3.5.97 \(\int x^2 (a+b \log (c (d+\frac {e}{\sqrt [3]{x}})^n))^2 \, dx\) [497]

Optimal. Leaf size=572 \[ \frac {481 b^2 e^8 n^2 \sqrt [3]{x}}{420 d^8}-\frac {341 b^2 e^7 n^2 x^{2/3}}{840 d^7}+\frac {743 b^2 e^6 n^2 x}{3780 d^6}-\frac {533 b^2 e^5 n^2 x^{4/3}}{5040 d^5}+\frac {73 b^2 e^4 n^2 x^{5/3}}{1260 d^4}-\frac {5 b^2 e^3 n^2 x^2}{168 d^3}+\frac {b^2 e^2 n^2 x^{7/3}}{84 d^2}-\frac {481 b^2 e^9 n^2 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{420 d^9}-\frac {2 b e^8 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^9}+\frac {b e^7 n x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^7}-\frac {2 b e^6 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^6}+\frac {b e^5 n x^{4/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{6 d^5}-\frac {2 b e^4 n x^{5/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{15 d^4}+\frac {b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^3}-\frac {2 b e^2 n x^{7/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{21 d^2}+\frac {b e n x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 d}-\frac {2 b e^9 n \log \left (1-\frac {d}{d+\frac {e}{\sqrt [3]{x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^9}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {761 b^2 e^9 n^2 \log (x)}{1260 d^9}+\frac {2 b^2 e^9 n^2 \text {Li}_2\left (\frac {d}{d+\frac {e}{\sqrt [3]{x}}}\right )}{3 d^9} \]

[Out]

481/420*b^2*e^8*n^2*x^(1/3)/d^8-341/840*b^2*e^7*n^2*x^(2/3)/d^7+743/3780*b^2*e^6*n^2*x/d^6-533/5040*b^2*e^5*n^
2*x^(4/3)/d^5+73/1260*b^2*e^4*n^2*x^(5/3)/d^4-5/168*b^2*e^3*n^2*x^2/d^3+1/84*b^2*e^2*n^2*x^(7/3)/d^2-481/420*b
^2*e^9*n^2*ln(d+e/x^(1/3))/d^9-2/3*b*e^8*n*(d+e/x^(1/3))*x^(1/3)*(a+b*ln(c*(d+e/x^(1/3))^n))/d^9+1/3*b*e^7*n*x
^(2/3)*(a+b*ln(c*(d+e/x^(1/3))^n))/d^7-2/9*b*e^6*n*x*(a+b*ln(c*(d+e/x^(1/3))^n))/d^6+1/6*b*e^5*n*x^(4/3)*(a+b*
ln(c*(d+e/x^(1/3))^n))/d^5-2/15*b*e^4*n*x^(5/3)*(a+b*ln(c*(d+e/x^(1/3))^n))/d^4+1/9*b*e^3*n*x^2*(a+b*ln(c*(d+e
/x^(1/3))^n))/d^3-2/21*b*e^2*n*x^(7/3)*(a+b*ln(c*(d+e/x^(1/3))^n))/d^2+1/12*b*e*n*x^(8/3)*(a+b*ln(c*(d+e/x^(1/
3))^n))/d-2/3*b*e^9*n*ln(1-d/(d+e/x^(1/3)))*(a+b*ln(c*(d+e/x^(1/3))^n))/d^9+1/3*x^3*(a+b*ln(c*(d+e/x^(1/3))^n)
)^2-761/1260*b^2*e^9*n^2*ln(x)/d^9+2/3*b^2*e^9*n^2*polylog(2,d/(d+e/x^(1/3)))/d^9

________________________________________________________________________________________

Rubi [A]
time = 1.08, antiderivative size = 572, normalized size of antiderivative = 1.00, number of steps used = 36, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2504, 2445, 2458, 2389, 2379, 2438, 2351, 31, 2356, 46} \begin {gather*} \frac {2 b^2 e^9 n^2 \text {PolyLog}\left (2,\frac {d}{d+\frac {e}{\sqrt [3]{x}}}\right )}{3 d^9}-\frac {2 b e^9 n \log \left (1-\frac {d}{d+\frac {e}{\sqrt [3]{x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^9}-\frac {2 b e^8 n \sqrt [3]{x} \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^9}+\frac {b e^7 n x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^7}-\frac {2 b e^6 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^6}+\frac {b e^5 n x^{4/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{6 d^5}-\frac {2 b e^4 n x^{5/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{15 d^4}+\frac {b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^3}-\frac {2 b e^2 n x^{7/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{21 d^2}+\frac {b e n x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 d}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {481 b^2 e^9 n^2 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{420 d^9}-\frac {761 b^2 e^9 n^2 \log (x)}{1260 d^9}+\frac {481 b^2 e^8 n^2 \sqrt [3]{x}}{420 d^8}-\frac {341 b^2 e^7 n^2 x^{2/3}}{840 d^7}+\frac {743 b^2 e^6 n^2 x}{3780 d^6}-\frac {533 b^2 e^5 n^2 x^{4/3}}{5040 d^5}+\frac {73 b^2 e^4 n^2 x^{5/3}}{1260 d^4}-\frac {5 b^2 e^3 n^2 x^2}{168 d^3}+\frac {b^2 e^2 n^2 x^{7/3}}{84 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(481*b^2*e^8*n^2*x^(1/3))/(420*d^8) - (341*b^2*e^7*n^2*x^(2/3))/(840*d^7) + (743*b^2*e^6*n^2*x)/(3780*d^6) - (
533*b^2*e^5*n^2*x^(4/3))/(5040*d^5) + (73*b^2*e^4*n^2*x^(5/3))/(1260*d^4) - (5*b^2*e^3*n^2*x^2)/(168*d^3) + (b
^2*e^2*n^2*x^(7/3))/(84*d^2) - (481*b^2*e^9*n^2*Log[d + e/x^(1/3)])/(420*d^9) - (2*b*e^8*n*(d + e/x^(1/3))*x^(
1/3)*(a + b*Log[c*(d + e/x^(1/3))^n]))/(3*d^9) + (b*e^7*n*x^(2/3)*(a + b*Log[c*(d + e/x^(1/3))^n]))/(3*d^7) -
(2*b*e^6*n*x*(a + b*Log[c*(d + e/x^(1/3))^n]))/(9*d^6) + (b*e^5*n*x^(4/3)*(a + b*Log[c*(d + e/x^(1/3))^n]))/(6
*d^5) - (2*b*e^4*n*x^(5/3)*(a + b*Log[c*(d + e/x^(1/3))^n]))/(15*d^4) + (b*e^3*n*x^2*(a + b*Log[c*(d + e/x^(1/
3))^n]))/(9*d^3) - (2*b*e^2*n*x^(7/3)*(a + b*Log[c*(d + e/x^(1/3))^n]))/(21*d^2) + (b*e*n*x^(8/3)*(a + b*Log[c
*(d + e/x^(1/3))^n]))/(12*d) - (2*b*e^9*n*Log[1 - d/(d + e/x^(1/3))]*(a + b*Log[c*(d + e/x^(1/3))^n]))/(3*d^9)
+ (x^3*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/3 - (761*b^2*e^9*n^2*Log[x])/(1260*d^9) + (2*b^2*e^9*n^2*PolyLog[2
, d/(d + e/x^(1/3))])/(3*d^9)

Rule 31

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

Rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 2351

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp[x*(d + e*x^r)^(q +
1)*((a + b*Log[c*x^n])/d), x] - Dist[b*(n/d), Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q,
r}, x] && EqQ[r*(q + 1) + 1, 0]

Rule 2356

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)
*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Dist[b*n*(p/(e*(q + 1))), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2379

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

Rule 2389

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[(d
+ e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x), x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

Rule 2438

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

Rule 2445

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[(f
+ g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])^p/(g*(q + 1))), x] - Dist[b*e*n*(p/(g*(q + 1))), Int[(f + g*x)^(q
+ 1)*((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*
f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && IntegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2458

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 2504

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

Rubi steps

\begin {align*} \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \, dx &=-\left (3 \text {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^{10}} \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\right )\\ &=\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {1}{3} (2 b e n) \text {Subst}\left (\int \frac {a+b \log \left (c (d+e x)^n\right )}{x^9 (d+e x)} \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\\ &=\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {1}{3} (2 b n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^9} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )\\ &=\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {(2 b n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^9} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{3 d}+\frac {(2 b e n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^8} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{3 d}\\ &=\frac {b e n x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 d}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2+\frac {(2 b e n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^8} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{3 d^2}-\frac {\left (2 b e^2 n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^7} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{3 d^2}-\frac {\left (b^2 e n^2\right ) \text {Subst}\left (\int \frac {1}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^8} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{12 d}\\ &=-\frac {2 b e^2 n x^{7/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{21 d^2}+\frac {b e n x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 d}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {\left (2 b e^2 n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^7} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{3 d^3}+\frac {\left (2 b e^3 n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^6} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{3 d^3}-\frac {\left (b^2 e n^2\right ) \text {Subst}\left (\int \left (\frac {e^8}{d (d-x)^8}+\frac {e^8}{d^2 (d-x)^7}+\frac {e^8}{d^3 (d-x)^6}+\frac {e^8}{d^4 (d-x)^5}+\frac {e^8}{d^5 (d-x)^4}+\frac {e^8}{d^6 (d-x)^3}+\frac {e^8}{d^7 (d-x)^2}+\frac {e^8}{d^8 (d-x)}+\frac {e^8}{d^8 x}\right ) \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{12 d}+\frac {\left (2 b^2 e^2 n^2\right ) \text {Subst}\left (\int \frac {1}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^7} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{21 d^2}\\ &=\frac {b^2 e^8 n^2 \sqrt [3]{x}}{12 d^8}-\frac {b^2 e^7 n^2 x^{2/3}}{24 d^7}+\frac {b^2 e^6 n^2 x}{36 d^6}-\frac {b^2 e^5 n^2 x^{4/3}}{48 d^5}+\frac {b^2 e^4 n^2 x^{5/3}}{60 d^4}-\frac {b^2 e^3 n^2 x^2}{72 d^3}+\frac {b^2 e^2 n^2 x^{7/3}}{84 d^2}-\frac {b^2 e^9 n^2 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{12 d^9}+\frac {b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^3}-\frac {2 b e^2 n x^{7/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{21 d^2}+\frac {b e n x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 d}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {b^2 e^9 n^2 \log (x)}{36 d^9}+\frac {\left (2 b e^3 n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^6} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{3 d^4}-\frac {\left (2 b e^4 n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^5} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{3 d^4}+\frac {\left (2 b^2 e^2 n^2\right ) \text {Subst}\left (\int \left (-\frac {e^7}{d (d-x)^7}-\frac {e^7}{d^2 (d-x)^6}-\frac {e^7}{d^3 (d-x)^5}-\frac {e^7}{d^4 (d-x)^4}-\frac {e^7}{d^5 (d-x)^3}-\frac {e^7}{d^6 (d-x)^2}-\frac {e^7}{d^7 (d-x)}-\frac {e^7}{d^7 x}\right ) \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{21 d^2}-\frac {\left (b^2 e^3 n^2\right ) \text {Subst}\left (\int \frac {1}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^6} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{9 d^3}\\ &=\frac {5 b^2 e^8 n^2 \sqrt [3]{x}}{28 d^8}-\frac {5 b^2 e^7 n^2 x^{2/3}}{56 d^7}+\frac {5 b^2 e^6 n^2 x}{84 d^6}-\frac {5 b^2 e^5 n^2 x^{4/3}}{112 d^5}+\frac {b^2 e^4 n^2 x^{5/3}}{28 d^4}-\frac {5 b^2 e^3 n^2 x^2}{168 d^3}+\frac {b^2 e^2 n^2 x^{7/3}}{84 d^2}-\frac {5 b^2 e^9 n^2 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{28 d^9}-\frac {2 b e^4 n x^{5/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{15 d^4}+\frac {b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^3}-\frac {2 b e^2 n x^{7/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{21 d^2}+\frac {b e n x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 d}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {5 b^2 e^9 n^2 \log (x)}{84 d^9}-\frac {\left (2 b e^4 n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^5} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{3 d^5}+\frac {\left (2 b e^5 n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^4} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{3 d^5}-\frac {\left (b^2 e^3 n^2\right ) \text {Subst}\left (\int \left (\frac {e^6}{d (d-x)^6}+\frac {e^6}{d^2 (d-x)^5}+\frac {e^6}{d^3 (d-x)^4}+\frac {e^6}{d^4 (d-x)^3}+\frac {e^6}{d^5 (d-x)^2}+\frac {e^6}{d^6 (d-x)}+\frac {e^6}{d^6 x}\right ) \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{9 d^3}+\frac {\left (2 b^2 e^4 n^2\right ) \text {Subst}\left (\int \frac {1}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^5} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{15 d^4}\\ &=\frac {73 b^2 e^8 n^2 \sqrt [3]{x}}{252 d^8}-\frac {73 b^2 e^7 n^2 x^{2/3}}{504 d^7}+\frac {73 b^2 e^6 n^2 x}{756 d^6}-\frac {73 b^2 e^5 n^2 x^{4/3}}{1008 d^5}+\frac {73 b^2 e^4 n^2 x^{5/3}}{1260 d^4}-\frac {5 b^2 e^3 n^2 x^2}{168 d^3}+\frac {b^2 e^2 n^2 x^{7/3}}{84 d^2}-\frac {73 b^2 e^9 n^2 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{252 d^9}+\frac {b e^5 n x^{4/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{6 d^5}-\frac {2 b e^4 n x^{5/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{15 d^4}+\frac {b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^3}-\frac {2 b e^2 n x^{7/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{21 d^2}+\frac {b e n x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 d}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {73 b^2 e^9 n^2 \log (x)}{756 d^9}+\frac {\left (2 b e^5 n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^4} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{3 d^6}-\frac {\left (2 b e^6 n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{3 d^6}+\frac {\left (2 b^2 e^4 n^2\right ) \text {Subst}\left (\int \left (-\frac {e^5}{d (d-x)^5}-\frac {e^5}{d^2 (d-x)^4}-\frac {e^5}{d^3 (d-x)^3}-\frac {e^5}{d^4 (d-x)^2}-\frac {e^5}{d^5 (d-x)}-\frac {e^5}{d^5 x}\right ) \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{15 d^4}-\frac {\left (b^2 e^5 n^2\right ) \text {Subst}\left (\int \frac {1}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^4} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{6 d^5}\\ &=\frac {533 b^2 e^8 n^2 \sqrt [3]{x}}{1260 d^8}-\frac {533 b^2 e^7 n^2 x^{2/3}}{2520 d^7}+\frac {533 b^2 e^6 n^2 x}{3780 d^6}-\frac {533 b^2 e^5 n^2 x^{4/3}}{5040 d^5}+\frac {73 b^2 e^4 n^2 x^{5/3}}{1260 d^4}-\frac {5 b^2 e^3 n^2 x^2}{168 d^3}+\frac {b^2 e^2 n^2 x^{7/3}}{84 d^2}-\frac {533 b^2 e^9 n^2 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{1260 d^9}-\frac {2 b e^6 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^6}+\frac {b e^5 n x^{4/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{6 d^5}-\frac {2 b e^4 n x^{5/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{15 d^4}+\frac {b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^3}-\frac {2 b e^2 n x^{7/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{21 d^2}+\frac {b e n x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 d}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {533 b^2 e^9 n^2 \log (x)}{3780 d^9}-\frac {\left (2 b e^6 n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{3 d^7}+\frac {\left (2 b e^7 n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{3 d^7}-\frac {\left (b^2 e^5 n^2\right ) \text {Subst}\left (\int \left (\frac {e^4}{d (d-x)^4}+\frac {e^4}{d^2 (d-x)^3}+\frac {e^4}{d^3 (d-x)^2}+\frac {e^4}{d^4 (d-x)}+\frac {e^4}{d^4 x}\right ) \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{6 d^5}+\frac {\left (2 b^2 e^6 n^2\right ) \text {Subst}\left (\int \frac {1}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{9 d^6}\\ &=\frac {743 b^2 e^8 n^2 \sqrt [3]{x}}{1260 d^8}-\frac {743 b^2 e^7 n^2 x^{2/3}}{2520 d^7}+\frac {743 b^2 e^6 n^2 x}{3780 d^6}-\frac {533 b^2 e^5 n^2 x^{4/3}}{5040 d^5}+\frac {73 b^2 e^4 n^2 x^{5/3}}{1260 d^4}-\frac {5 b^2 e^3 n^2 x^2}{168 d^3}+\frac {b^2 e^2 n^2 x^{7/3}}{84 d^2}-\frac {743 b^2 e^9 n^2 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{1260 d^9}+\frac {b e^7 n x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^7}-\frac {2 b e^6 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^6}+\frac {b e^5 n x^{4/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{6 d^5}-\frac {2 b e^4 n x^{5/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{15 d^4}+\frac {b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^3}-\frac {2 b e^2 n x^{7/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{21 d^2}+\frac {b e n x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 d}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {743 b^2 e^9 n^2 \log (x)}{3780 d^9}+\frac {\left (2 b e^7 n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{3 d^8}-\frac {\left (2 b e^8 n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{3 d^8}+\frac {\left (2 b^2 e^6 n^2\right ) \text {Subst}\left (\int \left (-\frac {e^3}{d (d-x)^3}-\frac {e^3}{d^2 (d-x)^2}-\frac {e^3}{d^3 (d-x)}-\frac {e^3}{d^3 x}\right ) \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{9 d^6}-\frac {\left (b^2 e^7 n^2\right ) \text {Subst}\left (\int \frac {1}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{3 d^7}\\ &=\frac {341 b^2 e^8 n^2 \sqrt [3]{x}}{420 d^8}-\frac {341 b^2 e^7 n^2 x^{2/3}}{840 d^7}+\frac {743 b^2 e^6 n^2 x}{3780 d^6}-\frac {533 b^2 e^5 n^2 x^{4/3}}{5040 d^5}+\frac {73 b^2 e^4 n^2 x^{5/3}}{1260 d^4}-\frac {5 b^2 e^3 n^2 x^2}{168 d^3}+\frac {b^2 e^2 n^2 x^{7/3}}{84 d^2}-\frac {341 b^2 e^9 n^2 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{420 d^9}-\frac {2 b e^8 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^9}+\frac {b e^7 n x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^7}-\frac {2 b e^6 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^6}+\frac {b e^5 n x^{4/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{6 d^5}-\frac {2 b e^4 n x^{5/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{15 d^4}+\frac {b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^3}-\frac {2 b e^2 n x^{7/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{21 d^2}+\frac {b e n x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 d}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {341 b^2 e^9 n^2 \log (x)}{1260 d^9}-\frac {\left (2 b e^8 n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{3 d^9}+\frac {\left (2 b e^9 n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{3 d^9}-\frac {\left (b^2 e^7 n^2\right ) \text {Subst}\left (\int \left (\frac {e^2}{d (d-x)^2}+\frac {e^2}{d^2 (d-x)}+\frac {e^2}{d^2 x}\right ) \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{3 d^7}+\frac {\left (2 b^2 e^8 n^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{3 d^9}\\ &=\frac {481 b^2 e^8 n^2 \sqrt [3]{x}}{420 d^8}-\frac {341 b^2 e^7 n^2 x^{2/3}}{840 d^7}+\frac {743 b^2 e^6 n^2 x}{3780 d^6}-\frac {533 b^2 e^5 n^2 x^{4/3}}{5040 d^5}+\frac {73 b^2 e^4 n^2 x^{5/3}}{1260 d^4}-\frac {5 b^2 e^3 n^2 x^2}{168 d^3}+\frac {b^2 e^2 n^2 x^{7/3}}{84 d^2}-\frac {481 b^2 e^9 n^2 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{420 d^9}-\frac {2 b e^8 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^9}+\frac {b e^7 n x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^7}-\frac {2 b e^6 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^6}+\frac {b e^5 n x^{4/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{6 d^5}-\frac {2 b e^4 n x^{5/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{15 d^4}+\frac {b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^3}-\frac {2 b e^2 n x^{7/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{21 d^2}+\frac {b e n x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 d}+\frac {e^9 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{3 d^9}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {2 b e^9 n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \log \left (-\frac {e}{d \sqrt [3]{x}}\right )}{3 d^9}-\frac {761 b^2 e^9 n^2 \log (x)}{1260 d^9}+\frac {\left (2 b^2 e^9 n^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{d}\right )}{x} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{3 d^9}\\ &=\frac {481 b^2 e^8 n^2 \sqrt [3]{x}}{420 d^8}-\frac {341 b^2 e^7 n^2 x^{2/3}}{840 d^7}+\frac {743 b^2 e^6 n^2 x}{3780 d^6}-\frac {533 b^2 e^5 n^2 x^{4/3}}{5040 d^5}+\frac {73 b^2 e^4 n^2 x^{5/3}}{1260 d^4}-\frac {5 b^2 e^3 n^2 x^2}{168 d^3}+\frac {b^2 e^2 n^2 x^{7/3}}{84 d^2}-\frac {481 b^2 e^9 n^2 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{420 d^9}-\frac {2 b e^8 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^9}+\frac {b e^7 n x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^7}-\frac {2 b e^6 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^6}+\frac {b e^5 n x^{4/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{6 d^5}-\frac {2 b e^4 n x^{5/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{15 d^4}+\frac {b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^3}-\frac {2 b e^2 n x^{7/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{21 d^2}+\frac {b e n x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 d}+\frac {e^9 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{3 d^9}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {2 b e^9 n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \log \left (-\frac {e}{d \sqrt [3]{x}}\right )}{3 d^9}-\frac {761 b^2 e^9 n^2 \log (x)}{1260 d^9}-\frac {2 b^2 e^9 n^2 \text {Li}_2\left (1+\frac {e}{d \sqrt [3]{x}}\right )}{3 d^9}\\ \end {align*}

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Mathematica [A]
time = 1.06, size = 495, normalized size = 0.87 \begin {gather*} -\frac {-5040 d^9 x^3 \left (a-b n \log \left (d+\frac {e}{\sqrt [3]{x}}\right )+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2+12 b n \left (a-b n \log \left (d+\frac {e}{\sqrt [3]{x}}\right )+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \left (d e \sqrt [3]{x} \left (840 e^7-420 d e^6 \sqrt [3]{x}+280 d^2 e^5 x^{2/3}-210 d^3 e^4 x+168 d^4 e^3 x^{4/3}-140 d^5 e^2 x^{5/3}+120 d^6 e x^2-105 d^7 x^{7/3}\right )-840 \left (e^9+d^9 x^3\right ) \log \left (d+\frac {e}{\sqrt [3]{x}}\right )+840 e^9 \log \left (\frac {e}{\sqrt [3]{x}}\right )\right )+b^2 n^2 \left (-5040 \left (e^9+d^9 x^3\right ) \log ^2\left (d+\frac {e}{\sqrt [3]{x}}\right )-e^2 \left (17316 d e^6 \sqrt [3]{x}-6138 d^2 e^5 x^{2/3}+2972 d^3 e^4 x-1599 d^4 e^3 x^{4/3}+876 d^5 e^2 x^{5/3}-450 d^6 e x^2+180 d^7 x^{7/3}+27396 e^7 \log \left (-\frac {e}{d \sqrt [3]{x}}\right )\right )+12 e \log \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (2283 e^8+840 d e^7 \sqrt [3]{x}-420 d^2 e^6 x^{2/3}+280 d^3 e^5 x-210 d^4 e^4 x^{4/3}+168 d^5 e^3 x^{5/3}-140 d^6 e^2 x^2+120 d^7 e x^{7/3}-105 d^8 x^{8/3}+840 e^8 \log \left (-\frac {e}{d \sqrt [3]{x}}\right )\right )+10080 e^9 \text {Li}_2\left (1+\frac {e}{d \sqrt [3]{x}}\right )\right )}{15120 d^9} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/15120*(-5040*d^9*x^3*(a - b*n*Log[d + e/x^(1/3)] + b*Log[c*(d + e/x^(1/3))^n])^2 + 12*b*n*(a - b*n*Log[d +
e/x^(1/3)] + b*Log[c*(d + e/x^(1/3))^n])*(d*e*x^(1/3)*(840*e^7 - 420*d*e^6*x^(1/3) + 280*d^2*e^5*x^(2/3) - 210
*d^3*e^4*x + 168*d^4*e^3*x^(4/3) - 140*d^5*e^2*x^(5/3) + 120*d^6*e*x^2 - 105*d^7*x^(7/3)) - 840*(e^9 + d^9*x^3
)*Log[d + e/x^(1/3)] + 840*e^9*Log[e/x^(1/3)]) + b^2*n^2*(-5040*(e^9 + d^9*x^3)*Log[d + e/x^(1/3)]^2 - e^2*(17
316*d*e^6*x^(1/3) - 6138*d^2*e^5*x^(2/3) + 2972*d^3*e^4*x - 1599*d^4*e^3*x^(4/3) + 876*d^5*e^2*x^(5/3) - 450*d
^6*e*x^2 + 180*d^7*x^(7/3) + 27396*e^7*Log[-(e/(d*x^(1/3)))]) + 12*e*Log[d + e/x^(1/3)]*(2283*e^8 + 840*d*e^7*
x^(1/3) - 420*d^2*e^6*x^(2/3) + 280*d^3*e^5*x - 210*d^4*e^4*x^(4/3) + 168*d^5*e^3*x^(5/3) - 140*d^6*e^2*x^2 +
120*d^7*e*x^(7/3) - 105*d^8*x^(8/3) + 840*e^8*Log[-(e/(d*x^(1/3)))]) + 10080*e^9*PolyLog[2, 1 + e/(d*x^(1/3))]
))/d^9

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int x^{2} \left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {1}{3}}}\right )^{n}\right )\right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*(d+e/x^(1/3))^n))^2,x, algorithm="maxima")

[Out]

1/3*b^2*x^3*log((d*x^(1/3) + e)^n)^2 - integrate(-1/9*(9*(b^2*log(c)^2 + 2*a*b*log(c) + a^2)*x^(8/3)*e + 9*(b^
2*d*log(c)^2 + 2*a*b*d*log(c) + a^2*d)*x^3 + 9*(b^2*d*x^3 + b^2*x^(8/3)*e)*log(x^(1/3*n))^2 - 2*(b^2*d*n*x^3 -
9*(b^2*log(c) + a*b)*x^(8/3)*e - 9*(b^2*d*log(c) + a*b*d)*x^3 + 9*(b^2*d*x^3 + b^2*x^(8/3)*e)*log(x^(1/3*n)))
*log((d*x^(1/3) + e)^n) - 18*((b^2*log(c) + a*b)*x^(8/3)*e + (b^2*d*log(c) + a*b*d)*x^3)*log(x^(1/3*n)))/(d*x
+ x^(2/3)*e), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*(d+e/x^(1/3))^n))^2,x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*(d+e/x^(1/3))^n))^2,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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