3.480 \(\int \frac{(a+b \log (c (d+e x^{2/3})^n))^2}{x^6} \, dx\)

Optimal. Leaf size=640 \[ -\frac{4 i b^2 e^{15/2} n^2 \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} \sqrt [3]{x}}\right )}{5 d^{15/2}}-\frac{4 b e^7 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{5 d^7 \sqrt [3]{x}}+\frac{4 b e^6 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 d^6 x}-\frac{4 b e^5 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{25 d^5 x^{5/3}}+\frac{4 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^4 x^{7/3}}-\frac{4 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{45 d^3 x^3}+\frac{4 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{55 d^2 x^{11/3}}-\frac{4 b e^{15/2} n \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{5 d^{15/2}}-\frac{4 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{65 d x^{13/3}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 x^5}+\frac{1216 b^2 e^5 n^2}{9009 d^5 x^{5/3}}-\frac{2872 b^2 e^4 n^2}{45045 d^4 x^{7/3}}+\frac{64 b^2 e^3 n^2}{2145 d^3 x^3}-\frac{8 b^2 e^2 n^2}{715 d^2 x^{11/3}}+\frac{344192 b^2 e^7 n^2}{225225 d^7 \sqrt [3]{x}}-\frac{224072 b^2 e^6 n^2}{675675 d^6 x}-\frac{4 i b^2 e^{15/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{5 d^{15/2}}+\frac{704552 b^2 e^{15/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{225225 d^{15/2}}-\frac{8 b^2 e^{15/2} n^2 \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} \sqrt [3]{x}}\right ) \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{5 d^{15/2}} \]

[Out]

(-8*b^2*e^2*n^2)/(715*d^2*x^(11/3)) + (64*b^2*e^3*n^2)/(2145*d^3*x^3) - (2872*b^2*e^4*n^2)/(45045*d^4*x^(7/3))
+ (1216*b^2*e^5*n^2)/(9009*d^5*x^(5/3)) - (224072*b^2*e^6*n^2)/(675675*d^6*x) + (344192*b^2*e^7*n^2)/(225225*
d^7*x^(1/3)) + (704552*b^2*e^(15/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]])/(225225*d^(15/2)) - (((4*I)/5)*b^2*
e^(15/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]^2)/d^(15/2) - (8*b^2*e^(15/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt
[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/(5*d^(15/2)) - (4*b*e*n*(a + b*Log[c*(d + e*x^(2/3))^n]))
/(65*d*x^(13/3)) + (4*b*e^2*n*(a + b*Log[c*(d + e*x^(2/3))^n]))/(55*d^2*x^(11/3)) - (4*b*e^3*n*(a + b*Log[c*(d
+ e*x^(2/3))^n]))/(45*d^3*x^3) + (4*b*e^4*n*(a + b*Log[c*(d + e*x^(2/3))^n]))/(35*d^4*x^(7/3)) - (4*b*e^5*n*(
a + b*Log[c*(d + e*x^(2/3))^n]))/(25*d^5*x^(5/3)) + (4*b*e^6*n*(a + b*Log[c*(d + e*x^(2/3))^n]))/(15*d^6*x) -
(4*b*e^7*n*(a + b*Log[c*(d + e*x^(2/3))^n]))/(5*d^7*x^(1/3)) - (4*b*e^(15/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d
]]*(a + b*Log[c*(d + e*x^(2/3))^n]))/(5*d^(15/2)) - (a + b*Log[c*(d + e*x^(2/3))^n])^2/(5*x^5) - (((4*I)/5)*b^
2*e^(15/2)*n^2*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/d^(15/2)

________________________________________________________________________________________

Rubi [A]  time = 0.944001, antiderivative size = 640, normalized size of antiderivative = 1., number of steps used = 45, number of rules used = 12, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2458, 2457, 2476, 2455, 325, 205, 2470, 12, 4920, 4854, 2402, 2315} \[ -\frac{4 i b^2 e^{15/2} n^2 \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} \sqrt [3]{x}}\right )}{5 d^{15/2}}-\frac{4 b e^7 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{5 d^7 \sqrt [3]{x}}+\frac{4 b e^6 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 d^6 x}-\frac{4 b e^5 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{25 d^5 x^{5/3}}+\frac{4 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^4 x^{7/3}}-\frac{4 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{45 d^3 x^3}+\frac{4 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{55 d^2 x^{11/3}}-\frac{4 b e^{15/2} n \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{5 d^{15/2}}-\frac{4 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{65 d x^{13/3}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 x^5}+\frac{1216 b^2 e^5 n^2}{9009 d^5 x^{5/3}}-\frac{2872 b^2 e^4 n^2}{45045 d^4 x^{7/3}}+\frac{64 b^2 e^3 n^2}{2145 d^3 x^3}-\frac{8 b^2 e^2 n^2}{715 d^2 x^{11/3}}+\frac{344192 b^2 e^7 n^2}{225225 d^7 \sqrt [3]{x}}-\frac{224072 b^2 e^6 n^2}{675675 d^6 x}-\frac{4 i b^2 e^{15/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{5 d^{15/2}}+\frac{704552 b^2 e^{15/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{225225 d^{15/2}}-\frac{8 b^2 e^{15/2} n^2 \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} \sqrt [3]{x}}\right ) \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{5 d^{15/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d + e*x^(2/3))^n])^2/x^6,x]

[Out]

(-8*b^2*e^2*n^2)/(715*d^2*x^(11/3)) + (64*b^2*e^3*n^2)/(2145*d^3*x^3) - (2872*b^2*e^4*n^2)/(45045*d^4*x^(7/3))
+ (1216*b^2*e^5*n^2)/(9009*d^5*x^(5/3)) - (224072*b^2*e^6*n^2)/(675675*d^6*x) + (344192*b^2*e^7*n^2)/(225225*
d^7*x^(1/3)) + (704552*b^2*e^(15/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]])/(225225*d^(15/2)) - (((4*I)/5)*b^2*
e^(15/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]^2)/d^(15/2) - (8*b^2*e^(15/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt
[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/(5*d^(15/2)) - (4*b*e*n*(a + b*Log[c*(d + e*x^(2/3))^n]))
/(65*d*x^(13/3)) + (4*b*e^2*n*(a + b*Log[c*(d + e*x^(2/3))^n]))/(55*d^2*x^(11/3)) - (4*b*e^3*n*(a + b*Log[c*(d
+ e*x^(2/3))^n]))/(45*d^3*x^3) + (4*b*e^4*n*(a + b*Log[c*(d + e*x^(2/3))^n]))/(35*d^4*x^(7/3)) - (4*b*e^5*n*(
a + b*Log[c*(d + e*x^(2/3))^n]))/(25*d^5*x^(5/3)) + (4*b*e^6*n*(a + b*Log[c*(d + e*x^(2/3))^n]))/(15*d^6*x) -
(4*b*e^7*n*(a + b*Log[c*(d + e*x^(2/3))^n]))/(5*d^7*x^(1/3)) - (4*b*e^(15/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d
]]*(a + b*Log[c*(d + e*x^(2/3))^n]))/(5*d^(15/2)) - (a + b*Log[c*(d + e*x^(2/3))^n])^2/(5*x^5) - (((4*I)/5)*b^
2*e^(15/2)*n^2*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/d^(15/2)

Rule 2458

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_)*(x_)^(m_.), x_Symbol] :> With[{k = Denomina
tor[n]}, Dist[k, Subst[Int[x^(k*(m + 1) - 1)*(a + b*Log[c*(d + e*x^(k*n))^p])^q, x], x, x^(1/k)], x]] /; FreeQ
[{a, b, c, d, e, m, p, q}, x] && FractionQ[n]

Rule 2457

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_)*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[((f*x
)^(m + 1)*(a + b*Log[c*(d + e*x^n)^p])^q)/(f*(m + 1)), x] - Dist[(b*e*n*p*q)/(f^n*(m + 1)), Int[((f*x)^(m + n)
*(a + b*Log[c*(d + e*x^n)^p])^(q - 1))/(d + e*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && IGtQ[q, 1]
&& IntegerQ[n] && NeQ[m, -1]

Rule 2476

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x^s)^r, x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, n, p, q, r, s}, x] && IGtQ[q, 0] && IntegerQ[m] && IntegerQ[r] && IntegerQ[s]

Rule 2455

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[((f*x)^(m
+ 1)*(a + b*Log[c*(d + e*x^n)^p]))/(f*(m + 1)), x] - Dist[(b*e*n*p)/(f*(m + 1)), Int[(x^(n - 1)*(f*x)^(m + 1))
/(d + e*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 2470

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_) + (g_.)*(x_)^2), x_Symbol] :> With[{u = In
tHide[1/(f + g*x^2), x]}, Simp[u*(a + b*Log[c*(d + e*x^n)^p]), x] - Dist[b*e*n*p, Int[(u*x^(n - 1))/(d + e*x^n
), x], x]] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IntegerQ[n]

Rule 12

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

Rule 4920

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(I*(a + b*ArcTan
[c*x])^(p + 1))/(b*e*(p + 1)), x] - Dist[1/(c*d), Int[(a + b*ArcTan[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b,
c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]

Rule 4854

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTan[c*x])^p*Lo
g[2/(1 + (e*x)/d)])/e, x] + Dist[(b*c*p)/e, Int[((a + b*ArcTan[c*x])^(p - 1)*Log[2/(1 + (e*x)/d)])/(1 + c^2*x^
2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]

Rule 2402

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> -Dist[e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^6} \, dx &=3 \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{x^{16}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 x^5}+\frac{1}{5} (4 b e n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+e x^2\right )^n\right )}{x^{14} \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 x^5}+\frac{1}{5} (4 b e n) \operatorname{Subst}\left (\int \left (\frac{a+b \log \left (c \left (d+e x^2\right )^n\right )}{d x^{14}}-\frac{e \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{d^2 x^{12}}+\frac{e^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{d^3 x^{10}}-\frac{e^3 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{d^4 x^8}+\frac{e^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{d^5 x^6}-\frac{e^5 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{d^6 x^4}+\frac{e^6 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{d^7 x^2}-\frac{e^7 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{d^7 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 x^5}+\frac{(4 b e n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+e x^2\right )^n\right )}{x^{14}} \, dx,x,\sqrt [3]{x}\right )}{5 d}-\frac{\left (4 b e^2 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+e x^2\right )^n\right )}{x^{12}} \, dx,x,\sqrt [3]{x}\right )}{5 d^2}+\frac{\left (4 b e^3 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+e x^2\right )^n\right )}{x^{10}} \, dx,x,\sqrt [3]{x}\right )}{5 d^3}-\frac{\left (4 b e^4 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+e x^2\right )^n\right )}{x^8} \, dx,x,\sqrt [3]{x}\right )}{5 d^4}+\frac{\left (4 b e^5 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+e x^2\right )^n\right )}{x^6} \, dx,x,\sqrt [3]{x}\right )}{5 d^5}-\frac{\left (4 b e^6 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+e x^2\right )^n\right )}{x^4} \, dx,x,\sqrt [3]{x}\right )}{5 d^6}+\frac{\left (4 b e^7 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+e x^2\right )^n\right )}{x^2} \, dx,x,\sqrt [3]{x}\right )}{5 d^7}-\frac{\left (4 b e^8 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+e x^2\right )^n\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{5 d^7}\\ &=-\frac{4 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{65 d x^{13/3}}+\frac{4 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{55 d^2 x^{11/3}}-\frac{4 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{45 d^3 x^3}+\frac{4 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^4 x^{7/3}}-\frac{4 b e^5 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{25 d^5 x^{5/3}}+\frac{4 b e^6 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 d^6 x}-\frac{4 b e^7 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{5 d^7 \sqrt [3]{x}}-\frac{4 b e^{15/2} n \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{5 d^{15/2}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 x^5}+\frac{\left (8 b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^{12} \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{65 d}-\frac{\left (8 b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^{10} \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{55 d^2}+\frac{\left (8 b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^8 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{45 d^3}-\frac{\left (8 b^2 e^5 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^6 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{35 d^4}+\frac{\left (8 b^2 e^6 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^4 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{25 d^5}-\frac{\left (8 b^2 e^7 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{15 d^6}+\frac{\left (8 b^2 e^8 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{5 d^7}+\frac{\left (8 b^2 e^9 n^2\right ) \operatorname{Subst}\left (\int \frac{x \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{e} \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{5 d^7}\\ &=-\frac{8 b^2 e^2 n^2}{715 d^2 x^{11/3}}+\frac{8 b^2 e^3 n^2}{495 d^3 x^3}-\frac{8 b^2 e^4 n^2}{315 d^4 x^{7/3}}+\frac{8 b^2 e^5 n^2}{175 d^5 x^{5/3}}-\frac{8 b^2 e^6 n^2}{75 d^6 x}+\frac{8 b^2 e^7 n^2}{15 d^7 \sqrt [3]{x}}+\frac{8 b^2 e^{15/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{5 d^{15/2}}-\frac{4 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{65 d x^{13/3}}+\frac{4 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{55 d^2 x^{11/3}}-\frac{4 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{45 d^3 x^3}+\frac{4 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^4 x^{7/3}}-\frac{4 b e^5 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{25 d^5 x^{5/3}}+\frac{4 b e^6 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 d^6 x}-\frac{4 b e^7 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{5 d^7 \sqrt [3]{x}}-\frac{4 b e^{15/2} n \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{5 d^{15/2}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 x^5}-\frac{\left (8 b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^{10} \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{65 d^2}+\frac{\left (8 b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^8 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{55 d^3}-\frac{\left (8 b^2 e^5 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^6 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{45 d^4}+\frac{\left (8 b^2 e^6 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^4 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{35 d^5}-\frac{\left (8 b^2 e^7 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{25 d^6}+\frac{\left (8 b^2 e^8 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{15 d^7}+\frac{\left (8 b^2 e^{17/2} n^2\right ) \operatorname{Subst}\left (\int \frac{x \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{5 d^{15/2}}\\ &=-\frac{8 b^2 e^2 n^2}{715 d^2 x^{11/3}}+\frac{64 b^2 e^3 n^2}{2145 d^3 x^3}-\frac{32 b^2 e^4 n^2}{693 d^4 x^{7/3}}+\frac{128 b^2 e^5 n^2}{1575 d^5 x^{5/3}}-\frac{32 b^2 e^6 n^2}{175 d^6 x}+\frac{64 b^2 e^7 n^2}{75 d^7 \sqrt [3]{x}}+\frac{32 b^2 e^{15/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{15 d^{15/2}}-\frac{4 i b^2 e^{15/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{5 d^{15/2}}-\frac{4 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{65 d x^{13/3}}+\frac{4 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{55 d^2 x^{11/3}}-\frac{4 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{45 d^3 x^3}+\frac{4 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^4 x^{7/3}}-\frac{4 b e^5 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{25 d^5 x^{5/3}}+\frac{4 b e^6 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 d^6 x}-\frac{4 b e^7 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{5 d^7 \sqrt [3]{x}}-\frac{4 b e^{15/2} n \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{5 d^{15/2}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 x^5}+\frac{\left (8 b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^8 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{65 d^3}-\frac{\left (8 b^2 e^5 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^6 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{55 d^4}+\frac{\left (8 b^2 e^6 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^4 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{45 d^5}-\frac{\left (8 b^2 e^7 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{35 d^6}-\frac{\left (8 b^2 e^8 n^2\right ) \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{i-\frac{\sqrt{e} x}{\sqrt{d}}} \, dx,x,\sqrt [3]{x}\right )}{5 d^8}+\frac{\left (8 b^2 e^8 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{25 d^7}\\ &=-\frac{8 b^2 e^2 n^2}{715 d^2 x^{11/3}}+\frac{64 b^2 e^3 n^2}{2145 d^3 x^3}-\frac{2872 b^2 e^4 n^2}{45045 d^4 x^{7/3}}+\frac{1912 b^2 e^5 n^2}{17325 d^5 x^{5/3}}-\frac{1144 b^2 e^6 n^2}{4725 d^6 x}+\frac{568 b^2 e^7 n^2}{525 d^7 \sqrt [3]{x}}+\frac{184 b^2 e^{15/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{75 d^{15/2}}-\frac{4 i b^2 e^{15/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{5 d^{15/2}}-\frac{8 b^2 e^{15/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} \sqrt [3]{x}}\right )}{5 d^{15/2}}-\frac{4 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{65 d x^{13/3}}+\frac{4 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{55 d^2 x^{11/3}}-\frac{4 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{45 d^3 x^3}+\frac{4 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^4 x^{7/3}}-\frac{4 b e^5 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{25 d^5 x^{5/3}}+\frac{4 b e^6 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 d^6 x}-\frac{4 b e^7 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{5 d^7 \sqrt [3]{x}}-\frac{4 b e^{15/2} n \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{5 d^{15/2}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 x^5}-\frac{\left (8 b^2 e^5 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^6 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{65 d^4}+\frac{\left (8 b^2 e^6 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^4 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{55 d^5}-\frac{\left (8 b^2 e^7 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{45 d^6}+\frac{\left (8 b^2 e^8 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1+\frac{i \sqrt{e} x}{\sqrt{d}}}\right )}{1+\frac{e x^2}{d}} \, dx,x,\sqrt [3]{x}\right )}{5 d^8}+\frac{\left (8 b^2 e^8 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{35 d^7}\\ &=-\frac{8 b^2 e^2 n^2}{715 d^2 x^{11/3}}+\frac{64 b^2 e^3 n^2}{2145 d^3 x^3}-\frac{2872 b^2 e^4 n^2}{45045 d^4 x^{7/3}}+\frac{1216 b^2 e^5 n^2}{9009 d^5 x^{5/3}}-\frac{15104 b^2 e^6 n^2}{51975 d^6 x}+\frac{1984 b^2 e^7 n^2}{1575 d^7 \sqrt [3]{x}}+\frac{1408 b^2 e^{15/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{525 d^{15/2}}-\frac{4 i b^2 e^{15/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{5 d^{15/2}}-\frac{8 b^2 e^{15/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} \sqrt [3]{x}}\right )}{5 d^{15/2}}-\frac{4 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{65 d x^{13/3}}+\frac{4 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{55 d^2 x^{11/3}}-\frac{4 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{45 d^3 x^3}+\frac{4 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^4 x^{7/3}}-\frac{4 b e^5 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{25 d^5 x^{5/3}}+\frac{4 b e^6 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 d^6 x}-\frac{4 b e^7 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{5 d^7 \sqrt [3]{x}}-\frac{4 b e^{15/2} n \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{5 d^{15/2}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 x^5}+\frac{\left (8 b^2 e^6 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^4 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{65 d^5}-\frac{\left (8 b^2 e^7 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{55 d^6}-\frac{\left (8 i b^2 e^{15/2} n^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+\frac{i \sqrt{e} \sqrt [3]{x}}{\sqrt{d}}}\right )}{5 d^{15/2}}+\frac{\left (8 b^2 e^8 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{45 d^7}\\ &=-\frac{8 b^2 e^2 n^2}{715 d^2 x^{11/3}}+\frac{64 b^2 e^3 n^2}{2145 d^3 x^3}-\frac{2872 b^2 e^4 n^2}{45045 d^4 x^{7/3}}+\frac{1216 b^2 e^5 n^2}{9009 d^5 x^{5/3}}-\frac{224072 b^2 e^6 n^2}{675675 d^6 x}+\frac{24344 b^2 e^7 n^2}{17325 d^7 \sqrt [3]{x}}+\frac{4504 b^2 e^{15/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{1575 d^{15/2}}-\frac{4 i b^2 e^{15/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{5 d^{15/2}}-\frac{8 b^2 e^{15/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} \sqrt [3]{x}}\right )}{5 d^{15/2}}-\frac{4 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{65 d x^{13/3}}+\frac{4 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{55 d^2 x^{11/3}}-\frac{4 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{45 d^3 x^3}+\frac{4 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^4 x^{7/3}}-\frac{4 b e^5 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{25 d^5 x^{5/3}}+\frac{4 b e^6 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 d^6 x}-\frac{4 b e^7 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{5 d^7 \sqrt [3]{x}}-\frac{4 b e^{15/2} n \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{5 d^{15/2}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 x^5}-\frac{4 i b^2 e^{15/2} n^2 \text{Li}_2\left (1-\frac{2}{1+\frac{i \sqrt{e} \sqrt [3]{x}}{\sqrt{d}}}\right )}{5 d^{15/2}}-\frac{\left (8 b^2 e^7 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{65 d^6}+\frac{\left (8 b^2 e^8 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{55 d^7}\\ &=-\frac{8 b^2 e^2 n^2}{715 d^2 x^{11/3}}+\frac{64 b^2 e^3 n^2}{2145 d^3 x^3}-\frac{2872 b^2 e^4 n^2}{45045 d^4 x^{7/3}}+\frac{1216 b^2 e^5 n^2}{9009 d^5 x^{5/3}}-\frac{224072 b^2 e^6 n^2}{675675 d^6 x}+\frac{344192 b^2 e^7 n^2}{225225 d^7 \sqrt [3]{x}}+\frac{52064 b^2 e^{15/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{17325 d^{15/2}}-\frac{4 i b^2 e^{15/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{5 d^{15/2}}-\frac{8 b^2 e^{15/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} \sqrt [3]{x}}\right )}{5 d^{15/2}}-\frac{4 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{65 d x^{13/3}}+\frac{4 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{55 d^2 x^{11/3}}-\frac{4 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{45 d^3 x^3}+\frac{4 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^4 x^{7/3}}-\frac{4 b e^5 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{25 d^5 x^{5/3}}+\frac{4 b e^6 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 d^6 x}-\frac{4 b e^7 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{5 d^7 \sqrt [3]{x}}-\frac{4 b e^{15/2} n \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{5 d^{15/2}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 x^5}-\frac{4 i b^2 e^{15/2} n^2 \text{Li}_2\left (1-\frac{2}{1+\frac{i \sqrt{e} \sqrt [3]{x}}{\sqrt{d}}}\right )}{5 d^{15/2}}+\frac{\left (8 b^2 e^8 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{65 d^7}\\ &=-\frac{8 b^2 e^2 n^2}{715 d^2 x^{11/3}}+\frac{64 b^2 e^3 n^2}{2145 d^3 x^3}-\frac{2872 b^2 e^4 n^2}{45045 d^4 x^{7/3}}+\frac{1216 b^2 e^5 n^2}{9009 d^5 x^{5/3}}-\frac{224072 b^2 e^6 n^2}{675675 d^6 x}+\frac{344192 b^2 e^7 n^2}{225225 d^7 \sqrt [3]{x}}+\frac{704552 b^2 e^{15/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{225225 d^{15/2}}-\frac{4 i b^2 e^{15/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{5 d^{15/2}}-\frac{8 b^2 e^{15/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} \sqrt [3]{x}}\right )}{5 d^{15/2}}-\frac{4 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{65 d x^{13/3}}+\frac{4 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{55 d^2 x^{11/3}}-\frac{4 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{45 d^3 x^3}+\frac{4 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^4 x^{7/3}}-\frac{4 b e^5 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{25 d^5 x^{5/3}}+\frac{4 b e^6 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 d^6 x}-\frac{4 b e^7 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{5 d^7 \sqrt [3]{x}}-\frac{4 b e^{15/2} n \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{5 d^{15/2}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 x^5}-\frac{4 i b^2 e^{15/2} n^2 \text{Li}_2\left (1-\frac{2}{1+\frac{i \sqrt{e} \sqrt [3]{x}}{\sqrt{d}}}\right )}{5 d^{15/2}}\\ \end{align*}

Mathematica [C]  time = 1.03363, size = 678, normalized size = 1.06 \[ -\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 x^5}+\frac{4}{5} b e n \left (-\frac{i b e^{13/2} n \left (\text{PolyLog}\left (2,\frac{\sqrt{e} \sqrt [3]{x}+i \sqrt{d}}{\sqrt{e} \sqrt [3]{x}-i \sqrt{d}}\right )+\tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (\tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )-2 i \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} \sqrt [3]{x}}\right )\right )\right )}{d^{15/2}}-\frac{e^6 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d^7 \sqrt [3]{x}}+\frac{e^5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 d^6 x}-\frac{e^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{5 d^5 x^{5/3}}+\frac{e^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{7 d^4 x^{7/3}}-\frac{e^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{9 d^3 x^3}-\frac{e^{13/2} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d^{15/2}}+\frac{e \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{11 d^2 x^{11/3}}-\frac{a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{13 d x^{13/3}}+\frac{2 b e^6 n \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};-\frac{e x^{2/3}}{d}\right )}{3 d^7 \sqrt [3]{x}}-\frac{2 b e^5 n \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};-\frac{e x^{2/3}}{d}\right )}{15 d^6 x}+\frac{2 b e^4 n \, _2F_1\left (-\frac{5}{2},1;-\frac{3}{2};-\frac{e x^{2/3}}{d}\right )}{35 d^5 x^{5/3}}-\frac{2 b e^3 n \, _2F_1\left (-\frac{7}{2},1;-\frac{5}{2};-\frac{e x^{2/3}}{d}\right )}{63 d^4 x^{7/3}}+\frac{2 b e^2 n \, _2F_1\left (-\frac{9}{2},1;-\frac{7}{2};-\frac{e x^{2/3}}{d}\right )}{99 d^3 x^3}+\frac{2 b e^{13/2} n \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{d^{15/2}}-\frac{2 b e n \, _2F_1\left (-\frac{11}{2},1;-\frac{9}{2};-\frac{e x^{2/3}}{d}\right )}{143 d^2 x^{11/3}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d + e*x^(2/3))^n])^2/x^6,x]

[Out]

-(a + b*Log[c*(d + e*x^(2/3))^n])^2/(5*x^5) + (4*b*e*n*((2*b*e^(13/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]])/d^(
15/2) - (2*b*e*n*Hypergeometric2F1[-11/2, 1, -9/2, -((e*x^(2/3))/d)])/(143*d^2*x^(11/3)) + (2*b*e^2*n*Hypergeo
metric2F1[-9/2, 1, -7/2, -((e*x^(2/3))/d)])/(99*d^3*x^3) - (2*b*e^3*n*Hypergeometric2F1[-7/2, 1, -5/2, -((e*x^
(2/3))/d)])/(63*d^4*x^(7/3)) + (2*b*e^4*n*Hypergeometric2F1[-5/2, 1, -3/2, -((e*x^(2/3))/d)])/(35*d^5*x^(5/3))
- (2*b*e^5*n*Hypergeometric2F1[-3/2, 1, -1/2, -((e*x^(2/3))/d)])/(15*d^6*x) + (2*b*e^6*n*Hypergeometric2F1[-1
/2, 1, 1/2, -((e*x^(2/3))/d)])/(3*d^7*x^(1/3)) - (a + b*Log[c*(d + e*x^(2/3))^n])/(13*d*x^(13/3)) + (e*(a + b*
Log[c*(d + e*x^(2/3))^n]))/(11*d^2*x^(11/3)) - (e^2*(a + b*Log[c*(d + e*x^(2/3))^n]))/(9*d^3*x^3) + (e^3*(a +
b*Log[c*(d + e*x^(2/3))^n]))/(7*d^4*x^(7/3)) - (e^4*(a + b*Log[c*(d + e*x^(2/3))^n]))/(5*d^5*x^(5/3)) + (e^5*(
a + b*Log[c*(d + e*x^(2/3))^n]))/(3*d^6*x) - (e^6*(a + b*Log[c*(d + e*x^(2/3))^n]))/(d^7*x^(1/3)) - (e^(13/2)*
ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*(a + b*Log[c*(d + e*x^(2/3))^n]))/d^(15/2) - (I*b*e^(13/2)*n*(ArcTan[(Sqrt[e
]*x^(1/3))/Sqrt[d]]*(ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]] - (2*I)*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])
+ PolyLog[2, (I*Sqrt[d] + Sqrt[e]*x^(1/3))/((-I)*Sqrt[d] + Sqrt[e]*x^(1/3))]))/d^(15/2)))/5

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Maple [F]  time = 0.351, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{6}} \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^6,x)

[Out]

int((a+b*ln(c*(d+e*x^(2/3))^n))^2/x^6,x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \log \left ({\left (e x^{\frac{2}{3}} + d\right )}^{n} c\right )^{2} + 2 \, a b \log \left ({\left (e x^{\frac{2}{3}} + d\right )}^{n} c\right ) + a^{2}}{x^{6}}, x\right ) \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^6,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^6, x)

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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**6,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 ({\left (e x^{\frac{2}{3}} + d\right )}^{n} c\right ) + a\right )}^{2}}{x^{6}}\,{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^6,x, algorithm="giac")

[Out]

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