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

Optimal. Leaf size=547 \[ \frac{4 i b^2 d^{9/2} n^2 \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} \sqrt [3]{x}}\right )}{3 e^{9/2}}+\frac{4 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{9 e^3}-\frac{4 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 e^2}+\frac{4 b d^{9/2} n \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 e^{9/2}}+\frac{4 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{21 e}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac{4}{27} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac{4 a b d^4 n \sqrt [3]{x}}{3 e^4}-\frac{4 b^2 d^4 n \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 e^4}+\frac{1144 b^2 d^2 n^2 x^{5/3}}{4725 e^2}+\frac{4504 b^2 d^4 n^2 \sqrt [3]{x}}{945 e^4}-\frac{1984 b^2 d^3 n^2 x}{2835 e^3}+\frac{4 i b^2 d^{9/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{3 e^{9/2}}-\frac{4504 b^2 d^{9/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{945 e^{9/2}}+\frac{8 b^2 d^{9/2} n^2 \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} \sqrt [3]{x}}\right ) \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{3 e^{9/2}}-\frac{128 b^2 d n^2 x^{7/3}}{1323 e}+\frac{8}{243} b^2 n^2 x^3 \]

[Out]

(-4*a*b*d^4*n*x^(1/3))/(3*e^4) + (4504*b^2*d^4*n^2*x^(1/3))/(945*e^4) - (1984*b^2*d^3*n^2*x)/(2835*e^3) + (114
4*b^2*d^2*n^2*x^(5/3))/(4725*e^2) - (128*b^2*d*n^2*x^(7/3))/(1323*e) + (8*b^2*n^2*x^3)/243 - (4504*b^2*d^(9/2)
*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]])/(945*e^(9/2)) + (((4*I)/3)*b^2*d^(9/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sq
rt[d]]^2)/e^(9/2) + (8*b^2*d^(9/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*
x^(1/3))])/(3*e^(9/2)) - (4*b^2*d^4*n*x^(1/3)*Log[c*(d + e*x^(2/3))^n])/(3*e^4) + (4*b*d^3*n*x*(a + b*Log[c*(d
+ e*x^(2/3))^n]))/(9*e^3) - (4*b*d^2*n*x^(5/3)*(a + b*Log[c*(d + e*x^(2/3))^n]))/(15*e^2) + (4*b*d*n*x^(7/3)*
(a + b*Log[c*(d + e*x^(2/3))^n]))/(21*e) - (4*b*n*x^3*(a + b*Log[c*(d + e*x^(2/3))^n]))/27 + (4*b*d^(9/2)*n*Ar
cTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*(a + b*Log[c*(d + e*x^(2/3))^n]))/(3*e^(9/2)) + (x^3*(a + b*Log[c*(d + e*x^(2/
3))^n])^2)/3 + (((4*I)/3)*b^2*d^(9/2)*n^2*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/e^(9/2)

________________________________________________________________________________________

Rubi [A]  time = 0.769075, antiderivative size = 547, normalized size of antiderivative = 1., number of steps used = 30, number of rules used = 14, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {2458, 2457, 2476, 2448, 321, 205, 2455, 302, 2470, 12, 4920, 4854, 2402, 2315} \[ \frac{4 i b^2 d^{9/2} n^2 \text{PolyLog}\left (2,1-\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} \sqrt [3]{x}}\right )}{3 e^{9/2}}+\frac{4 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{9 e^3}-\frac{4 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 e^2}+\frac{4 b d^{9/2} n \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 e^{9/2}}+\frac{4 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{21 e}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac{4}{27} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac{4 a b d^4 n \sqrt [3]{x}}{3 e^4}-\frac{4 b^2 d^4 n \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 e^4}+\frac{1144 b^2 d^2 n^2 x^{5/3}}{4725 e^2}+\frac{4504 b^2 d^4 n^2 \sqrt [3]{x}}{945 e^4}-\frac{1984 b^2 d^3 n^2 x}{2835 e^3}+\frac{4 i b^2 d^{9/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{3 e^{9/2}}-\frac{4504 b^2 d^{9/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{945 e^{9/2}}+\frac{8 b^2 d^{9/2} n^2 \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} \sqrt [3]{x}}\right ) \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{3 e^{9/2}}-\frac{128 b^2 d n^2 x^{7/3}}{1323 e}+\frac{8}{243} b^2 n^2 x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*a*b*d^4*n*x^(1/3))/(3*e^4) + (4504*b^2*d^4*n^2*x^(1/3))/(945*e^4) - (1984*b^2*d^3*n^2*x)/(2835*e^3) + (114
4*b^2*d^2*n^2*x^(5/3))/(4725*e^2) - (128*b^2*d*n^2*x^(7/3))/(1323*e) + (8*b^2*n^2*x^3)/243 - (4504*b^2*d^(9/2)
*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]])/(945*e^(9/2)) + (((4*I)/3)*b^2*d^(9/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sq
rt[d]]^2)/e^(9/2) + (8*b^2*d^(9/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*
x^(1/3))])/(3*e^(9/2)) - (4*b^2*d^4*n*x^(1/3)*Log[c*(d + e*x^(2/3))^n])/(3*e^4) + (4*b*d^3*n*x*(a + b*Log[c*(d
+ e*x^(2/3))^n]))/(9*e^3) - (4*b*d^2*n*x^(5/3)*(a + b*Log[c*(d + e*x^(2/3))^n]))/(15*e^2) + (4*b*d*n*x^(7/3)*
(a + b*Log[c*(d + e*x^(2/3))^n]))/(21*e) - (4*b*n*x^3*(a + b*Log[c*(d + e*x^(2/3))^n]))/27 + (4*b*d^(9/2)*n*Ar
cTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*(a + b*Log[c*(d + e*x^(2/3))^n]))/(3*e^(9/2)) + (x^3*(a + b*Log[c*(d + e*x^(2/
3))^n])^2)/3 + (((4*I)/3)*b^2*d^(9/2)*n^2*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/e^(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 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 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 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 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 x^2 \left (a+b \log \left (c \left (d+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+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+e x^{2/3}\right )^n\right )\right )^2-\frac{1}{3} (4 b e n) \operatorname{Subst}\left (\int \frac{x^{10} \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac{1}{3} (4 b e n) \operatorname{Subst}\left (\int \left (\frac{d^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^5}-\frac{d^3 x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^4}+\frac{d^2 x^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^3}-\frac{d x^6 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^2}+\frac{x^8 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e}-\frac{d^5 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^5 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac{1}{3} (4 b n) \operatorname{Subst}\left (\int x^8 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )-\frac{\left (4 b d^4 n\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{3 e^4}+\frac{\left (4 b d^5 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 )}{3 e^4}+\frac{\left (4 b d^3 n\right ) \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{3 e^3}-\frac{\left (4 b d^2 n\right ) \operatorname{Subst}\left (\int x^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{3 e^2}+\frac{(4 b d n) \operatorname{Subst}\left (\int x^6 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{3 e}\\ &=-\frac{4 a b d^4 n \sqrt [3]{x}}{3 e^4}+\frac{4 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{9 e^3}-\frac{4 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 e^2}+\frac{4 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{21 e}-\frac{4}{27} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac{4 b d^{9/2} 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 )}{3 e^{9/2}}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac{\left (4 b^2 d^4 n\right ) \operatorname{Subst}\left (\int \log \left (c \left (d+e x^2\right )^n\right ) \, dx,x,\sqrt [3]{x}\right )}{3 e^4}-\frac{1}{21} \left (8 b^2 d n^2\right ) \operatorname{Subst}\left (\int \frac{x^8}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )-\frac{\left (8 b^2 d^5 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 )}{3 e^3}-\frac{\left (8 b^2 d^3 n^2\right ) \operatorname{Subst}\left (\int \frac{x^4}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{9 e^2}+\frac{\left (8 b^2 d^2 n^2\right ) \operatorname{Subst}\left (\int \frac{x^6}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{15 e}+\frac{1}{27} \left (8 b^2 e n^2\right ) \operatorname{Subst}\left (\int \frac{x^{10}}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{4 a b d^4 n \sqrt [3]{x}}{3 e^4}-\frac{4 b^2 d^4 n \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 e^4}+\frac{4 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{9 e^3}-\frac{4 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 e^2}+\frac{4 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{21 e}-\frac{4}{27} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac{4 b d^{9/2} 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 )}{3 e^{9/2}}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac{1}{21} \left (8 b^2 d n^2\right ) \operatorname{Subst}\left (\int \left (-\frac{d^3}{e^4}+\frac{d^2 x^2}{e^3}-\frac{d x^4}{e^2}+\frac{x^6}{e}+\frac{d^4}{e^4 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )-\frac{\left (8 b^2 d^{9/2} 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 )}{3 e^{7/2}}+\frac{\left (8 b^2 d^4 n^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{3 e^3}-\frac{\left (8 b^2 d^3 n^2\right ) \operatorname{Subst}\left (\int \left (-\frac{d}{e^2}+\frac{x^2}{e}+\frac{d^2}{e^2 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{9 e^2}+\frac{\left (8 b^2 d^2 n^2\right ) \operatorname{Subst}\left (\int \left (\frac{d^2}{e^3}-\frac{d x^2}{e^2}+\frac{x^4}{e}-\frac{d^3}{e^3 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{15 e}+\frac{1}{27} \left (8 b^2 e n^2\right ) \operatorname{Subst}\left (\int \left (\frac{d^4}{e^5}-\frac{d^3 x^2}{e^4}+\frac{d^2 x^4}{e^3}-\frac{d x^6}{e^2}+\frac{x^8}{e}-\frac{d^5}{e^5 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{4 a b d^4 n \sqrt [3]{x}}{3 e^4}+\frac{4504 b^2 d^4 n^2 \sqrt [3]{x}}{945 e^4}-\frac{1984 b^2 d^3 n^2 x}{2835 e^3}+\frac{1144 b^2 d^2 n^2 x^{5/3}}{4725 e^2}-\frac{128 b^2 d n^2 x^{7/3}}{1323 e}+\frac{8}{243} b^2 n^2 x^3+\frac{4 i b^2 d^{9/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{3 e^{9/2}}-\frac{4 b^2 d^4 n \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 e^4}+\frac{4 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{9 e^3}-\frac{4 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 e^2}+\frac{4 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{21 e}-\frac{4}{27} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac{4 b d^{9/2} 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 )}{3 e^{9/2}}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac{\left (8 b^2 d^4 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 )}{3 e^4}-\frac{\left (8 b^2 d^5 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{27 e^4}-\frac{\left (8 b^2 d^5 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{21 e^4}-\frac{\left (8 b^2 d^5 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{15 e^4}-\frac{\left (8 b^2 d^5 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{9 e^4}-\frac{\left (8 b^2 d^5 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{3 e^4}\\ &=-\frac{4 a b d^4 n \sqrt [3]{x}}{3 e^4}+\frac{4504 b^2 d^4 n^2 \sqrt [3]{x}}{945 e^4}-\frac{1984 b^2 d^3 n^2 x}{2835 e^3}+\frac{1144 b^2 d^2 n^2 x^{5/3}}{4725 e^2}-\frac{128 b^2 d n^2 x^{7/3}}{1323 e}+\frac{8}{243} b^2 n^2 x^3-\frac{4504 b^2 d^{9/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{945 e^{9/2}}+\frac{4 i b^2 d^{9/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{3 e^{9/2}}+\frac{8 b^2 d^{9/2} 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 )}{3 e^{9/2}}-\frac{4 b^2 d^4 n \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 e^4}+\frac{4 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{9 e^3}-\frac{4 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 e^2}+\frac{4 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{21 e}-\frac{4}{27} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac{4 b d^{9/2} 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 )}{3 e^{9/2}}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac{\left (8 b^2 d^4 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 )}{3 e^4}\\ &=-\frac{4 a b d^4 n \sqrt [3]{x}}{3 e^4}+\frac{4504 b^2 d^4 n^2 \sqrt [3]{x}}{945 e^4}-\frac{1984 b^2 d^3 n^2 x}{2835 e^3}+\frac{1144 b^2 d^2 n^2 x^{5/3}}{4725 e^2}-\frac{128 b^2 d n^2 x^{7/3}}{1323 e}+\frac{8}{243} b^2 n^2 x^3-\frac{4504 b^2 d^{9/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{945 e^{9/2}}+\frac{4 i b^2 d^{9/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{3 e^{9/2}}+\frac{8 b^2 d^{9/2} 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 )}{3 e^{9/2}}-\frac{4 b^2 d^4 n \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 e^4}+\frac{4 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{9 e^3}-\frac{4 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 e^2}+\frac{4 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{21 e}-\frac{4}{27} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac{4 b d^{9/2} 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 )}{3 e^{9/2}}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac{\left (8 i b^2 d^{9/2} 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 )}{3 e^{9/2}}\\ &=-\frac{4 a b d^4 n \sqrt [3]{x}}{3 e^4}+\frac{4504 b^2 d^4 n^2 \sqrt [3]{x}}{945 e^4}-\frac{1984 b^2 d^3 n^2 x}{2835 e^3}+\frac{1144 b^2 d^2 n^2 x^{5/3}}{4725 e^2}-\frac{128 b^2 d n^2 x^{7/3}}{1323 e}+\frac{8}{243} b^2 n^2 x^3-\frac{4504 b^2 d^{9/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{945 e^{9/2}}+\frac{4 i b^2 d^{9/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{3 e^{9/2}}+\frac{8 b^2 d^{9/2} 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 )}{3 e^{9/2}}-\frac{4 b^2 d^4 n \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 e^4}+\frac{4 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{9 e^3}-\frac{4 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 e^2}+\frac{4 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{21 e}-\frac{4}{27} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac{4 b d^{9/2} 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 )}{3 e^{9/2}}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac{4 i b^2 d^{9/2} n^2 \text{Li}_2\left (1-\frac{2}{1+\frac{i \sqrt{e} \sqrt [3]{x}}{\sqrt{d}}}\right )}{3 e^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.485456, size = 438, normalized size = 0.8 \[ \frac{396900 i b^2 d^{9/2} n^2 \text{PolyLog}\left (2,\frac{\sqrt{e} \sqrt [3]{x}+i \sqrt{d}}{\sqrt{e} \sqrt [3]{x}-i \sqrt{d}}\right )+\sqrt{e} \sqrt [3]{x} \left (99225 a^2 e^4 x^{8/3}-630 b \left (2 b n \left (63 d^2 e^2 x^{4/3}-105 d^3 e x^{2/3}+315 d^4-45 d e^3 x^2+35 e^4 x^{8/3}\right )-315 a e^4 x^{8/3}\right ) \log \left (c \left (d+e x^{2/3}\right )^n\right )-1260 a b n \left (63 d^2 e^2 x^{4/3}-105 d^3 e x^{2/3}+315 d^4-45 d e^3 x^2+35 e^4 x^{8/3}\right )+99225 b^2 e^4 x^{8/3} \log ^2\left (c \left (d+e x^{2/3}\right )^n\right )+8 b^2 n^2 \left (9009 d^2 e^2 x^{4/3}-26040 d^3 e x^{2/3}+177345 d^4-3600 d e^3 x^2+1225 e^4 x^{8/3}\right )\right )+1260 b d^{9/2} n \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right ) \left (315 a+315 b \log \left (c \left (d+e x^{2/3}\right )^n\right )+630 b n \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} \sqrt [3]{x}}\right )-1126 b n\right )+396900 i b^2 d^{9/2} n^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )^2}{297675 e^{9/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((396900*I)*b^2*d^(9/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]^2 + 1260*b*d^(9/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sq
rt[d]]*(315*a - 1126*b*n + 630*b*n*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))] + 315*b*Log[c*(d + e*x^(2/3)
)^n]) + Sqrt[e]*x^(1/3)*(99225*a^2*e^4*x^(8/3) - 1260*a*b*n*(315*d^4 - 105*d^3*e*x^(2/3) + 63*d^2*e^2*x^(4/3)
- 45*d*e^3*x^2 + 35*e^4*x^(8/3)) + 8*b^2*n^2*(177345*d^4 - 26040*d^3*e*x^(2/3) + 9009*d^2*e^2*x^(4/3) - 3600*d
*e^3*x^2 + 1225*e^4*x^(8/3)) - 630*b*(-315*a*e^4*x^(8/3) + 2*b*n*(315*d^4 - 105*d^3*e*x^(2/3) + 63*d^2*e^2*x^(
4/3) - 45*d*e^3*x^2 + 35*e^4*x^(8/3)))*Log[c*(d + e*x^(2/3))^n] + 99225*b^2*e^4*x^(8/3)*Log[c*(d + e*x^(2/3))^
n]^2) + (396900*I)*b^2*d^(9/2)*n^2*PolyLog[2, (I*Sqrt[d] + Sqrt[e]*x^(1/3))/((-I)*Sqrt[d] + Sqrt[e]*x^(1/3))])
/(297675*e^(9/2))

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Maple [F]  time = 0.341, 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 ({\left (e x^{\frac{2}{3}} + d\right )}^{n} c\right )^{2} + 2 \, a b x^{2} \log \left ({\left (e x^{\frac{2}{3}} + d\right )}^{n} c\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((e*x^(2/3) + d)^n*c)^2 + 2*a*b*x^2*log((e*x^(2/3) + d)^n*c) + 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 ({\left (e x^{\frac{2}{3}} + d\right )}^{n} c\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((e*x^(2/3) + d)^n*c) + a)^2*x^2, x)