\(\int \frac {(a+b \log (c (d+e x^{2/3})^n))^2}{x^6} \, dx\) [480]

Optimal result

Integrand size = 24, antiderivative size = 640 \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^6} \, dx=-\frac {8 b^2 e^2 n^2}{715 d^2 x^{11/3}}+\frac {64 b^2 e^3 n^2}{2145 d^3 x^3}-\frac {2872 b^2 e^4 n^2}{45045 d^4 x^{7/3}}+\frac {1216 b^2 e^5 n^2}{9009 d^5 x^{5/3}}-\frac {224072 b^2 e^6 n^2}{675675 d^6 x}+\frac {344192 b^2 e^7 n^2}{225225 d^7 \sqrt [3]{x}}+\frac {704552 b^2 e^{15/2} n^2 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{225225 d^{15/2}}-\frac {4 i b^2 e^{15/2} n^2 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{5 d^{15/2}}-\frac {8 b^2 e^{15/2} n^2 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{5 d^{15/2}}-\frac {4 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{65 d x^{13/3}}+\frac {4 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{55 d^2 x^{11/3}}-\frac {4 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{45 d^3 x^3}+\frac {4 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^4 x^{7/3}}-\frac {4 b e^5 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{25 d^5 x^{5/3}}+\frac {4 b e^6 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 d^6 x}-\frac {4 b e^7 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{5 d^7 \sqrt [3]{x}}-\frac {4 b e^{15/2} n \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{5 d^{15/2}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 x^5}-\frac {4 i b^2 e^{15/2} n^2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{5 d^{15/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 640, normalized size of antiderivative = 1.00, number of steps used = 45, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2508, 2507, 2526, 2505, 331, 211, 2520, 12, 5040, 4964, 2449, 2352} \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^6} \, dx=-\frac {4 b e^{15/2} n \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{5 d^{15/2}}-\frac {4 b e^7 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{5 d^7 \sqrt [3]{x}}+\frac {4 b e^6 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 d^6 x}-\frac {4 b e^5 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{25 d^5 x^{5/3}}+\frac {4 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{35 d^4 x^{7/3}}-\frac {4 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{45 d^3 x^3}+\frac {4 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{55 d^2 x^{11/3}}-\frac {4 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{65 d x^{13/3}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 x^5}-\frac {4 i b^2 e^{15/2} n^2 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{5 d^{15/2}}+\frac {704552 b^2 e^{15/2} n^2 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{225225 d^{15/2}}-\frac {8 b^2 e^{15/2} n^2 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{5 d^{15/2}}-\frac {4 i b^2 e^{15/2} n^2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{5 d^{15/2}}+\frac {344192 b^2 e^7 n^2}{225225 d^7 \sqrt [3]{x}}-\frac {224072 b^2 e^6 n^2}{675675 d^6 x}+\frac {1216 b^2 e^5 n^2}{9009 d^5 x^{5/3}}-\frac {2872 b^2 e^4 n^2}{45045 d^4 x^{7/3}}+\frac {64 b^2 e^3 n^2}{2145 d^3 x^3}-\frac {8 b^2 e^2 n^2}{715 d^2 x^{11/3}} \]

[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 12

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

Rule 211

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

Rule 331

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 2352

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

Rule 2449

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 2505

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 2507

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 2508

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 2520

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 2526

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 4964

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x])^p)*(
Log[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 5040

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*((a + b*ArcT
an[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]

Rubi steps \begin{align*} \text {integral}& = 3 \text {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) \text {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) \text {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) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.76 (sec) , antiderivative size = 678, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^6} \, dx=-\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 {2 b e^{13/2} n \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{d^{15/2}}-\frac {2 b e n \operatorname {Hypergeometric2F1}\left (-\frac {11}{2},1,-\frac {9}{2},-\frac {e x^{2/3}}{d}\right )}{143 d^2 x^{11/3}}+\frac {2 b e^2 n \operatorname {Hypergeometric2F1}\left (-\frac {9}{2},1,-\frac {7}{2},-\frac {e x^{2/3}}{d}\right )}{99 d^3 x^3}-\frac {2 b e^3 n \operatorname {Hypergeometric2F1}\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^4 n \operatorname {Hypergeometric2F1}\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^5 n \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-\frac {e x^{2/3}}{d}\right )}{15 d^6 x}+\frac {2 b e^6 n \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-\frac {e x^{2/3}}{d}\right )}{3 d^7 \sqrt [3]{x}}-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{13 d x^{13/3}}+\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 {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^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^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^5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 d^6 x}-\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^{13/2} \arctan \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 {i b e^{13/2} n \left (\arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (\arctan \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 )+\operatorname {PolyLog}\left (2,\frac {i \sqrt {d}+\sqrt {e} \sqrt [3]{x}}{-i \sqrt {d}+\sqrt {e} \sqrt [3]{x}}\right )\right )}{d^{15/2}}\right ) \]

[In]

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

[Out]

-1/5*(a + b*Log[c*(d + e*x^(2/3))^n])^2/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

Maple [F]

\[\int \frac {{\left (a +b \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )^{n}\right )\right )}^{2}}{x^{6}}d x\]

[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)

Fricas [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^6} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a\right )}^{2}}{x^{6}} \,d x } \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^6} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^6} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^6} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a\right )}^{2}}{x^{6}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^6} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )\right )}^2}{x^6} \,d x \]

[In]

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

[Out]

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