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

Optimal. Leaf size=490 \[ -\frac{4 i b^2 e^{9/2} n^2 \text{PolyLog}\left (2,-1+\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\right )}{3 d^{9/2}}+\frac{4 b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{9 d^3}-\frac{4 b e^2 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{15 d^2}+\frac{4 b e^{9/2} n \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{3 d^{9/2}}+\frac{4 b e n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{21 d}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2-\frac{4 a b e^4 n \sqrt [3]{x}}{3 d^4}-\frac{4 b^2 e^4 n \sqrt [3]{x} \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{3 d^4}+\frac{8 b^2 e^2 n^2 x^{5/3}}{105 d^2}+\frac{568 b^2 e^4 n^2 \sqrt [3]{x}}{315 d^4}-\frac{32 b^2 e^3 n^2 x}{105 d^3}-\frac{4 i b^2 e^{9/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )^2}{3 d^{9/2}}-\frac{1408 b^2 e^{9/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{315 d^{9/2}}+\frac{8 b^2 e^{9/2} n^2 \log \left (2-\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\right ) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{3 d^{9/2}} \]

[Out]

(-4*a*b*e^4*n*x^(1/3))/(3*d^4) + (568*b^2*e^4*n^2*x^(1/3))/(315*d^4) - (32*b^2*e^3*n^2*x)/(105*d^3) + (8*b^2*e
^2*n^2*x^(5/3))/(105*d^2) - (1408*b^2*e^(9/2)*n^2*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]])/(315*d^(9/2)) - (((4*I)/3
)*b^2*e^(9/2)*n^2*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]^2)/d^(9/2) + (8*b^2*e^(9/2)*n^2*ArcTan[(Sqrt[d]*x^(1/3))/S
qrt[e]]*Log[2 - (2*Sqrt[e])/(Sqrt[e] - I*Sqrt[d]*x^(1/3))])/(3*d^(9/2)) - (4*b^2*e^4*n*x^(1/3)*Log[c*(d + e/x^
(2/3))^n])/(3*d^4) + (4*b*e^3*n*x*(a + b*Log[c*(d + e/x^(2/3))^n]))/(9*d^3) - (4*b*e^2*n*x^(5/3)*(a + b*Log[c*
(d + e/x^(2/3))^n]))/(15*d^2) + (4*b*e*n*x^(7/3)*(a + b*Log[c*(d + e/x^(2/3))^n]))/(21*d) + (4*b*e^(9/2)*n*Arc
Tan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*(a + b*Log[c*(d + e/x^(2/3))^n]))/(3*d^(9/2)) + (x^3*(a + b*Log[c*(d + e/x^(2/3
))^n])^2)/3 - (((4*I)/3)*b^2*e^(9/2)*n^2*PolyLog[2, -1 + (2*Sqrt[e])/(Sqrt[e] - I*Sqrt[d]*x^(1/3))])/d^(9/2)

________________________________________________________________________________________

Rubi [A]  time = 0.806912, antiderivative size = 490, normalized size of antiderivative = 1., number of steps used = 28, number of rules used = 17, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.708, Rules used = {2458, 2457, 2476, 2448, 263, 205, 2455, 193, 321, 302, 2470, 12, 260, 6688, 4924, 4868, 2447} \[ -\frac{4 i b^2 e^{9/2} n^2 \text{PolyLog}\left (2,-1+\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\right )}{3 d^{9/2}}+\frac{4 b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{9 d^3}-\frac{4 b e^2 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{15 d^2}+\frac{4 b e^{9/2} n \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{3 d^{9/2}}+\frac{4 b e n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{21 d}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2-\frac{4 a b e^4 n \sqrt [3]{x}}{3 d^4}-\frac{4 b^2 e^4 n \sqrt [3]{x} \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{3 d^4}+\frac{8 b^2 e^2 n^2 x^{5/3}}{105 d^2}+\frac{568 b^2 e^4 n^2 \sqrt [3]{x}}{315 d^4}-\frac{32 b^2 e^3 n^2 x}{105 d^3}-\frac{4 i b^2 e^{9/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )^2}{3 d^{9/2}}-\frac{1408 b^2 e^{9/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{315 d^{9/2}}+\frac{8 b^2 e^{9/2} n^2 \log \left (2-\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\right ) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{3 d^{9/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*a*b*e^4*n*x^(1/3))/(3*d^4) + (568*b^2*e^4*n^2*x^(1/3))/(315*d^4) - (32*b^2*e^3*n^2*x)/(105*d^3) + (8*b^2*e
^2*n^2*x^(5/3))/(105*d^2) - (1408*b^2*e^(9/2)*n^2*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]])/(315*d^(9/2)) - (((4*I)/3
)*b^2*e^(9/2)*n^2*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]^2)/d^(9/2) + (8*b^2*e^(9/2)*n^2*ArcTan[(Sqrt[d]*x^(1/3))/S
qrt[e]]*Log[2 - (2*Sqrt[e])/(Sqrt[e] - I*Sqrt[d]*x^(1/3))])/(3*d^(9/2)) - (4*b^2*e^4*n*x^(1/3)*Log[c*(d + e/x^
(2/3))^n])/(3*d^4) + (4*b*e^3*n*x*(a + b*Log[c*(d + e/x^(2/3))^n]))/(9*d^3) - (4*b*e^2*n*x^(5/3)*(a + b*Log[c*
(d + e/x^(2/3))^n]))/(15*d^2) + (4*b*e*n*x^(7/3)*(a + b*Log[c*(d + e/x^(2/3))^n]))/(21*d) + (4*b*e^(9/2)*n*Arc
Tan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*(a + b*Log[c*(d + e/x^(2/3))^n]))/(3*d^(9/2)) + (x^3*(a + b*Log[c*(d + e/x^(2/3
))^n])^2)/3 - (((4*I)/3)*b^2*e^(9/2)*n^2*PolyLog[2, -1 + (2*Sqrt[e])/(Sqrt[e] - I*Sqrt[d]*x^(1/3))])/d^(9/2)

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 2448

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

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 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 193

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b}, x] && LtQ[n, 0]
&& IntegerQ[p]

Rule 321

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

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

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 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 4924

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

Rule 4868

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Simp[((a + b*ArcTan[c*x]
)^p*Log[2 - 2/(1 + (e*x)/d)])/d, x] - Dist[(b*c*p)/d, Int[((a + b*ArcTan[c*x])^(p - 1)*Log[2 - 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 2447

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[(Pq^m*(1 - u))/D[u, x]]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rubi steps

\begin{align*} \int x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2 \, dx &=3 \operatorname{Subst}\left (\int x^8 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2+\frac{1}{3} (4 b e n) \operatorname{Subst}\left (\int \frac{x^6 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{d+\frac{e}{x^2}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2+\frac{1}{3} (4 b e n) \operatorname{Subst}\left (\int \left (-\frac{e^3 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{d^4}+\frac{e^2 x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{d^3}-\frac{e x^4 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{d^2}+\frac{x^6 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{d}+\frac{e^4 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{d^4 \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2+\frac{(4 b e n) \operatorname{Subst}\left (\int x^6 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{3 d}-\frac{\left (4 b e^2 n\right ) \operatorname{Subst}\left (\int x^4 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{3 d^2}+\frac{\left (4 b e^3 n\right ) \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{3 d^3}-\frac{\left (4 b e^4 n\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{3 d^4}+\frac{\left (4 b e^5 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{3 d^4}\\ &=-\frac{4 a b e^4 n \sqrt [3]{x}}{3 d^4}+\frac{4 b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{9 d^3}-\frac{4 b e^2 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{15 d^2}+\frac{4 b e n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{21 d}+\frac{4 b e^{9/2} n \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{3 d^{9/2}}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2-\frac{\left (4 b^2 e^4 n\right ) \operatorname{Subst}\left (\int \log \left (c \left (d+\frac{e}{x^2}\right )^n\right ) \, dx,x,\sqrt [3]{x}\right )}{3 d^4}+\frac{\left (8 b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \frac{x^4}{d+\frac{e}{x^2}} \, dx,x,\sqrt [3]{x}\right )}{21 d}-\frac{\left (8 b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{d+\frac{e}{x^2}} \, dx,x,\sqrt [3]{x}\right )}{15 d^2}+\frac{\left (8 b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{d+\frac{e}{x^2}} \, dx,x,\sqrt [3]{x}\right )}{9 d^3}+\frac{\left (8 b^2 e^6 n^2\right ) \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{e}}\right )}{\sqrt{d} \sqrt{e} \left (d+\frac{e}{x^2}\right ) x^3} \, dx,x,\sqrt [3]{x}\right )}{3 d^4}\\ &=-\frac{4 a b e^4 n \sqrt [3]{x}}{3 d^4}-\frac{4 b^2 e^4 n \sqrt [3]{x} \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{3 d^4}+\frac{4 b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{9 d^3}-\frac{4 b e^2 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{15 d^2}+\frac{4 b e n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{21 d}+\frac{4 b e^{9/2} n \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{3 d^{9/2}}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2+\frac{\left (8 b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \frac{x^6}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{21 d}-\frac{\left (8 b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \frac{x^4}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{15 d^2}+\frac{\left (8 b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{9 d^3}-\frac{\left (8 b^2 e^5 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{\left (d+\frac{e}{x^2}\right ) x^2} \, dx,x,\sqrt [3]{x}\right )}{3 d^4}+\frac{\left (8 b^2 e^{11/2} n^2\right ) \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{e}}\right )}{\left (d+\frac{e}{x^2}\right ) x^3} \, dx,x,\sqrt [3]{x}\right )}{3 d^{9/2}}\\ &=-\frac{4 a b e^4 n \sqrt [3]{x}}{3 d^4}+\frac{8 b^2 e^4 n^2 \sqrt [3]{x}}{9 d^4}-\frac{4 b^2 e^4 n \sqrt [3]{x} \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{3 d^4}+\frac{4 b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{9 d^3}-\frac{4 b e^2 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{15 d^2}+\frac{4 b e n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{21 d}+\frac{4 b e^{9/2} n \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{3 d^{9/2}}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2+\frac{\left (8 b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \left (\frac{e^2}{d^3}-\frac{e x^2}{d^2}+\frac{x^4}{d}-\frac{e^3}{d^3 \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{21 d}-\frac{\left (8 b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \left (-\frac{e}{d^2}+\frac{x^2}{d}+\frac{e^2}{d^2 \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{15 d^2}-\frac{\left (8 b^2 e^5 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{9 d^4}-\frac{\left (8 b^2 e^5 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{3 d^4}+\frac{\left (8 b^2 e^{11/2} n^2\right ) \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{e}}\right )}{x \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{3 d^{9/2}}\\ &=-\frac{4 a b e^4 n \sqrt [3]{x}}{3 d^4}+\frac{568 b^2 e^4 n^2 \sqrt [3]{x}}{315 d^4}-\frac{32 b^2 e^3 n^2 x}{105 d^3}+\frac{8 b^2 e^2 n^2 x^{5/3}}{105 d^2}-\frac{32 b^2 e^{9/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{9 d^{9/2}}-\frac{4 i b^2 e^{9/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )^2}{3 d^{9/2}}-\frac{4 b^2 e^4 n \sqrt [3]{x} \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{3 d^4}+\frac{4 b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{9 d^3}-\frac{4 b e^2 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{15 d^2}+\frac{4 b e n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{21 d}+\frac{4 b e^{9/2} n \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{3 d^{9/2}}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2+\frac{\left (8 i b^2 e^{9/2} n^2\right ) \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{e}}\right )}{x \left (i+\frac{\sqrt{d} x}{\sqrt{e}}\right )} \, dx,x,\sqrt [3]{x}\right )}{3 d^{9/2}}-\frac{\left (8 b^2 e^5 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{21 d^4}-\frac{\left (8 b^2 e^5 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{15 d^4}\\ &=-\frac{4 a b e^4 n \sqrt [3]{x}}{3 d^4}+\frac{568 b^2 e^4 n^2 \sqrt [3]{x}}{315 d^4}-\frac{32 b^2 e^3 n^2 x}{105 d^3}+\frac{8 b^2 e^2 n^2 x^{5/3}}{105 d^2}-\frac{1408 b^2 e^{9/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{315 d^{9/2}}-\frac{4 i b^2 e^{9/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )^2}{3 d^{9/2}}+\frac{8 b^2 e^{9/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \log \left (2-\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\right )}{3 d^{9/2}}-\frac{4 b^2 e^4 n \sqrt [3]{x} \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{3 d^4}+\frac{4 b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{9 d^3}-\frac{4 b e^2 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{15 d^2}+\frac{4 b e n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{21 d}+\frac{4 b e^{9/2} n \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{3 d^{9/2}}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2-\frac{\left (8 b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (2-\frac{2}{1-\frac{i \sqrt{d} x}{\sqrt{e}}}\right )}{1+\frac{d x^2}{e}} \, dx,x,\sqrt [3]{x}\right )}{3 d^4}\\ &=-\frac{4 a b e^4 n \sqrt [3]{x}}{3 d^4}+\frac{568 b^2 e^4 n^2 \sqrt [3]{x}}{315 d^4}-\frac{32 b^2 e^3 n^2 x}{105 d^3}+\frac{8 b^2 e^2 n^2 x^{5/3}}{105 d^2}-\frac{1408 b^2 e^{9/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{315 d^{9/2}}-\frac{4 i b^2 e^{9/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )^2}{3 d^{9/2}}+\frac{8 b^2 e^{9/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \log \left (2-\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\right )}{3 d^{9/2}}-\frac{4 b^2 e^4 n \sqrt [3]{x} \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{3 d^4}+\frac{4 b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{9 d^3}-\frac{4 b e^2 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{15 d^2}+\frac{4 b e n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{21 d}+\frac{4 b e^{9/2} n \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{3 d^{9/2}}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2-\frac{4 i b^2 e^{9/2} n^2 \text{Li}_2\left (-1+\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\right )}{3 d^{9/2}}\\ \end{align*}

Mathematica [C]  time = 2.22864, size = 735, normalized size = 1.5 \[ \frac{1}{3} \left (x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2-4 b e n \left (-\frac{b e^{7/2} n \left (-4 \text{PolyLog}\left (2,1-\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )+2 \text{PolyLog}\left (2,\frac{1}{2}-\frac{\sqrt{-d} \sqrt [3]{x}}{2 \sqrt{e}}\right )+\log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right ) \left (\log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right )+2 \log \left (\frac{1}{2} \left (\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}+1\right )\right )-4 \log \left (\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )\right )\right )}{4 (-d)^{9/2}}+\frac{b e^{7/2} n \left (2 \text{PolyLog}\left (2,\frac{1}{2} \left (\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}+1\right )\right )-4 \text{PolyLog}\left (2,\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}+1\right )+\log \left (\sqrt{-d} \sqrt [3]{x}+\sqrt{e}\right ) \left (\log \left (\sqrt{-d} \sqrt [3]{x}+\sqrt{e}\right )+2 \log \left (\frac{1}{2}-\frac{\sqrt{-d} \sqrt [3]{x}}{2 \sqrt{e}}\right )-4 \log \left (-\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )\right )\right )}{4 (-d)^{9/2}}-\frac{e^2 x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{3 d^3}+\frac{e x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{5 d^2}+\frac{e^{7/2} \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{2 (-d)^{9/2}}-\frac{e^{7/2} \log \left (\sqrt{-d} \sqrt [3]{x}+\sqrt{e}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{2 (-d)^{9/2}}-\frac{x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{7 d}+\frac{a e^3 \sqrt [3]{x}}{d^4}+\frac{b e^3 \sqrt [3]{x} \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{d^4}-\frac{2 b e^3 n \sqrt [3]{x} \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};-\frac{e}{d x^{2/3}}\right )}{3 d^4}+\frac{2 b e^2 n x \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};-\frac{e}{d x^{2/3}}\right )}{15 d^3}-\frac{2 b e^{7/2} n \tan ^{-1}\left (\frac{\sqrt{e}}{\sqrt{d} \sqrt [3]{x}}\right )}{d^{9/2}}-\frac{2 b e n x^{5/3} \, _2F_1\left (-\frac{5}{2},1;-\frac{3}{2};-\frac{e}{d x^{2/3}}\right )}{35 d^2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^3*(a + b*Log[c*(d + e/x^(2/3))^n])^2 - 4*b*e*n*((a*e^3*x^(1/3))/d^4 - (2*b*e^(7/2)*n*ArcTan[Sqrt[e]/(Sqrt[d
]*x^(1/3))])/d^(9/2) - (2*b*e*n*x^(5/3)*Hypergeometric2F1[-5/2, 1, -3/2, -(e/(d*x^(2/3)))])/(35*d^2) + (2*b*e^
2*n*x*Hypergeometric2F1[-3/2, 1, -1/2, -(e/(d*x^(2/3)))])/(15*d^3) - (2*b*e^3*n*x^(1/3)*Hypergeometric2F1[-1/2
, 1, 1/2, -(e/(d*x^(2/3)))])/(3*d^4) + (b*e^3*x^(1/3)*Log[c*(d + e/x^(2/3))^n])/d^4 - (e^2*x*(a + b*Log[c*(d +
e/x^(2/3))^n]))/(3*d^3) + (e*x^(5/3)*(a + b*Log[c*(d + e/x^(2/3))^n]))/(5*d^2) - (x^(7/3)*(a + b*Log[c*(d + e
/x^(2/3))^n]))/(7*d) + (e^(7/2)*(a + b*Log[c*(d + e/x^(2/3))^n])*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)])/(2*(-d)^(9/2
)) - (e^(7/2)*(a + b*Log[c*(d + e/x^(2/3))^n])*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)])/(2*(-d)^(9/2)) - (b*e^(7/2)*n*
(Log[Sqrt[e] - Sqrt[-d]*x^(1/3)]*(Log[Sqrt[e] - Sqrt[-d]*x^(1/3)] + 2*Log[(1 + (Sqrt[-d]*x^(1/3))/Sqrt[e])/2]
- 4*Log[(Sqrt[-d]*x^(1/3))/Sqrt[e]]) - 4*PolyLog[2, 1 - (Sqrt[-d]*x^(1/3))/Sqrt[e]] + 2*PolyLog[2, 1/2 - (Sqrt
[-d]*x^(1/3))/(2*Sqrt[e])]))/(4*(-d)^(9/2)) + (b*e^(7/2)*n*(Log[Sqrt[e] + Sqrt[-d]*x^(1/3)]*(Log[Sqrt[e] + Sqr
t[-d]*x^(1/3)] + 2*Log[1/2 - (Sqrt[-d]*x^(1/3))/(2*Sqrt[e])] - 4*Log[-((Sqrt[-d]*x^(1/3))/Sqrt[e])]) + 2*PolyL
og[2, (1 + (Sqrt[-d]*x^(1/3))/Sqrt[e])/2] - 4*PolyLog[2, 1 + (Sqrt[-d]*x^(1/3))/Sqrt[e]]))/(4*(-d)^(9/2))))/3

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Maple [F]  time = 0.347, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \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(x^2*(a+b*ln(c*(d+e/x^(2/3))^n))^2,x)

[Out]

int(x^2*(a+b*ln(c*(d+e/x^(2/3))^n))^2,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(x^2*(a+b*log(c*(d+e/x^(2/3))^n))^2,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 (b^{2} x^{2} \log \left (c \left (\frac{d x + e x^{\frac{1}{3}}}{x}\right )^{n}\right )^{2} + 2 \, a b x^{2} \log \left (c \left (\frac{d x + e x^{\frac{1}{3}}}{x}\right )^{n}\right ) + a^{2} x^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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