\right )\right ) d^6+7200 b n x^4 \text{PolyLog}\left (2,\frac{\sqrt [3]{x} \sqrt{-d}}{\sqrt{e}}+1\right ) d^6-3600 a e x^{10/3} d^5+8820 b e n x^{10/3} d^5-3600 b e x^{10/3} \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right ) d^5+1800 a e^2 x^{8/3} d^4-2610 b e^2 n x^{8/3} d^4+1800 b e^2 x^{8/3} \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right ) d^4-1200 a e^3 x^2 d^3+1140 b e^3 n x^2 d^3-1200 b e^3 x^2 \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right ) d^3+900 a e^4 x^{4/3} d^2-555 b e^4 n x^{4/3} d^2+900 b e^4 x^{4/3} \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right ) d^2-720 b e^5 x^{2/3} \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right ) d-720 a e^5 x^{2/3} d+264 b e^5 n x^{2/3} d+600 a e^6-100 b e^6 n+600 b e^6 \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e^6}-1800 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{7200 x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-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) - 5220*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*b*d^6*x^4*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[S
qrt[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*a*d^6*x^4*Log[-(e/(d*x^(2/3)))] + 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]] + 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 + (Sqrt[-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)/(7200*x^4)

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Maple [F]  time = 0.352, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{5}} \left ( a+b\ln \left ( c \left ( d+{e{x}^{-{\frac{2}{3}}}} \right ) ^{n} \right ) \right ) ^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [A]  time = 1.09712, size = 536, normalized size = 1.11 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.87401, size = 1137, normalized size = 2.36 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)