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

Optimal. Leaf size=1835 \[ \text{result too large to display} \]

[Out]

(-99*b^3*d^10*n^3*(d + e*x^(1/3))^2)/(8*e^12) + (110*b^3*d^9*n^3*(d + e*x^(1/3))^3)/(9*e^12) - (1485*b^3*d^8*n
^3*(d + e*x^(1/3))^4)/(128*e^12) + (1188*b^3*d^7*n^3*(d + e*x^(1/3))^5)/(125*e^12) - (77*b^3*d^6*n^3*(d + e*x^
(1/3))^6)/(12*e^12) + (1188*b^3*d^5*n^3*(d + e*x^(1/3))^7)/(343*e^12) - (1485*b^3*d^4*n^3*(d + e*x^(1/3))^8)/(
1024*e^12) + (110*b^3*d^3*n^3*(d + e*x^(1/3))^9)/(243*e^12) - (99*b^3*d^2*n^3*(d + e*x^(1/3))^10)/(1000*e^12)
+ (18*b^3*d*n^3*(d + e*x^(1/3))^11)/(1331*e^12) - (b^3*n^3*(d + e*x^(1/3))^12)/(1152*e^12) - (18*a*b^2*d^11*n^
2*x^(1/3))/e^11 + (18*b^3*d^11*n^3*x^(1/3))/e^11 - (18*b^3*d^11*n^2*(d + e*x^(1/3))*Log[c*(d + e*x^(1/3))^n])/
e^12 + (99*b^2*d^10*n^2*(d + e*x^(1/3))^2*(a + b*Log[c*(d + e*x^(1/3))^n]))/(4*e^12) - (110*b^2*d^9*n^2*(d + e
*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^n]))/(3*e^12) + (1485*b^2*d^8*n^2*(d + e*x^(1/3))^4*(a + b*Log[c*(d +
e*x^(1/3))^n]))/(32*e^12) - (1188*b^2*d^7*n^2*(d + e*x^(1/3))^5*(a + b*Log[c*(d + e*x^(1/3))^n]))/(25*e^12) +
(77*b^2*d^6*n^2*(d + e*x^(1/3))^6*(a + b*Log[c*(d + e*x^(1/3))^n]))/(2*e^12) - (1188*b^2*d^5*n^2*(d + e*x^(1/
3))^7*(a + b*Log[c*(d + e*x^(1/3))^n]))/(49*e^12) + (1485*b^2*d^4*n^2*(d + e*x^(1/3))^8*(a + b*Log[c*(d + e*x^
(1/3))^n]))/(128*e^12) - (110*b^2*d^3*n^2*(d + e*x^(1/3))^9*(a + b*Log[c*(d + e*x^(1/3))^n]))/(27*e^12) + (99*
b^2*d^2*n^2*(d + e*x^(1/3))^10*(a + b*Log[c*(d + e*x^(1/3))^n]))/(100*e^12) - (18*b^2*d*n^2*(d + e*x^(1/3))^11
*(a + b*Log[c*(d + e*x^(1/3))^n]))/(121*e^12) + (b^2*n^2*(d + e*x^(1/3))^12*(a + b*Log[c*(d + e*x^(1/3))^n]))/
(96*e^12) + (9*b*d^11*n*(d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/e^12 - (99*b*d^10*n*(d + e*x^(1/3)
)^2*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(4*e^12) + (55*b*d^9*n*(d + e*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^
n])^2)/e^12 - (1485*b*d^8*n*(d + e*x^(1/3))^4*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(16*e^12) + (594*b*d^7*n*(d
+ e*x^(1/3))^5*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(5*e^12) - (231*b*d^6*n*(d + e*x^(1/3))^6*(a + b*Log[c*(d +
e*x^(1/3))^n])^2)/(2*e^12) + (594*b*d^5*n*(d + e*x^(1/3))^7*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(7*e^12) - (1
485*b*d^4*n*(d + e*x^(1/3))^8*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(32*e^12) + (55*b*d^3*n*(d + e*x^(1/3))^9*(a
+ b*Log[c*(d + e*x^(1/3))^n])^2)/(3*e^12) - (99*b*d^2*n*(d + e*x^(1/3))^10*(a + b*Log[c*(d + e*x^(1/3))^n])^2
)/(20*e^12) + (9*b*d*n*(d + e*x^(1/3))^11*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(11*e^12) - (b*n*(d + e*x^(1/3))
^12*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(16*e^12) - (3*d^11*(d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n])^3
)/e^12 + (33*d^10*(d + e*x^(1/3))^2*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/(2*e^12) - (55*d^9*(d + e*x^(1/3))^3*(
a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^12 + (495*d^8*(d + e*x^(1/3))^4*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/(4*e^
12) - (198*d^7*(d + e*x^(1/3))^5*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^12 + (231*d^6*(d + e*x^(1/3))^6*(a + b*
Log[c*(d + e*x^(1/3))^n])^3)/e^12 - (198*d^5*(d + e*x^(1/3))^7*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^12 + (495
*d^4*(d + e*x^(1/3))^8*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/(4*e^12) - (55*d^3*(d + e*x^(1/3))^9*(a + b*Log[c*(
d + e*x^(1/3))^n])^3)/e^12 + (33*d^2*(d + e*x^(1/3))^10*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/(2*e^12) - (3*d*(d
+ e*x^(1/3))^11*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^12 + ((d + e*x^(1/3))^12*(a + b*Log[c*(d + e*x^(1/3))^n
])^3)/(4*e^12)

________________________________________________________________________________________

Rubi [A]  time = 2.26676, antiderivative size = 1835, normalized size of antiderivative = 1., number of steps used = 52, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2454, 2401, 2389, 2296, 2295, 2390, 2305, 2304} \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-99*b^3*d^10*n^3*(d + e*x^(1/3))^2)/(8*e^12) + (110*b^3*d^9*n^3*(d + e*x^(1/3))^3)/(9*e^12) - (1485*b^3*d^8*n
^3*(d + e*x^(1/3))^4)/(128*e^12) + (1188*b^3*d^7*n^3*(d + e*x^(1/3))^5)/(125*e^12) - (77*b^3*d^6*n^3*(d + e*x^
(1/3))^6)/(12*e^12) + (1188*b^3*d^5*n^3*(d + e*x^(1/3))^7)/(343*e^12) - (1485*b^3*d^4*n^3*(d + e*x^(1/3))^8)/(
1024*e^12) + (110*b^3*d^3*n^3*(d + e*x^(1/3))^9)/(243*e^12) - (99*b^3*d^2*n^3*(d + e*x^(1/3))^10)/(1000*e^12)
+ (18*b^3*d*n^3*(d + e*x^(1/3))^11)/(1331*e^12) - (b^3*n^3*(d + e*x^(1/3))^12)/(1152*e^12) - (18*a*b^2*d^11*n^
2*x^(1/3))/e^11 + (18*b^3*d^11*n^3*x^(1/3))/e^11 - (18*b^3*d^11*n^2*(d + e*x^(1/3))*Log[c*(d + e*x^(1/3))^n])/
e^12 + (99*b^2*d^10*n^2*(d + e*x^(1/3))^2*(a + b*Log[c*(d + e*x^(1/3))^n]))/(4*e^12) - (110*b^2*d^9*n^2*(d + e
*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^n]))/(3*e^12) + (1485*b^2*d^8*n^2*(d + e*x^(1/3))^4*(a + b*Log[c*(d +
e*x^(1/3))^n]))/(32*e^12) - (1188*b^2*d^7*n^2*(d + e*x^(1/3))^5*(a + b*Log[c*(d + e*x^(1/3))^n]))/(25*e^12) +
(77*b^2*d^6*n^2*(d + e*x^(1/3))^6*(a + b*Log[c*(d + e*x^(1/3))^n]))/(2*e^12) - (1188*b^2*d^5*n^2*(d + e*x^(1/
3))^7*(a + b*Log[c*(d + e*x^(1/3))^n]))/(49*e^12) + (1485*b^2*d^4*n^2*(d + e*x^(1/3))^8*(a + b*Log[c*(d + e*x^
(1/3))^n]))/(128*e^12) - (110*b^2*d^3*n^2*(d + e*x^(1/3))^9*(a + b*Log[c*(d + e*x^(1/3))^n]))/(27*e^12) + (99*
b^2*d^2*n^2*(d + e*x^(1/3))^10*(a + b*Log[c*(d + e*x^(1/3))^n]))/(100*e^12) - (18*b^2*d*n^2*(d + e*x^(1/3))^11
*(a + b*Log[c*(d + e*x^(1/3))^n]))/(121*e^12) + (b^2*n^2*(d + e*x^(1/3))^12*(a + b*Log[c*(d + e*x^(1/3))^n]))/
(96*e^12) + (9*b*d^11*n*(d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/e^12 - (99*b*d^10*n*(d + e*x^(1/3)
)^2*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(4*e^12) + (55*b*d^9*n*(d + e*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^
n])^2)/e^12 - (1485*b*d^8*n*(d + e*x^(1/3))^4*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(16*e^12) + (594*b*d^7*n*(d
+ e*x^(1/3))^5*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(5*e^12) - (231*b*d^6*n*(d + e*x^(1/3))^6*(a + b*Log[c*(d +
e*x^(1/3))^n])^2)/(2*e^12) + (594*b*d^5*n*(d + e*x^(1/3))^7*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(7*e^12) - (1
485*b*d^4*n*(d + e*x^(1/3))^8*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(32*e^12) + (55*b*d^3*n*(d + e*x^(1/3))^9*(a
+ b*Log[c*(d + e*x^(1/3))^n])^2)/(3*e^12) - (99*b*d^2*n*(d + e*x^(1/3))^10*(a + b*Log[c*(d + e*x^(1/3))^n])^2
)/(20*e^12) + (9*b*d*n*(d + e*x^(1/3))^11*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(11*e^12) - (b*n*(d + e*x^(1/3))
^12*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(16*e^12) - (3*d^11*(d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n])^3
)/e^12 + (33*d^10*(d + e*x^(1/3))^2*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/(2*e^12) - (55*d^9*(d + e*x^(1/3))^3*(
a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^12 + (495*d^8*(d + e*x^(1/3))^4*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/(4*e^
12) - (198*d^7*(d + e*x^(1/3))^5*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^12 + (231*d^6*(d + e*x^(1/3))^6*(a + b*
Log[c*(d + e*x^(1/3))^n])^3)/e^12 - (198*d^5*(d + e*x^(1/3))^7*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^12 + (495
*d^4*(d + e*x^(1/3))^8*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/(4*e^12) - (55*d^3*(d + e*x^(1/3))^9*(a + b*Log[c*(
d + e*x^(1/3))^n])^3)/e^12 + (33*d^2*(d + e*x^(1/3))^10*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/(2*e^12) - (3*d*(d
+ e*x^(1/3))^11*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^12 + ((d + e*x^(1/3))^12*(a + b*Log[c*(d + e*x^(1/3))^n
])^3)/(4*e^12)

Rule 2454

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])

Rule 2401

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

Rule 2389

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

Rule 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2390

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

Rule 2305

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

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rubi steps

\begin{align*} \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx &=3 \operatorname{Subst}\left (\int x^{11} \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (-\frac{d^{11} \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^{11}}+\frac{11 d^{10} (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^{11}}-\frac{55 d^9 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^{11}}+\frac{165 d^8 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^{11}}-\frac{330 d^7 (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^{11}}+\frac{462 d^6 (d+e x)^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^{11}}-\frac{462 d^5 (d+e x)^6 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^{11}}+\frac{330 d^4 (d+e x)^7 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^{11}}-\frac{165 d^3 (d+e x)^8 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^{11}}+\frac{55 d^2 (d+e x)^9 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^{11}}-\frac{11 d (d+e x)^{10} \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^{11}}+\frac{(d+e x)^{11} \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^{11}}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{3 \operatorname{Subst}\left (\int (d+e x)^{11} \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right )}{e^{11}}-\frac{(33 d) \operatorname{Subst}\left (\int (d+e x)^{10} \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right )}{e^{11}}+\frac{\left (165 d^2\right ) \operatorname{Subst}\left (\int (d+e x)^9 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right )}{e^{11}}-\frac{\left (495 d^3\right ) \operatorname{Subst}\left (\int (d+e x)^8 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right )}{e^{11}}+\frac{\left (990 d^4\right ) \operatorname{Subst}\left (\int (d+e x)^7 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right )}{e^{11}}-\frac{\left (1386 d^5\right ) \operatorname{Subst}\left (\int (d+e x)^6 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right )}{e^{11}}+\frac{\left (1386 d^6\right ) \operatorname{Subst}\left (\int (d+e x)^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right )}{e^{11}}-\frac{\left (990 d^7\right ) \operatorname{Subst}\left (\int (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right )}{e^{11}}+\frac{\left (495 d^8\right ) \operatorname{Subst}\left (\int (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right )}{e^{11}}-\frac{\left (165 d^9\right ) \operatorname{Subst}\left (\int (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right )}{e^{11}}+\frac{\left (33 d^{10}\right ) \operatorname{Subst}\left (\int (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right )}{e^{11}}-\frac{\left (3 d^{11}\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right )}{e^{11}}\\ &=\frac{3 \operatorname{Subst}\left (\int x^{11} \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}-\frac{(33 d) \operatorname{Subst}\left (\int x^{10} \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}+\frac{\left (165 d^2\right ) \operatorname{Subst}\left (\int x^9 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}-\frac{\left (495 d^3\right ) \operatorname{Subst}\left (\int x^8 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}+\frac{\left (990 d^4\right ) \operatorname{Subst}\left (\int x^7 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}-\frac{\left (1386 d^5\right ) \operatorname{Subst}\left (\int x^6 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}+\frac{\left (1386 d^6\right ) \operatorname{Subst}\left (\int x^5 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}-\frac{\left (990 d^7\right ) \operatorname{Subst}\left (\int x^4 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}+\frac{\left (495 d^8\right ) \operatorname{Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}-\frac{\left (165 d^9\right ) \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}+\frac{\left (33 d^{10}\right ) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}-\frac{\left (3 d^{11}\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}\\ &=-\frac{3 d^{11} \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^{12}}+\frac{33 d^{10} \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 e^{12}}-\frac{55 d^9 \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^{12}}+\frac{495 d^8 \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{4 e^{12}}-\frac{198 d^7 \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^{12}}+\frac{231 d^6 \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^{12}}-\frac{198 d^5 \left (d+e \sqrt [3]{x}\right )^7 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^{12}}+\frac{495 d^4 \left (d+e \sqrt [3]{x}\right )^8 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{4 e^{12}}-\frac{55 d^3 \left (d+e \sqrt [3]{x}\right )^9 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^{12}}+\frac{33 d^2 \left (d+e \sqrt [3]{x}\right )^{10} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 e^{12}}-\frac{3 d \left (d+e \sqrt [3]{x}\right )^{11} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^{12}}+\frac{\left (d+e \sqrt [3]{x}\right )^{12} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{4 e^{12}}-\frac{(3 b n) \operatorname{Subst}\left (\int x^{11} \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt [3]{x}\right )}{4 e^{12}}+\frac{(9 b d n) \operatorname{Subst}\left (\int x^{10} \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}-\frac{\left (99 b d^2 n\right ) \operatorname{Subst}\left (\int x^9 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt [3]{x}\right )}{2 e^{12}}+\frac{\left (165 b d^3 n\right ) \operatorname{Subst}\left (\int x^8 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}-\frac{\left (1485 b d^4 n\right ) \operatorname{Subst}\left (\int x^7 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt [3]{x}\right )}{4 e^{12}}+\frac{\left (594 b d^5 n\right ) \operatorname{Subst}\left (\int x^6 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}-\frac{\left (693 b d^6 n\right ) \operatorname{Subst}\left (\int x^5 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}+\frac{\left (594 b d^7 n\right ) \operatorname{Subst}\left (\int x^4 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}-\frac{\left (1485 b d^8 n\right ) \operatorname{Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt [3]{x}\right )}{4 e^{12}}+\frac{\left (165 b d^9 n\right ) \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}-\frac{\left (99 b d^{10} n\right ) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt [3]{x}\right )}{2 e^{12}}+\frac{\left (9 b d^{11} n\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}\\ &=\frac{9 b d^{11} n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{e^{12}}-\frac{99 b d^{10} n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{4 e^{12}}+\frac{55 b d^9 n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{e^{12}}-\frac{1485 b d^8 n \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{16 e^{12}}+\frac{594 b d^7 n \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{5 e^{12}}-\frac{231 b d^6 n \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 e^{12}}+\frac{594 b d^5 n \left (d+e \sqrt [3]{x}\right )^7 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{7 e^{12}}-\frac{1485 b d^4 n \left (d+e \sqrt [3]{x}\right )^8 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{32 e^{12}}+\frac{55 b d^3 n \left (d+e \sqrt [3]{x}\right )^9 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{3 e^{12}}-\frac{99 b d^2 n \left (d+e \sqrt [3]{x}\right )^{10} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{20 e^{12}}+\frac{9 b d n \left (d+e \sqrt [3]{x}\right )^{11} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{11 e^{12}}-\frac{b n \left (d+e \sqrt [3]{x}\right )^{12} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{16 e^{12}}-\frac{3 d^{11} \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^{12}}+\frac{33 d^{10} \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 e^{12}}-\frac{55 d^9 \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^{12}}+\frac{495 d^8 \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{4 e^{12}}-\frac{198 d^7 \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^{12}}+\frac{231 d^6 \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^{12}}-\frac{198 d^5 \left (d+e \sqrt [3]{x}\right )^7 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^{12}}+\frac{495 d^4 \left (d+e \sqrt [3]{x}\right )^8 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{4 e^{12}}-\frac{55 d^3 \left (d+e \sqrt [3]{x}\right )^9 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^{12}}+\frac{33 d^2 \left (d+e \sqrt [3]{x}\right )^{10} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 e^{12}}-\frac{3 d \left (d+e \sqrt [3]{x}\right )^{11} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^{12}}+\frac{\left (d+e \sqrt [3]{x}\right )^{12} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{4 e^{12}}+\frac{\left (b^2 n^2\right ) \operatorname{Subst}\left (\int x^{11} \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{8 e^{12}}-\frac{\left (18 b^2 d n^2\right ) \operatorname{Subst}\left (\int x^{10} \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{11 e^{12}}+\frac{\left (99 b^2 d^2 n^2\right ) \operatorname{Subst}\left (\int x^9 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{10 e^{12}}-\frac{\left (110 b^2 d^3 n^2\right ) \operatorname{Subst}\left (\int x^8 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{3 e^{12}}+\frac{\left (1485 b^2 d^4 n^2\right ) \operatorname{Subst}\left (\int x^7 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{16 e^{12}}-\frac{\left (1188 b^2 d^5 n^2\right ) \operatorname{Subst}\left (\int x^6 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{7 e^{12}}+\frac{\left (231 b^2 d^6 n^2\right ) \operatorname{Subst}\left (\int x^5 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}-\frac{\left (1188 b^2 d^7 n^2\right ) \operatorname{Subst}\left (\int x^4 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{5 e^{12}}+\frac{\left (1485 b^2 d^8 n^2\right ) \operatorname{Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{8 e^{12}}-\frac{\left (110 b^2 d^9 n^2\right ) \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}+\frac{\left (99 b^2 d^{10} n^2\right ) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{2 e^{12}}-\frac{\left (18 b^2 d^{11} n^2\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}\\ &=-\frac{99 b^3 d^{10} n^3 \left (d+e \sqrt [3]{x}\right )^2}{8 e^{12}}+\frac{110 b^3 d^9 n^3 \left (d+e \sqrt [3]{x}\right )^3}{9 e^{12}}-\frac{1485 b^3 d^8 n^3 \left (d+e \sqrt [3]{x}\right )^4}{128 e^{12}}+\frac{1188 b^3 d^7 n^3 \left (d+e \sqrt [3]{x}\right )^5}{125 e^{12}}-\frac{77 b^3 d^6 n^3 \left (d+e \sqrt [3]{x}\right )^6}{12 e^{12}}+\frac{1188 b^3 d^5 n^3 \left (d+e \sqrt [3]{x}\right )^7}{343 e^{12}}-\frac{1485 b^3 d^4 n^3 \left (d+e \sqrt [3]{x}\right )^8}{1024 e^{12}}+\frac{110 b^3 d^3 n^3 \left (d+e \sqrt [3]{x}\right )^9}{243 e^{12}}-\frac{99 b^3 d^2 n^3 \left (d+e \sqrt [3]{x}\right )^{10}}{1000 e^{12}}+\frac{18 b^3 d n^3 \left (d+e \sqrt [3]{x}\right )^{11}}{1331 e^{12}}-\frac{b^3 n^3 \left (d+e \sqrt [3]{x}\right )^{12}}{1152 e^{12}}-\frac{18 a b^2 d^{11} n^2 \sqrt [3]{x}}{e^{11}}+\frac{99 b^2 d^{10} n^2 \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{4 e^{12}}-\frac{110 b^2 d^9 n^2 \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^{12}}+\frac{1485 b^2 d^8 n^2 \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{32 e^{12}}-\frac{1188 b^2 d^7 n^2 \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{25 e^{12}}+\frac{77 b^2 d^6 n^2 \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{2 e^{12}}-\frac{1188 b^2 d^5 n^2 \left (d+e \sqrt [3]{x}\right )^7 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{49 e^{12}}+\frac{1485 b^2 d^4 n^2 \left (d+e \sqrt [3]{x}\right )^8 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{128 e^{12}}-\frac{110 b^2 d^3 n^2 \left (d+e \sqrt [3]{x}\right )^9 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{27 e^{12}}+\frac{99 b^2 d^2 n^2 \left (d+e \sqrt [3]{x}\right )^{10} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{100 e^{12}}-\frac{18 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^{11} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{121 e^{12}}+\frac{b^2 n^2 \left (d+e \sqrt [3]{x}\right )^{12} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{96 e^{12}}+\frac{9 b d^{11} n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{e^{12}}-\frac{99 b d^{10} n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{4 e^{12}}+\frac{55 b d^9 n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{e^{12}}-\frac{1485 b d^8 n \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{16 e^{12}}+\frac{594 b d^7 n \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{5 e^{12}}-\frac{231 b d^6 n \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 e^{12}}+\frac{594 b d^5 n \left (d+e \sqrt [3]{x}\right )^7 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{7 e^{12}}-\frac{1485 b d^4 n \left (d+e \sqrt [3]{x}\right )^8 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{32 e^{12}}+\frac{55 b d^3 n \left (d+e \sqrt [3]{x}\right )^9 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{3 e^{12}}-\frac{99 b d^2 n \left (d+e \sqrt [3]{x}\right )^{10} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{20 e^{12}}+\frac{9 b d n \left (d+e \sqrt [3]{x}\right )^{11} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{11 e^{12}}-\frac{b n \left (d+e \sqrt [3]{x}\right )^{12} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{16 e^{12}}-\frac{3 d^{11} \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^{12}}+\frac{33 d^{10} \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 e^{12}}-\frac{55 d^9 \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^{12}}+\frac{495 d^8 \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{4 e^{12}}-\frac{198 d^7 \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^{12}}+\frac{231 d^6 \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^{12}}-\frac{198 d^5 \left (d+e \sqrt [3]{x}\right )^7 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^{12}}+\frac{495 d^4 \left (d+e \sqrt [3]{x}\right )^8 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{4 e^{12}}-\frac{55 d^3 \left (d+e \sqrt [3]{x}\right )^9 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^{12}}+\frac{33 d^2 \left (d+e \sqrt [3]{x}\right )^{10} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 e^{12}}-\frac{3 d \left (d+e \sqrt [3]{x}\right )^{11} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^{12}}+\frac{\left (d+e \sqrt [3]{x}\right )^{12} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{4 e^{12}}-\frac{\left (18 b^3 d^{11} n^2\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}\\ &=-\frac{99 b^3 d^{10} n^3 \left (d+e \sqrt [3]{x}\right )^2}{8 e^{12}}+\frac{110 b^3 d^9 n^3 \left (d+e \sqrt [3]{x}\right )^3}{9 e^{12}}-\frac{1485 b^3 d^8 n^3 \left (d+e \sqrt [3]{x}\right )^4}{128 e^{12}}+\frac{1188 b^3 d^7 n^3 \left (d+e \sqrt [3]{x}\right )^5}{125 e^{12}}-\frac{77 b^3 d^6 n^3 \left (d+e \sqrt [3]{x}\right )^6}{12 e^{12}}+\frac{1188 b^3 d^5 n^3 \left (d+e \sqrt [3]{x}\right )^7}{343 e^{12}}-\frac{1485 b^3 d^4 n^3 \left (d+e \sqrt [3]{x}\right )^8}{1024 e^{12}}+\frac{110 b^3 d^3 n^3 \left (d+e \sqrt [3]{x}\right )^9}{243 e^{12}}-\frac{99 b^3 d^2 n^3 \left (d+e \sqrt [3]{x}\right )^{10}}{1000 e^{12}}+\frac{18 b^3 d n^3 \left (d+e \sqrt [3]{x}\right )^{11}}{1331 e^{12}}-\frac{b^3 n^3 \left (d+e \sqrt [3]{x}\right )^{12}}{1152 e^{12}}-\frac{18 a b^2 d^{11} n^2 \sqrt [3]{x}}{e^{11}}+\frac{18 b^3 d^{11} n^3 \sqrt [3]{x}}{e^{11}}-\frac{18 b^3 d^{11} n^2 \left (d+e \sqrt [3]{x}\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{e^{12}}+\frac{99 b^2 d^{10} n^2 \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{4 e^{12}}-\frac{110 b^2 d^9 n^2 \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^{12}}+\frac{1485 b^2 d^8 n^2 \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{32 e^{12}}-\frac{1188 b^2 d^7 n^2 \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{25 e^{12}}+\frac{77 b^2 d^6 n^2 \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{2 e^{12}}-\frac{1188 b^2 d^5 n^2 \left (d+e \sqrt [3]{x}\right )^7 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{49 e^{12}}+\frac{1485 b^2 d^4 n^2 \left (d+e \sqrt [3]{x}\right )^8 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{128 e^{12}}-\frac{110 b^2 d^3 n^2 \left (d+e \sqrt [3]{x}\right )^9 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{27 e^{12}}+\frac{99 b^2 d^2 n^2 \left (d+e \sqrt [3]{x}\right )^{10} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{100 e^{12}}-\frac{18 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^{11} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{121 e^{12}}+\frac{b^2 n^2 \left (d+e \sqrt [3]{x}\right )^{12} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{96 e^{12}}+\frac{9 b d^{11} n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{e^{12}}-\frac{99 b d^{10} n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{4 e^{12}}+\frac{55 b d^9 n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{e^{12}}-\frac{1485 b d^8 n \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{16 e^{12}}+\frac{594 b d^7 n \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{5 e^{12}}-\frac{231 b d^6 n \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 e^{12}}+\frac{594 b d^5 n \left (d+e \sqrt [3]{x}\right )^7 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{7 e^{12}}-\frac{1485 b d^4 n \left (d+e \sqrt [3]{x}\right )^8 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{32 e^{12}}+\frac{55 b d^3 n \left (d+e \sqrt [3]{x}\right )^9 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{3 e^{12}}-\frac{99 b d^2 n \left (d+e \sqrt [3]{x}\right )^{10} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{20 e^{12}}+\frac{9 b d n \left (d+e \sqrt [3]{x}\right )^{11} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{11 e^{12}}-\frac{b n \left (d+e \sqrt [3]{x}\right )^{12} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{16 e^{12}}-\frac{3 d^{11} \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^{12}}+\frac{33 d^{10} \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 e^{12}}-\frac{55 d^9 \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^{12}}+\frac{495 d^8 \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{4 e^{12}}-\frac{198 d^7 \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^{12}}+\frac{231 d^6 \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^{12}}-\frac{198 d^5 \left (d+e \sqrt [3]{x}\right )^7 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^{12}}+\frac{495 d^4 \left (d+e \sqrt [3]{x}\right )^8 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{4 e^{12}}-\frac{55 d^3 \left (d+e \sqrt [3]{x}\right )^9 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^{12}}+\frac{33 d^2 \left (d+e \sqrt [3]{x}\right )^{10} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 e^{12}}-\frac{3 d \left (d+e \sqrt [3]{x}\right )^{11} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^{12}}+\frac{\left (d+e \sqrt [3]{x}\right )^{12} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{4 e^{12}}\\ \end{align*}

Mathematica [A]  time = 1.25083, size = 1009, normalized size = 0.55 \[ \frac{-3550000608000 b^3 \left (d^{12}-e^{12} x^4\right ) \log ^3\left (c \left (d+e \sqrt [3]{x}\right )^n\right )-384199200 b^2 \left (27720 a \left (d^{12}-e^{12} x^4\right )+b n \left (-86021 d^{12}-27720 e \sqrt [3]{x} d^{11}+13860 e^2 x^{2/3} d^{10}-9240 e^3 x d^9+6930 e^4 x^{4/3} d^8-5544 e^5 x^{5/3} d^7+4620 e^6 x^2 d^6-3960 e^7 x^{7/3} d^5+3465 e^8 x^{8/3} d^4-3080 e^9 x^3 d^3+2772 e^{10} x^{10/3} d^2-2520 e^{11} x^{11/3} d+2310 e^{12} x^4\right )\right ) \log ^2\left (c \left (d+e \sqrt [3]{x}\right )^n\right )-27720 b \left (384199200 \left (d^{12}-e^{12} x^4\right ) a^2-27720 b n \left (86021 d^{12}+27720 e \sqrt [3]{x} d^{11}-13860 e^2 x^{2/3} d^{10}+9240 e^3 x d^9-6930 e^4 x^{4/3} d^8+5544 e^5 x^{5/3} d^7-4620 e^6 x^2 d^6+3960 e^7 x^{7/3} d^5-3465 e^8 x^{8/3} d^4+3080 e^9 x^3 d^3-2772 e^{10} x^{10/3} d^2+2520 e^{11} x^{11/3} d-2310 e^{12} x^4\right ) a+b^2 n^2 \left (4301068993 d^{12}[phone] e \sqrt [3]{x} d^{11}-808051860 e^2 x^{2/3} d^{10}[phone] e^3 x d^9-243942930 e^4 x^{4/3} d^8+156734424 e^5 x^{5/3} d^7-104998740 e^6 x^2 d^6+71703720 e^7 x^{7/3} d^5-49019355 e^8 x^{8/3} d^4+32900560 e^9 x^3 d^3-21072744 e^{10} x^{10/3} d^2+12171600 e^{11} x^{11/3} d-5336100 e^{12} x^4\right )\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )+e \sqrt [3]{x} \left (3550000608000 a^3 x^{11/3} e^{11}+b^3 n^3 \left (119225632485960 d^{11}-26563616859780 e \sqrt [3]{x} d^{10}[phone] e^2 x^{2/3} d^9-4836309598890 e^3 x d^8+2516628075192 e^4 x^{4/3} d^7-1373077023780 e^5 x^{5/3} d^6+761128152840 e^6 x^2 d^5-417533743935 e^7 x^{7/3} d^4+220161492320 e^8 x^{8/3} d^3-106944990768 e^9 x^3 d^2+44119404000 e^{10} x^{10/3} d-12326391000 e^{11} x^{11/3}\right )-27720 a b^2 n^2 \left (2384502120 d^{11}-808051860 e \sqrt [3]{x} d^{10}[phone] e^2 x^{2/3} d^9-243942930 e^3 x d^8+156734424 e^4 x^{4/3} d^7-104998740 e^5 x^{5/3} d^6+71703720 e^6 x^2 d^5-49019355 e^7 x^{7/3} d^4+32900560 e^8 x^{8/3} d^3-21072744 e^9 x^3 d^2+12171600 e^{10} x^{10/3} d-5336100 e^{11} x^{11/3}\right )[phone] a^2 b n \left (27720 d^{11}-13860 e \sqrt [3]{x} d^{10}+9240 e^2 x^{2/3} d^9-6930 e^3 x d^8+5544 e^4 x^{4/3} d^7-4620 e^5 x^{5/3} d^6+3960 e^6 x^2 d^5-3465 e^7 x^{7/3} d^4+3080 e^8 x^{8/3} d^3-2772 e^9 x^3 d^2+2520 e^{10} x^{10/3} d-2310 e^{11} x^{11/3}\right )\right )}{14200002432000 e^{12}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e*x^(1/3)*(3550000608000*a^3*e^11*x^(11/3) + b^3*n^3*(119225632485960*d^11 - 26563616859780*d^10*e*x^(1/3) +
10242678720120*d^9*e^2*x^(2/3) - 4836309598890*d^8*e^3*x + 2516628075192*d^7*e^4*x^(4/3) - 1373077023780*d^6*e
^5*x^(5/3) + 761128152840*d^5*e^6*x^2 - 417533743935*d^4*e^7*x^(7/3) + 220161492320*d^3*e^8*x^(8/3) - 10694499
0768*d^2*e^9*x^3 + 44119404000*d*e^10*x^(10/3) - 12326391000*e^11*x^(11/3)) - 27720*a*b^2*n^2*(2384502120*d^11
- 808051860*d^10*e*x^(1/3) + 410634840*d^9*e^2*x^(2/3) - 243942930*d^8*e^3*x + 156734424*d^7*e^4*x^(4/3) - 10
4998740*d^6*e^5*x^(5/3) + 71703720*d^5*e^6*x^2 - 49019355*d^4*e^7*x^(7/3) + 32900560*d^3*e^8*x^(8/3) - 2107274
4*d^2*e^9*x^3 + 12171600*d*e^10*x^(10/3) - 5336100*e^11*x^(11/3)) + 384199200*a^2*b*n*(27720*d^11 - 13860*d^10
*e*x^(1/3) + 9240*d^9*e^2*x^(2/3) - 6930*d^8*e^3*x + 5544*d^7*e^4*x^(4/3) - 4620*d^6*e^5*x^(5/3) + 3960*d^5*e^
6*x^2 - 3465*d^4*e^7*x^(7/3) + 3080*d^3*e^8*x^(8/3) - 2772*d^2*e^9*x^3 + 2520*d*e^10*x^(10/3) - 2310*e^11*x^(1
1/3))) - 27720*b*(b^2*n^2*(4301068993*d^12 + 2384502120*d^11*e*x^(1/3) - 808051860*d^10*e^2*x^(2/3) + 41063484
0*d^9*e^3*x - 243942930*d^8*e^4*x^(4/3) + 156734424*d^7*e^5*x^(5/3) - 104998740*d^6*e^6*x^2 + 71703720*d^5*e^7
*x^(7/3) - 49019355*d^4*e^8*x^(8/3) + 32900560*d^3*e^9*x^3 - 21072744*d^2*e^10*x^(10/3) + 12171600*d*e^11*x^(1
1/3) - 5336100*e^12*x^4) - 27720*a*b*n*(86021*d^12 + 27720*d^11*e*x^(1/3) - 13860*d^10*e^2*x^(2/3) + 9240*d^9*
e^3*x - 6930*d^8*e^4*x^(4/3) + 5544*d^7*e^5*x^(5/3) - 4620*d^6*e^6*x^2 + 3960*d^5*e^7*x^(7/3) - 3465*d^4*e^8*x
^(8/3) + 3080*d^3*e^9*x^3 - 2772*d^2*e^10*x^(10/3) + 2520*d*e^11*x^(11/3) - 2310*e^12*x^4) + 384199200*a^2*(d^
12 - e^12*x^4))*Log[c*(d + e*x^(1/3))^n] - 384199200*b^2*(27720*a*(d^12 - e^12*x^4) + b*n*(-86021*d^12 - 27720
*d^11*e*x^(1/3) + 13860*d^10*e^2*x^(2/3) - 9240*d^9*e^3*x + 6930*d^8*e^4*x^(4/3) - 5544*d^7*e^5*x^(5/3) + 4620
*d^6*e^6*x^2 - 3960*d^5*e^7*x^(7/3) + 3465*d^4*e^8*x^(8/3) - 3080*d^3*e^9*x^3 + 2772*d^2*e^10*x^(10/3) - 2520*
d*e^11*x^(11/3) + 2310*e^12*x^4))*Log[c*(d + e*x^(1/3))^n]^2 - 3550000608000*b^3*(d^12 - e^12*x^4)*Log[c*(d +
e*x^(1/3))^n]^3)/(14200002432000*e^12)

________________________________________________________________________________________

Maple [F]  time = 0.095, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ( a+b\ln \left ( c \left ( d+e\sqrt [3]{x} \right ) ^{n} \right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.15272, size = 1436, normalized size = 0.78 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b^3*x^4*log((e*x^(1/3) + d)^n*c)^3 + 3/4*a*b^2*x^4*log((e*x^(1/3) + d)^n*c)^2 + 3/4*a^2*b*x^4*log((e*x^(1/
3) + d)^n*c) + 1/4*a^3*x^4 - 1/36960*a^2*b*e*n*(27720*d^12*log(e*x^(1/3) + d)/e^13 + (2310*e^11*x^4 - 2520*d*e
^10*x^(11/3) + 2772*d^2*e^9*x^(10/3) - 3080*d^3*e^8*x^3 + 3465*d^4*e^7*x^(8/3) - 3960*d^5*e^6*x^(7/3) + 4620*d
^6*e^5*x^2 - 5544*d^7*e^4*x^(5/3) + 6930*d^8*e^3*x^(4/3) - 9240*d^9*e^2*x + 13860*d^10*e*x^(2/3) - 27720*d^11*
x^(1/3))/e^12) - 1/512265600*(27720*e*n*(27720*d^12*log(e*x^(1/3) + d)/e^13 + (2310*e^11*x^4 - 2520*d*e^10*x^(
11/3) + 2772*d^2*e^9*x^(10/3) - 3080*d^3*e^8*x^3 + 3465*d^4*e^7*x^(8/3) - 3960*d^5*e^6*x^(7/3) + 4620*d^6*e^5*
x^2 - 5544*d^7*e^4*x^(5/3) + 6930*d^8*e^3*x^(4/3) - 9240*d^9*e^2*x + 13860*d^10*e*x^(2/3) - 27720*d^11*x^(1/3)
)/e^12)*log((e*x^(1/3) + d)^n*c) - (5336100*e^12*x^4 - 12171600*d*e^11*x^(11/3) + 21072744*d^2*e^10*x^(10/3) -
32900560*d^3*e^9*x^3 + 49019355*d^4*e^8*x^(8/3) - 71703720*d^5*e^7*x^(7/3) + 104998740*d^6*e^6*x^2 + 38419920
0*d^12*log(e*x^(1/3) + d)^2 - 156734424*d^7*e^5*x^(5/3) + 243942930*d^8*e^4*x^(4/3) - 410634840*d^9*e^3*x + 23
84502120*d^12*log(e*x^(1/3) + d) + 808051860*d^10*e^2*x^(2/3) - 2384502120*d^11*e*x^(1/3))*n^2/e^12)*a*b^2 - 1
/14200002432000*(384199200*e*n*(27720*d^12*log(e*x^(1/3) + d)/e^13 + (2310*e^11*x^4 - 2520*d*e^10*x^(11/3) + 2
772*d^2*e^9*x^(10/3) - 3080*d^3*e^8*x^3 + 3465*d^4*e^7*x^(8/3) - 3960*d^5*e^6*x^(7/3) + 4620*d^6*e^5*x^2 - 554
4*d^7*e^4*x^(5/3) + 6930*d^8*e^3*x^(4/3) - 9240*d^9*e^2*x + 13860*d^10*e*x^(2/3) - 27720*d^11*x^(1/3))/e^12)*l
og((e*x^(1/3) + d)^n*c)^2 + e*n*((12326391000*e^12*x^4 - 44119404000*d*e^11*x^(11/3) + 106944990768*d^2*e^10*x
^(10/3) - 220161492320*d^3*e^9*x^3 + 3550000608000*d^12*log(e*x^(1/3) + d)^3 + 417533743935*d^4*e^8*x^(8/3) -
761128152840*d^5*e^7*x^(7/3) + 1373077023780*d^6*e^6*x^2 + 33049199383200*d^12*log(e*x^(1/3) + d)^2 - 25166280
75192*d^7*e^5*x^(5/3) + 4836309598890*d^8*e^4*x^(4/3) - 10242678720120*d^9*e^3*x + 119225632485960*d^12*log(e*
x^(1/3) + d) + 26563616859780*d^10*e^2*x^(2/3) - 119225632485960*d^11*e*x^(1/3))*n^2/e^13 - 27720*(5336100*e^1
2*x^4 - 12171600*d*e^11*x^(11/3) + 21072744*d^2*e^10*x^(10/3) - 32900560*d^3*e^9*x^3 + 49019355*d^4*e^8*x^(8/3
) - 71703720*d^5*e^7*x^(7/3) + 104998740*d^6*e^6*x^2 + 384199200*d^12*log(e*x^(1/3) + d)^2 - 156734424*d^7*e^5
*x^(5/3) + 243942930*d^8*e^4*x^(4/3) - 410634840*d^9*e^3*x + 2384502120*d^12*log(e*x^(1/3) + d) + 808051860*d^
10*e^2*x^(2/3) - 2384502120*d^11*e*x^(1/3))*n*log((e*x^(1/3) + d)^n*c)/e^13))*b^3

________________________________________________________________________________________

Fricas [A]  time = 4.33575, size = 5488, normalized size = 2.99 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14200002432000*(3550000608000*b^3*e^12*x^4*log(c)^3 - 12326391000*(b^3*e^12*n^3 - 12*a*b^2*e^12*n^2 + 72*a^2
*b*e^12*n - 288*a^3*e^12)*x^4 + 603680*(364699*b^3*d^3*e^9*n^3 - 1510740*a*b^2*d^3*e^9*n^2 + 1960200*a^2*b*d^3
*e^9*n)*x^3 + 3550000608000*(b^3*e^12*n^3*x^4 - b^3*d^12*n^3)*log(e*x^(1/3) + d)^3 - 4620*(297202819*b^3*d^6*e
^6*n^3 - 629992440*a*b^2*d^6*e^6*n^2 + 384199200*a^2*b*d^6*e^6*n)*x^2 + 384199200*(3080*b^3*d^3*e^9*n^3*x^3 -
4620*b^3*d^6*e^6*n^3*x^2 + 9240*b^3*d^9*e^3*n^3*x + 86021*b^3*d^12*n^3 - 27720*a*b^2*d^12*n^2 - 2310*(b^3*e^12
*n^3 - 12*a*b^2*e^12*n^2)*x^4 + 27720*(b^3*e^12*n^2*x^4 - b^3*d^12*n^2)*log(c) + 63*(40*b^3*d*e^11*n^3*x^3 - 5
5*b^3*d^4*e^8*n^3*x^2 + 88*b^3*d^7*e^5*n^3*x - 220*b^3*d^10*e^2*n^3)*x^(2/3) - 198*(14*b^3*d^2*e^10*n^3*x^3 -
20*b^3*d^5*e^7*n^3*x^2 + 35*b^3*d^8*e^4*n^3*x - 140*b^3*d^11*e*n^3)*x^(1/3))*log(e*x^(1/3) + d)^2 + 2958333840
00*(4*b^3*d^3*e^9*n*x^3 - 6*b^3*d^6*e^6*n*x^2 + 12*b^3*d^9*e^3*n*x - 3*(b^3*e^12*n - 12*a*b^2*e^12)*x^4)*log(c
)^2 + 9240*(1108515013*b^3*d^9*e^3*n^3 - 1231904520*a*b^2*d^9*e^3*n^2 + 384199200*a^2*b*d^9*e^3*n)*x - 27720*(
4301068993*b^3*d^12*n^3 - 2384502120*a*b^2*d^12*n^2 + 384199200*a^2*b*d^12*n - 5336100*(b^3*e^12*n^3 - 12*a*b^
2*e^12*n^2 + 72*a^2*b*e^12*n)*x^4 + 43120*(763*b^3*d^3*e^9*n^3 - 1980*a*b^2*d^3*e^9*n^2)*x^3 - 4620*(22727*b^3
*d^6*e^6*n^3 - 27720*a*b^2*d^6*e^6*n^2)*x^2 - 384199200*(b^3*e^12*n*x^4 - b^3*d^12*n)*log(c)^2 + 9240*(44441*b
^3*d^9*e^3*n^3 - 27720*a*b^2*d^9*e^3*n^2)*x - 27720*(3080*b^3*d^3*e^9*n^2*x^3 - 4620*b^3*d^6*e^6*n^2*x^2 + 924
0*b^3*d^9*e^3*n^2*x + 86021*b^3*d^12*n^2 - 27720*a*b^2*d^12*n - 2310*(b^3*e^12*n^2 - 12*a*b^2*e^12*n)*x^4)*log
(c) - 63*(12826220*b^3*d^10*e^2*n^3 - 6098400*a*b^2*d^10*e^2*n^2 - 8400*(23*b^3*d*e^11*n^3 - 132*a*b^2*d*e^11*
n^2)*x^3 + 385*(2021*b^3*d^4*e^8*n^3 - 3960*a*b^2*d^4*e^8*n^2)*x^2 - 88*(28271*b^3*d^7*e^5*n^3 - 27720*a*b^2*d
^7*e^5*n^2)*x + 27720*(40*b^3*d*e^11*n^2*x^3 - 55*b^3*d^4*e^8*n^2*x^2 + 88*b^3*d^7*e^5*n^2*x - 220*b^3*d^10*e^
2*n^2)*log(c))*x^(2/3) + 198*(12042940*b^3*d^11*e*n^3 - 3880800*a*b^2*d^11*e*n^2 - 588*(181*b^3*d^2*e^10*n^3 -
660*a*b^2*d^2*e^10*n^2)*x^3 + 20*(18107*b^3*d^5*e^7*n^3 - 27720*a*b^2*d^5*e^7*n^2)*x^2 - 35*(35201*b^3*d^8*e^
4*n^3 - 27720*a*b^2*d^8*e^4*n^2)*x + 27720*(14*b^3*d^2*e^10*n^2*x^3 - 20*b^3*d^5*e^7*n^2*x^2 + 35*b^3*d^8*e^4*
n^2*x - 140*b^3*d^11*e*n^2)*log(c))*x^(1/3))*log(e*x^(1/3) + d) + 42688800*(3465*(b^3*e^12*n^2 - 12*a*b^2*e^12
*n + 72*a^2*b*e^12)*x^4 - 28*(763*b^3*d^3*e^9*n^2 - 1980*a*b^2*d^3*e^9*n)*x^3 + 3*(22727*b^3*d^6*e^6*n^2 - 277
20*a*b^2*d^6*e^6*n)*x^2 - 6*(44441*b^3*d^9*e^3*n^2 - 27720*a*b^2*d^9*e^3*n)*x)*log(c) - 63*(421644712060*b^3*d
^10*e^2*n^3 - 355542818400*a*b^2*d^10*e^2*n^2 + 84523824000*a^2*b*d^10*e^2*n - 1764000*(397*b^3*d*e^11*n^3 - 3
036*a*b^2*d*e^11*n^2 + 8712*a^2*b*d*e^11*n)*x^3 + 2695*(2459191*b^3*d^4*e^8*n^3 - 8003160*a*b^2*d^4*e^8*n^2 +
7840800*a^2*b*d^4*e^8*n)*x^2 - 384199200*(40*b^3*d*e^11*n*x^3 - 55*b^3*d^4*e^8*n*x^2 + 88*b^3*d^7*e^5*n*x - 22
0*b^3*d^10*e^2*n)*log(c)^2 - 88*(453937243*b^3*d^7*e^5*n^3 - 783672120*a*b^2*d^7*e^5*n^2 + 384199200*a^2*b*d^7
*e^5*n)*x - 27720*(12826220*b^3*d^10*e^2*n^2 - 6098400*a*b^2*d^10*e^2*n - 8400*(23*b^3*d*e^11*n^2 - 132*a*b^2*
d*e^11*n)*x^3 + 385*(2021*b^3*d^4*e^8*n^2 - 3960*a*b^2*d^4*e^8*n)*x^2 - 88*(28271*b^3*d^7*e^5*n^2 - 27720*a*b^
2*d^7*e^5*n)*x)*log(c))*x^(2/3) + 198*(602149659020*b^3*d^11*e*n^3 - 333830296800*a*b^2*d^11*e*n^2 + 537878880
00*a^2*b*d^11*e*n - 24696*(21871*b^3*d^2*e^10*n^3 - 119460*a*b^2*d^2*e^10*n^2 + 217800*a^2*b*d^2*e^10*n)*x^3 +
20*(192204079*b^3*d^5*e^7*n^3 - 501926040*a*b^2*d^5*e^7*n^2 + 384199200*a^2*b*d^5*e^7*n)*x^2 - 384199200*(14*
b^3*d^2*e^10*n*x^3 - 20*b^3*d^5*e^7*n*x^2 + 35*b^3*d^8*e^4*n*x - 140*b^3*d^11*e*n)*log(c)^2 - 35*(697880173*b^
3*d^8*e^4*n^3 - 975771720*a*b^2*d^8*e^4*n^2 + 384199200*a^2*b*d^8*e^4*n)*x - 27720*(12042940*b^3*d^11*e*n^2 -
3880800*a*b^2*d^11*e*n - 588*(181*b^3*d^2*e^10*n^2 - 660*a*b^2*d^2*e^10*n)*x^3 + 20*(18107*b^3*d^5*e^7*n^2 - 2
7720*a*b^2*d^5*e^7*n)*x^2 - 35*(35201*b^3*d^8*e^4*n^2 - 27720*a*b^2*d^8*e^4*n)*x)*log(c))*x^(1/3))/e^12

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.62097, size = 5998, normalized size = 3.27 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14200002432000*(3550000608000*b^3*x^4*e*log(c)^3 + 10650001824000*a*b^2*x^4*e*log(c)^2 + 10650001824000*a^2*
b*x^4*e*log(c) + 3550000608000*a^3*x^4*e + (3550000608000*(x^(1/3)*e + d)^12*e^(-11)*log(x^(1/3)*e + d)^3 - 42
600007296000*(x^(1/3)*e + d)^11*d*e^(-11)*log(x^(1/3)*e + d)^3 + 234300040128000*(x^(1/3)*e + d)^10*d^2*e^(-11
)*log(x^(1/3)*e + d)^3 - 781000133760000*(x^(1/3)*e + d)^9*d^3*e^(-11)*log(x^(1/3)*e + d)^3 + 1757250300960000
*(x^(1/3)*e + d)^8*d^4*e^(-11)*log(x^(1/3)*e + d)^3 - 2811600481536000*(x^(1/3)*e + d)^7*d^5*e^(-11)*log(x^(1/
3)*e + d)^3 + 3280200561792000*(x^(1/3)*e + d)^6*d^6*e^(-11)*log(x^(1/3)*e + d)^3 - 2811600481536000*(x^(1/3)*
e + d)^5*d^7*e^(-11)*log(x^(1/3)*e + d)^3 + 1757250300960000*(x^(1/3)*e + d)^4*d^8*e^(-11)*log(x^(1/3)*e + d)^
3 - 781000133760000*(x^(1/3)*e + d)^3*d^9*e^(-11)*log(x^(1/3)*e + d)^3 + 234300040128000*(x^(1/3)*e + d)^2*d^1
0*e^(-11)*log(x^(1/3)*e + d)^3 - 42600007296000*(x^(1/3)*e + d)*d^11*e^(-11)*log(x^(1/3)*e + d)^3 - 8875001520
00*(x^(1/3)*e + d)^12*e^(-11)*log(x^(1/3)*e + d)^2 + 11618183808000*(x^(1/3)*e + d)^11*d*e^(-11)*log(x^(1/3)*e
+ d)^2 - 70290012038400*(x^(1/3)*e + d)^10*d^2*e^(-11)*log(x^(1/3)*e + d)^2 + 260333377920000*(x^(1/3)*e + d)
^9*d^3*e^(-11)*log(x^(1/3)*e + d)^2 - 658968862860000*(x^(1/3)*e + d)^8*d^4*e^(-11)*log(x^(1/3)*e + d)^2 + 120
4971634944000*(x^(1/3)*e + d)^7*d^5*e^(-11)*log(x^(1/3)*e + d)^2 - 1640100280896000*(x^(1/3)*e + d)^6*d^6*e^(-
11)*log(x^(1/3)*e + d)^2 + 1686960288921600*(x^(1/3)*e + d)^5*d^7*e^(-11)*log(x^(1/3)*e + d)^2 - 1317937725720
000*(x^(1/3)*e + d)^4*d^8*e^(-11)*log(x^(1/3)*e + d)^2 + 781000133760000*(x^(1/3)*e + d)^3*d^9*e^(-11)*log(x^(
1/3)*e + d)^2 - 351450060192000*(x^(1/3)*e + d)^2*d^10*e^(-11)*log(x^(1/3)*e + d)^2 + 127800021888000*(x^(1/3)
*e + d)*d^11*e^(-11)*log(x^(1/3)*e + d)^2 + 147916692000*(x^(1/3)*e + d)^12*e^(-11)*log(x^(1/3)*e + d) - 21123
97056000*(x^(1/3)*e + d)^11*d*e^(-11)*log(x^(1/3)*e + d) + 14058002407680*(x^(1/3)*e + d)^10*d^2*e^(-11)*log(x
^(1/3)*e + d) - 57851861760000*(x^(1/3)*e + d)^9*d^3*e^(-11)*log(x^(1/3)*e + d) + 164742215715000*(x^(1/3)*e +
d)^8*d^4*e^(-11)*log(x^(1/3)*e + d) - 344277609984000*(x^(1/3)*e + d)^7*d^5*e^(-11)*log(x^(1/3)*e + d) + 5467
00093632000*(x^(1/3)*e + d)^6*d^6*e^(-11)*log(x^(1/3)*e + d) - 674784115568640*(x^(1/3)*e + d)^5*d^7*e^(-11)*l
og(x^(1/3)*e + d) + 658968862860000*(x^(1/3)*e + d)^4*d^8*e^(-11)*log(x^(1/3)*e + d) - 520666755840000*(x^(1/3
)*e + d)^3*d^9*e^(-11)*log(x^(1/3)*e + d) + 351450060192000*(x^(1/3)*e + d)^2*d^10*e^(-11)*log(x^(1/3)*e + d)
- 255600043776000*(x^(1/3)*e + d)*d^11*e^(-11)*log(x^(1/3)*e + d) - 12326391000*(x^(1/3)*e + d)^12*e^(-11) + 1
92036096000*(x^(1/3)*e + d)^11*d*e^(-11) - 1405800240768*(x^(1/3)*e + d)^10*d^2*e^(-11) + 6427984640000*(x^(1/
3)*e + d)^9*d^3*e^(-11) - 20592776964375*(x^(1/3)*e + d)^8*d^4*e^(-11) + 49182515712000*(x^(1/3)*e + d)^7*d^5*
e^(-11) - 91116682272000*(x^(1/3)*e + d)^6*d^6*e^(-11) + 134956823113728*(x^(1/3)*e + d)^5*d^7*e^(-11) - 16474
2215715000*(x^(1/3)*e + d)^4*d^8*e^(-11) + 173555585280000*(x^(1/3)*e + d)^3*d^9*e^(-11) - 175725030096000*(x^
(1/3)*e + d)^2*d^10*e^(-11) + 255600043776000*(x^(1/3)*e + d)*d^11*e^(-11))*b^3*n^3 + 27720*(384199200*(x^(1/3
)*e + d)^12*e^(-11)*log(x^(1/3)*e + d)^2 - 4610390400*(x^(1/3)*e + d)^11*d*e^(-11)*log(x^(1/3)*e + d)^2 + 2535
7147200*(x^(1/3)*e + d)^10*d^2*e^(-11)*log(x^(1/3)*e + d)^2 - 84523824000*(x^(1/3)*e + d)^9*d^3*e^(-11)*log(x^
(1/3)*e + d)^2 + 190178604000*(x^(1/3)*e + d)^8*d^4*e^(-11)*log(x^(1/3)*e + d)^2 - 304285766400*(x^(1/3)*e + d
)^7*d^5*e^(-11)*log(x^(1/3)*e + d)^2 + 355000060800*(x^(1/3)*e + d)^6*d^6*e^(-11)*log(x^(1/3)*e + d)^2 - 30428
5766400*(x^(1/3)*e + d)^5*d^7*e^(-11)*log(x^(1/3)*e + d)^2 + 190178604000*(x^(1/3)*e + d)^4*d^8*e^(-11)*log(x^
(1/3)*e + d)^2 - 84523824000*(x^(1/3)*e + d)^3*d^9*e^(-11)*log(x^(1/3)*e + d)^2 + 25357147200*(x^(1/3)*e + d)^
2*d^10*e^(-11)*log(x^(1/3)*e + d)^2 - 4610390400*(x^(1/3)*e + d)*d^11*e^(-11)*log(x^(1/3)*e + d)^2 - 64033200*
(x^(1/3)*e + d)^12*e^(-11)*log(x^(1/3)*e + d) + 838252800*(x^(1/3)*e + d)^11*d*e^(-11)*log(x^(1/3)*e + d) - 50
71429440*(x^(1/3)*e + d)^10*d^2*e^(-11)*log(x^(1/3)*e + d) + 18783072000*(x^(1/3)*e + d)^9*d^3*e^(-11)*log(x^(
1/3)*e + d) - 47544651000*(x^(1/3)*e + d)^8*d^4*e^(-11)*log(x^(1/3)*e + d) + 86938790400*(x^(1/3)*e + d)^7*d^5
*e^(-11)*log(x^(1/3)*e + d) - 118333353600*(x^(1/3)*e + d)^6*d^6*e^(-11)*log(x^(1/3)*e + d) + 121714306560*(x^
(1/3)*e + d)^5*d^7*e^(-11)*log(x^(1/3)*e + d) - 95089302000*(x^(1/3)*e + d)^4*d^8*e^(-11)*log(x^(1/3)*e + d) +
56349216000*(x^(1/3)*e + d)^3*d^9*e^(-11)*log(x^(1/3)*e + d) - 25357147200*(x^(1/3)*e + d)^2*d^10*e^(-11)*log
(x^(1/3)*e + d) + 9220780800*(x^(1/3)*e + d)*d^11*e^(-11)*log(x^(1/3)*e + d) + 5336100*(x^(1/3)*e + d)^12*e^(-
11) - 76204800*(x^(1/3)*e + d)^11*d*e^(-11) + 507142944*(x^(1/3)*e + d)^10*d^2*e^(-11) - 2087008000*(x^(1/3)*e
+ d)^9*d^3*e^(-11) + 5943081375*(x^(1/3)*e + d)^8*d^4*e^(-11) - 12419827200*(x^(1/3)*e + d)^7*d^5*e^(-11) + 1
9722225600*(x^(1/3)*e + d)^6*d^6*e^(-11) - 24342861312*(x^(1/3)*e + d)^5*d^7*e^(-11) + 23772325500*(x^(1/3)*e
+ d)^4*d^8*e^(-11) - 18783072000*(x^(1/3)*e + d)^3*d^9*e^(-11) + 12678573600*(x^(1/3)*e + d)^2*d^10*e^(-11) -
9220780800*(x^(1/3)*e + d)*d^11*e^(-11))*b^3*n^2*log(c) + 384199200*(27720*(x^(1/3)*e + d)^12*e^(-11)*log(x^(1
/3)*e + d) - 332640*(x^(1/3)*e + d)^11*d*e^(-11)*log(x^(1/3)*e + d) + 1829520*(x^(1/3)*e + d)^10*d^2*e^(-11)*l
og(x^(1/3)*e + d) - 6098400*(x^(1/3)*e + d)^9*d^3*e^(-11)*log(x^(1/3)*e + d) + 13721400*(x^(1/3)*e + d)^8*d^4*
e^(-11)*log(x^(1/3)*e + d) - 21954240*(x^(1/3)*e + d)^7*d^5*e^(-11)*log(x^(1/3)*e + d) + 25613280*(x^(1/3)*e +
d)^6*d^6*e^(-11)*log(x^(1/3)*e + d) - 21954240*(x^(1/3)*e + d)^5*d^7*e^(-11)*log(x^(1/3)*e + d) + 13721400*(x
^(1/3)*e + d)^4*d^8*e^(-11)*log(x^(1/3)*e + d) - 6098400*(x^(1/3)*e + d)^3*d^9*e^(-11)*log(x^(1/3)*e + d) + 18
29520*(x^(1/3)*e + d)^2*d^10*e^(-11)*log(x^(1/3)*e + d) - 332640*(x^(1/3)*e + d)*d^11*e^(-11)*log(x^(1/3)*e +
d) - 2310*(x^(1/3)*e + d)^12*e^(-11) + 30240*(x^(1/3)*e + d)^11*d*e^(-11) - 182952*(x^(1/3)*e + d)^10*d^2*e^(-
11) + 677600*(x^(1/3)*e + d)^9*d^3*e^(-11) - 1715175*(x^(1/3)*e + d)^8*d^4*e^(-11) + 3136320*(x^(1/3)*e + d)^7
*d^5*e^(-11) - 4268880*(x^(1/3)*e + d)^6*d^6*e^(-11) + 4390848*(x^(1/3)*e + d)^5*d^7*e^(-11) - 3430350*(x^(1/3
)*e + d)^4*d^8*e^(-11) + 2032800*(x^(1/3)*e + d)^3*d^9*e^(-11) - 914760*(x^(1/3)*e + d)^2*d^10*e^(-11) + 33264
0*(x^(1/3)*e + d)*d^11*e^(-11))*b^3*n*log(c)^2 + 27720*(384199200*(x^(1/3)*e + d)^12*e^(-11)*log(x^(1/3)*e + d
)^2 - 4610390400*(x^(1/3)*e + d)^11*d*e^(-11)*log(x^(1/3)*e + d)^2 + 25357147200*(x^(1/3)*e + d)^10*d^2*e^(-11
)*log(x^(1/3)*e + d)^2 - 84523824000*(x^(1/3)*e + d)^9*d^3*e^(-11)*log(x^(1/3)*e + d)^2 + 190178604000*(x^(1/3
)*e + d)^8*d^4*e^(-11)*log(x^(1/3)*e + d)^2 - 304285766400*(x^(1/3)*e + d)^7*d^5*e^(-11)*log(x^(1/3)*e + d)^2
+ 355000060800*(x^(1/3)*e + d)^6*d^6*e^(-11)*log(x^(1/3)*e + d)^2 - 304285766400*(x^(1/3)*e + d)^5*d^7*e^(-11)
*log(x^(1/3)*e + d)^2 + 190178604000*(x^(1/3)*e + d)^4*d^8*e^(-11)*log(x^(1/3)*e + d)^2 - 84523824000*(x^(1/3)
*e + d)^3*d^9*e^(-11)*log(x^(1/3)*e + d)^2 + 25357147200*(x^(1/3)*e + d)^2*d^10*e^(-11)*log(x^(1/3)*e + d)^2 -
4610390400*(x^(1/3)*e + d)*d^11*e^(-11)*log(x^(1/3)*e + d)^2 - 64033200*(x^(1/3)*e + d)^12*e^(-11)*log(x^(1/3
)*e + d) + 838252800*(x^(1/3)*e + d)^11*d*e^(-11)*log(x^(1/3)*e + d) - 5071429440*(x^(1/3)*e + d)^10*d^2*e^(-1
1)*log(x^(1/3)*e + d) + 18783072000*(x^(1/3)*e + d)^9*d^3*e^(-11)*log(x^(1/3)*e + d) - 47544651000*(x^(1/3)*e
+ d)^8*d^4*e^(-11)*log(x^(1/3)*e + d) + 86938790400*(x^(1/3)*e + d)^7*d^5*e^(-11)*log(x^(1/3)*e + d) - 1183333
53600*(x^(1/3)*e + d)^6*d^6*e^(-11)*log(x^(1/3)*e + d) + 121714306560*(x^(1/3)*e + d)^5*d^7*e^(-11)*log(x^(1/3
)*e + d) - 95089302000*(x^(1/3)*e + d)^4*d^8*e^(-11)*log(x^(1/3)*e + d) + 56349216000*(x^(1/3)*e + d)^3*d^9*e^
(-11)*log(x^(1/3)*e + d) - 25357147200*(x^(1/3)*e + d)^2*d^10*e^(-11)*log(x^(1/3)*e + d) + 9220780800*(x^(1/3)
*e + d)*d^11*e^(-11)*log(x^(1/3)*e + d) + 5336100*(x^(1/3)*e + d)^12*e^(-11) - 76204800*(x^(1/3)*e + d)^11*d*e
^(-11) + 507142944*(x^(1/3)*e + d)^10*d^2*e^(-11) - 2087008000*(x^(1/3)*e + d)^9*d^3*e^(-11) + 5943081375*(x^(
1/3)*e + d)^8*d^4*e^(-11) - 12419827200*(x^(1/3)*e + d)^7*d^5*e^(-11) + 19722225600*(x^(1/3)*e + d)^6*d^6*e^(-
11) - 24342861312*(x^(1/3)*e + d)^5*d^7*e^(-11) + 23772325500*(x^(1/3)*e + d)^4*d^8*e^(-11) - 18783072000*(x^(
1/3)*e + d)^3*d^9*e^(-11) + 12678573600*(x^(1/3)*e + d)^2*d^10*e^(-11) - 9220780800*(x^(1/3)*e + d)*d^11*e^(-1
1))*a*b^2*n^2 + 768398400*(27720*(x^(1/3)*e + d)^12*e^(-11)*log(x^(1/3)*e + d) - 332640*(x^(1/3)*e + d)^11*d*e
^(-11)*log(x^(1/3)*e + d) + 1829520*(x^(1/3)*e + d)^10*d^2*e^(-11)*log(x^(1/3)*e + d) - 6098400*(x^(1/3)*e + d
)^9*d^3*e^(-11)*log(x^(1/3)*e + d) + 13721400*(x^(1/3)*e + d)^8*d^4*e^(-11)*log(x^(1/3)*e + d) - 21954240*(x^(
1/3)*e + d)^7*d^5*e^(-11)*log(x^(1/3)*e + d) + 25613280*(x^(1/3)*e + d)^6*d^6*e^(-11)*log(x^(1/3)*e + d) - 219
54240*(x^(1/3)*e + d)^5*d^7*e^(-11)*log(x^(1/3)*e + d) + 13721400*(x^(1/3)*e + d)^4*d^8*e^(-11)*log(x^(1/3)*e
+ d) - 6098400*(x^(1/3)*e + d)^3*d^9*e^(-11)*log(x^(1/3)*e + d) + 1829520*(x^(1/3)*e + d)^2*d^10*e^(-11)*log(x
^(1/3)*e + d) - 332640*(x^(1/3)*e + d)*d^11*e^(-11)*log(x^(1/3)*e + d) - 2310*(x^(1/3)*e + d)^12*e^(-11) + 302
40*(x^(1/3)*e + d)^11*d*e^(-11) - 182952*(x^(1/3)*e + d)^10*d^2*e^(-11) + 677600*(x^(1/3)*e + d)^9*d^3*e^(-11)
- 1715175*(x^(1/3)*e + d)^8*d^4*e^(-11) + 3136320*(x^(1/3)*e + d)^7*d^5*e^(-11) - 4268880*(x^(1/3)*e + d)^6*d
^6*e^(-11) + 4390848*(x^(1/3)*e + d)^5*d^7*e^(-11) - 3430350*(x^(1/3)*e + d)^4*d^8*e^(-11) + 2032800*(x^(1/3)*
e + d)^3*d^9*e^(-11) - 914760*(x^(1/3)*e + d)^2*d^10*e^(-11) + 332640*(x^(1/3)*e + d)*d^11*e^(-11))*a*b^2*n*lo
g(c) + 384199200*(27720*(x^(1/3)*e + d)^12*e^(-11)*log(x^(1/3)*e + d) - 332640*(x^(1/3)*e + d)^11*d*e^(-11)*lo
g(x^(1/3)*e + d) + 1829520*(x^(1/3)*e + d)^10*d^2*e^(-11)*log(x^(1/3)*e + d) - 6098400*(x^(1/3)*e + d)^9*d^3*e
^(-11)*log(x^(1/3)*e + d) + 13721400*(x^(1/3)*e + d)^8*d^4*e^(-11)*log(x^(1/3)*e + d) - 21954240*(x^(1/3)*e +
d)^7*d^5*e^(-11)*log(x^(1/3)*e + d) + 25613280*(x^(1/3)*e + d)^6*d^6*e^(-11)*log(x^(1/3)*e + d) - 21954240*(x^
(1/3)*e + d)^5*d^7*e^(-11)*log(x^(1/3)*e + d) + 13721400*(x^(1/3)*e + d)^4*d^8*e^(-11)*log(x^(1/3)*e + d) - 60
98400*(x^(1/3)*e + d)^3*d^9*e^(-11)*log(x^(1/3)*e + d) + 1829520*(x^(1/3)*e + d)^2*d^10*e^(-11)*log(x^(1/3)*e
+ d) - 332640*(x^(1/3)*e + d)*d^11*e^(-11)*log(x^(1/3)*e + d) - 2310*(x^(1/3)*e + d)^12*e^(-11) + 30240*(x^(1/
3)*e + d)^11*d*e^(-11) - 182952*(x^(1/3)*e + d)^10*d^2*e^(-11) + 677600*(x^(1/3)*e + d)^9*d^3*e^(-11) - 171517
5*(x^(1/3)*e + d)^8*d^4*e^(-11) + 3136320*(x^(1/3)*e + d)^7*d^5*e^(-11) - 4268880*(x^(1/3)*e + d)^6*d^6*e^(-11
) + 4390848*(x^(1/3)*e + d)^5*d^7*e^(-11) - 3430350*(x^(1/3)*e + d)^4*d^8*e^(-11) + 2032800*(x^(1/3)*e + d)^3*
d^9*e^(-11) - 914760*(x^(1/3)*e + d)^2*d^10*e^(-11) + 332640*(x^(1/3)*e + d)*d^11*e^(-11))*a^2*b*n)*e^(-1)