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

Optimal. Leaf size=1357 \[ \text{result too large to display} \]

[Out]

(9*b^3*d^7*n^3*(d + e*x^(1/3))^2)/e^9 - (56*b^3*d^6*n^3*(d + e*x^(1/3))^3)/(9*e^9) + (63*b^3*d^5*n^3*(d + e*x^
(1/3))^4)/(16*e^9) - (252*b^3*d^4*n^3*(d + e*x^(1/3))^5)/(125*e^9) + (7*b^3*d^3*n^3*(d + e*x^(1/3))^6)/(9*e^9)
- (72*b^3*d^2*n^3*(d + e*x^(1/3))^7)/(343*e^9) + (9*b^3*d*n^3*(d + e*x^(1/3))^8)/(256*e^9) - (2*b^3*n^3*(d +
e*x^(1/3))^9)/(729*e^9) + (18*a*b^2*d^8*n^2*x^(1/3))/e^8 - (18*b^3*d^8*n^3*x^(1/3))/e^8 + (18*b^3*d^8*n^2*(d +
e*x^(1/3))*Log[c*(d + e*x^(1/3))^n])/e^9 - (18*b^2*d^7*n^2*(d + e*x^(1/3))^2*(a + b*Log[c*(d + e*x^(1/3))^n])
)/e^9 + (56*b^2*d^6*n^2*(d + e*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^n]))/(3*e^9) - (63*b^2*d^5*n^2*(d + e*x
^(1/3))^4*(a + b*Log[c*(d + e*x^(1/3))^n]))/(4*e^9) + (252*b^2*d^4*n^2*(d + e*x^(1/3))^5*(a + b*Log[c*(d + e*x
^(1/3))^n]))/(25*e^9) - (14*b^2*d^3*n^2*(d + e*x^(1/3))^6*(a + b*Log[c*(d + e*x^(1/3))^n]))/(3*e^9) + (72*b^2*
d^2*n^2*(d + e*x^(1/3))^7*(a + b*Log[c*(d + e*x^(1/3))^n]))/(49*e^9) - (9*b^2*d*n^2*(d + e*x^(1/3))^8*(a + b*L
og[c*(d + e*x^(1/3))^n]))/(32*e^9) + (2*b^2*n^2*(d + e*x^(1/3))^9*(a + b*Log[c*(d + e*x^(1/3))^n]))/(81*e^9) -
(9*b*d^8*n*(d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/e^9 + (18*b*d^7*n*(d + e*x^(1/3))^2*(a + b*Log
[c*(d + e*x^(1/3))^n])^2)/e^9 - (28*b*d^6*n*(d + e*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/e^9 + (63*b*
d^5*n*(d + e*x^(1/3))^4*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(2*e^9) - (126*b*d^4*n*(d + e*x^(1/3))^5*(a + b*Lo
g[c*(d + e*x^(1/3))^n])^2)/(5*e^9) + (14*b*d^3*n*(d + e*x^(1/3))^6*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/e^9 - (
36*b*d^2*n*(d + e*x^(1/3))^7*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(7*e^9) + (9*b*d*n*(d + e*x^(1/3))^8*(a + b*L
og[c*(d + e*x^(1/3))^n])^2)/(8*e^9) - (b*n*(d + e*x^(1/3))^9*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(9*e^9) + (3*
d^8*(d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^9 - (12*d^7*(d + e*x^(1/3))^2*(a + b*Log[c*(d + e*x^
(1/3))^n])^3)/e^9 + (28*d^6*(d + e*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^9 - (42*d^5*(d + e*x^(1/3)
)^4*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^9 + (42*d^4*(d + e*x^(1/3))^5*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^
9 - (28*d^3*(d + e*x^(1/3))^6*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^9 + (12*d^2*(d + e*x^(1/3))^7*(a + b*Log[c
*(d + e*x^(1/3))^n])^3)/e^9 - (3*d*(d + e*x^(1/3))^8*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^9 + ((d + e*x^(1/3)
)^9*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/(3*e^9)

________________________________________________________________________________________

Rubi [A]  time = 1.5733, antiderivative size = 1357, normalized size of antiderivative = 1., number of steps used = 40, 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^2*(a + b*Log[c*(d + e*x^(1/3))^n])^3,x]

[Out]

(9*b^3*d^7*n^3*(d + e*x^(1/3))^2)/e^9 - (56*b^3*d^6*n^3*(d + e*x^(1/3))^3)/(9*e^9) + (63*b^3*d^5*n^3*(d + e*x^
(1/3))^4)/(16*e^9) - (252*b^3*d^4*n^3*(d + e*x^(1/3))^5)/(125*e^9) + (7*b^3*d^3*n^3*(d + e*x^(1/3))^6)/(9*e^9)
- (72*b^3*d^2*n^3*(d + e*x^(1/3))^7)/(343*e^9) + (9*b^3*d*n^3*(d + e*x^(1/3))^8)/(256*e^9) - (2*b^3*n^3*(d +
e*x^(1/3))^9)/(729*e^9) + (18*a*b^2*d^8*n^2*x^(1/3))/e^8 - (18*b^3*d^8*n^3*x^(1/3))/e^8 + (18*b^3*d^8*n^2*(d +
e*x^(1/3))*Log[c*(d + e*x^(1/3))^n])/e^9 - (18*b^2*d^7*n^2*(d + e*x^(1/3))^2*(a + b*Log[c*(d + e*x^(1/3))^n])
)/e^9 + (56*b^2*d^6*n^2*(d + e*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^n]))/(3*e^9) - (63*b^2*d^5*n^2*(d + e*x
^(1/3))^4*(a + b*Log[c*(d + e*x^(1/3))^n]))/(4*e^9) + (252*b^2*d^4*n^2*(d + e*x^(1/3))^5*(a + b*Log[c*(d + e*x
^(1/3))^n]))/(25*e^9) - (14*b^2*d^3*n^2*(d + e*x^(1/3))^6*(a + b*Log[c*(d + e*x^(1/3))^n]))/(3*e^9) + (72*b^2*
d^2*n^2*(d + e*x^(1/3))^7*(a + b*Log[c*(d + e*x^(1/3))^n]))/(49*e^9) - (9*b^2*d*n^2*(d + e*x^(1/3))^8*(a + b*L
og[c*(d + e*x^(1/3))^n]))/(32*e^9) + (2*b^2*n^2*(d + e*x^(1/3))^9*(a + b*Log[c*(d + e*x^(1/3))^n]))/(81*e^9) -
(9*b*d^8*n*(d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/e^9 + (18*b*d^7*n*(d + e*x^(1/3))^2*(a + b*Log
[c*(d + e*x^(1/3))^n])^2)/e^9 - (28*b*d^6*n*(d + e*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/e^9 + (63*b*
d^5*n*(d + e*x^(1/3))^4*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(2*e^9) - (126*b*d^4*n*(d + e*x^(1/3))^5*(a + b*Lo
g[c*(d + e*x^(1/3))^n])^2)/(5*e^9) + (14*b*d^3*n*(d + e*x^(1/3))^6*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/e^9 - (
36*b*d^2*n*(d + e*x^(1/3))^7*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(7*e^9) + (9*b*d*n*(d + e*x^(1/3))^8*(a + b*L
og[c*(d + e*x^(1/3))^n])^2)/(8*e^9) - (b*n*(d + e*x^(1/3))^9*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(9*e^9) + (3*
d^8*(d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^9 - (12*d^7*(d + e*x^(1/3))^2*(a + b*Log[c*(d + e*x^
(1/3))^n])^3)/e^9 + (28*d^6*(d + e*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^9 - (42*d^5*(d + e*x^(1/3)
)^4*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^9 + (42*d^4*(d + e*x^(1/3))^5*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^
9 - (28*d^3*(d + e*x^(1/3))^6*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^9 + (12*d^2*(d + e*x^(1/3))^7*(a + b*Log[c
*(d + e*x^(1/3))^n])^3)/e^9 - (3*d*(d + e*x^(1/3))^8*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^9 + ((d + e*x^(1/3)
)^9*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/(3*e^9)

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^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx &=3 \operatorname{Subst}\left (\int x^8 \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^8 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^8}-\frac{8 d^7 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^8}+\frac{28 d^6 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^8}-\frac{56 d^5 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^8}+\frac{70 d^4 (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^8}-\frac{56 d^3 (d+e x)^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^8}+\frac{28 d^2 (d+e x)^6 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^8}-\frac{8 d (d+e x)^7 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^8}+\frac{(d+e x)^8 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^8}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{3 \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^8}-\frac{(24 d) \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^8}+\frac{\left (84 d^2\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^8}-\frac{\left (168 d^3\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^8}+\frac{\left (210 d^4\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^8}-\frac{\left (168 d^5\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^8}+\frac{\left (84 d^6\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^8}-\frac{\left (24 d^7\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^8}+\frac{\left (3 d^8\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^8}\\ &=\frac{3 \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^9}-\frac{(24 d) \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^9}+\frac{\left (84 d^2\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^9}-\frac{\left (168 d^3\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^9}+\frac{\left (210 d^4\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^9}-\frac{\left (168 d^5\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^9}+\frac{\left (84 d^6\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^9}-\frac{\left (24 d^7\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^9}+\frac{\left (3 d^8\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^9}\\ &=\frac{3 d^8 \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^9}-\frac{12 d^7 \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}{e^9}+\frac{28 d^6 \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^9}-\frac{42 d^5 \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}{e^9}+\frac{42 d^4 \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^9}-\frac{28 d^3 \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^9}+\frac{12 d^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 )^3}{e^9}-\frac{3 d \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}{e^9}+\frac{\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}{3 e^9}-\frac{(b n) \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^9}+\frac{(9 b d n) \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 )}{e^9}-\frac{\left (36 b d^2 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^9}+\frac{\left (84 b d^3 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^9}-\frac{\left (126 b d^4 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^9}+\frac{\left (126 b d^5 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 )}{e^9}-\frac{\left (84 b d^6 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^9}+\frac{\left (36 b d^7 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 )}{e^9}-\frac{\left (9 b d^8 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^9}\\ &=-\frac{9 b d^8 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^9}+\frac{18 b d^7 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}{e^9}-\frac{28 b d^6 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^9}+\frac{63 b d^5 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}{2 e^9}-\frac{126 b d^4 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^9}+\frac{14 b d^3 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}{e^9}-\frac{36 b d^2 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^9}+\frac{9 b d 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}{8 e^9}-\frac{b 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}{9 e^9}+\frac{3 d^8 \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^9}-\frac{12 d^7 \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}{e^9}+\frac{28 d^6 \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^9}-\frac{42 d^5 \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}{e^9}+\frac{42 d^4 \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^9}-\frac{28 d^3 \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^9}+\frac{12 d^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 )^3}{e^9}-\frac{3 d \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}{e^9}+\frac{\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}{3 e^9}+\frac{\left (2 b^2 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 )}{9 e^9}-\frac{\left (9 b^2 d 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 )}{4 e^9}+\frac{\left (72 b^2 d^2 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^9}-\frac{\left (28 b^2 d^3 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^9}+\frac{\left (252 b^2 d^4 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^9}-\frac{\left (63 b^2 d^5 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 )}{e^9}+\frac{\left (56 b^2 d^6 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^9}-\frac{\left (36 b^2 d^7 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 )}{e^9}+\frac{\left (18 b^2 d^8 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^9}\\ &=\frac{9 b^3 d^7 n^3 \left (d+e \sqrt [3]{x}\right )^2}{e^9}-\frac{56 b^3 d^6 n^3 \left (d+e \sqrt [3]{x}\right )^3}{9 e^9}+\frac{63 b^3 d^5 n^3 \left (d+e \sqrt [3]{x}\right )^4}{16 e^9}-\frac{252 b^3 d^4 n^3 \left (d+e \sqrt [3]{x}\right )^5}{125 e^9}+\frac{7 b^3 d^3 n^3 \left (d+e \sqrt [3]{x}\right )^6}{9 e^9}-\frac{72 b^3 d^2 n^3 \left (d+e \sqrt [3]{x}\right )^7}{343 e^9}+\frac{9 b^3 d n^3 \left (d+e \sqrt [3]{x}\right )^8}{256 e^9}-\frac{2 b^3 n^3 \left (d+e \sqrt [3]{x}\right )^9}{729 e^9}+\frac{18 a b^2 d^8 n^2 \sqrt [3]{x}}{e^8}-\frac{18 b^2 d^7 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 )}{e^9}+\frac{56 b^2 d^6 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^9}-\frac{63 b^2 d^5 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 )}{4 e^9}+\frac{252 b^2 d^4 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^9}-\frac{14 b^2 d^3 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 )}{3 e^9}+\frac{72 b^2 d^2 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^9}-\frac{9 b^2 d 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 )}{32 e^9}+\frac{2 b^2 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 )}{81 e^9}-\frac{9 b d^8 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^9}+\frac{18 b d^7 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}{e^9}-\frac{28 b d^6 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^9}+\frac{63 b d^5 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}{2 e^9}-\frac{126 b d^4 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^9}+\frac{14 b d^3 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}{e^9}-\frac{36 b d^2 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^9}+\frac{9 b d 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}{8 e^9}-\frac{b 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}{9 e^9}+\frac{3 d^8 \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^9}-\frac{12 d^7 \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}{e^9}+\frac{28 d^6 \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^9}-\frac{42 d^5 \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}{e^9}+\frac{42 d^4 \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^9}-\frac{28 d^3 \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^9}+\frac{12 d^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 )^3}{e^9}-\frac{3 d \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}{e^9}+\frac{\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}{3 e^9}+\frac{\left (18 b^3 d^8 n^2\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{e^9}\\ &=\frac{9 b^3 d^7 n^3 \left (d+e \sqrt [3]{x}\right )^2}{e^9}-\frac{56 b^3 d^6 n^3 \left (d+e \sqrt [3]{x}\right )^3}{9 e^9}+\frac{63 b^3 d^5 n^3 \left (d+e \sqrt [3]{x}\right )^4}{16 e^9}-\frac{252 b^3 d^4 n^3 \left (d+e \sqrt [3]{x}\right )^5}{125 e^9}+\frac{7 b^3 d^3 n^3 \left (d+e \sqrt [3]{x}\right )^6}{9 e^9}-\frac{72 b^3 d^2 n^3 \left (d+e \sqrt [3]{x}\right )^7}{343 e^9}+\frac{9 b^3 d n^3 \left (d+e \sqrt [3]{x}\right )^8}{256 e^9}-\frac{2 b^3 n^3 \left (d+e \sqrt [3]{x}\right )^9}{729 e^9}+\frac{18 a b^2 d^8 n^2 \sqrt [3]{x}}{e^8}-\frac{18 b^3 d^8 n^3 \sqrt [3]{x}}{e^8}+\frac{18 b^3 d^8 n^2 \left (d+e \sqrt [3]{x}\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{e^9}-\frac{18 b^2 d^7 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 )}{e^9}+\frac{56 b^2 d^6 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^9}-\frac{63 b^2 d^5 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 )}{4 e^9}+\frac{252 b^2 d^4 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^9}-\frac{14 b^2 d^3 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 )}{3 e^9}+\frac{72 b^2 d^2 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^9}-\frac{9 b^2 d 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 )}{32 e^9}+\frac{2 b^2 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 )}{81 e^9}-\frac{9 b d^8 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^9}+\frac{18 b d^7 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}{e^9}-\frac{28 b d^6 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^9}+\frac{63 b d^5 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}{2 e^9}-\frac{126 b d^4 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^9}+\frac{14 b d^3 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}{e^9}-\frac{36 b d^2 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^9}+\frac{9 b d 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}{8 e^9}-\frac{b 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}{9 e^9}+\frac{3 d^8 \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^9}-\frac{12 d^7 \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}{e^9}+\frac{28 d^6 \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^9}-\frac{42 d^5 \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}{e^9}+\frac{42 d^4 \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^9}-\frac{28 d^3 \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^9}+\frac{12 d^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 )^3}{e^9}-\frac{3 d \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}{e^9}+\frac{\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}{3 e^9}\\ \end{align*}

Mathematica [A]  time = 0.771382, size = 808, normalized size = 0.6 \[ \frac{2667168000 \left (d^9+e^9 x^3\right ) a^3-3175200 b n \left (7129 d^9+2520 e \sqrt [3]{x} d^8-1260 e^2 x^{2/3} d^7+840 e^3 x d^6-630 e^4 x^{4/3} d^5+504 e^5 x^{5/3} d^4-420 e^6 x^2 d^3+360 e^7 x^{7/3} d^2-315 e^8 x^{8/3} d+280 e^9 x^3\right ) a^2-2520 b^2 n^2 \left (26853209 d^9-17965080 e \sqrt [3]{x} d^8+5807340 e^2 x^{2/3} d^7-2813160 e^3 x d^6+1580670 e^4 x^{4/3} d^5-947016 e^5 x^{5/3} d^4+577500 e^6 x^2 d^3-343800 e^7 x^{7/3} d^2+187425 e^8 x^{8/3} d-78400 e^9 x^3\right ) a+2667168000 b^3 \left (d^9+e^9 x^3\right ) \log ^3\left (c \left (d+e \sqrt [3]{x}\right )^n\right )+3175200 b^2 \left (2520 a \left (d^9+e^9 x^3\right )-b n \left (7129 d^9+2520 e \sqrt [3]{x} d^8-1260 e^2 x^{2/3} d^7+840 e^3 x d^6-630 e^4 x^{4/3} d^5+504 e^5 x^{5/3} d^4-420 e^6 x^2 d^3+360 e^7 x^{7/3} d^2-315 e^8 x^{8/3} d+280 e^9 x^3\right )\right ) \log ^2\left (c \left (d+e \sqrt [3]{x}\right )^n\right )+b^3 e n^3 \sqrt [3]{x} \left (-76356985320 d^8+15542491860 e \sqrt [3]{x} d^7-5483495640 e^2 x^{2/3} d^6+2340330930 e^3 x d^5-1075607064 e^4 x^{4/3} d^4+498592500 e^5 x^{5/3} d^3-219465000 e^6 x^2 d^2+83734875 e^7 x^{7/3} d-21952000 e^8 x^{8/3}\right )+2520 b \left (3175200 \left (d^9+e^9 x^3\right ) a^2-2520 b n \left (7129 d^9+2520 e \sqrt [3]{x} d^8-1260 e^2 x^{2/3} d^7+840 e^3 x d^6-630 e^4 x^{4/3} d^5+504 e^5 x^{5/3} d^4-420 e^6 x^2 d^3+360 e^7 x^{7/3} d^2-315 e^8 x^{8/3} d+280 e^9 x^3\right ) a+b^2 n^2 \left (30300391 d^9+17965080 e \sqrt [3]{x} d^8-5807340 e^2 x^{2/3} d^7+2813160 e^3 x d^6-1580670 e^4 x^{4/3} d^5+947016 e^5 x^{5/3} d^4-577500 e^6 x^2 d^3+343800 e^7 x^{7/3} d^2-187425 e^8 x^{8/3} d+78400 e^9 x^3\right )\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{8001504000 e^9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b^3*e*n^3*x^(1/3)*(-76356985320*d^8 + 15542491860*d^7*e*x^(1/3) - 5483495640*d^6*e^2*x^(2/3) + 2340330930*d^5
*e^3*x - 1075607064*d^4*e^4*x^(4/3) + 498592500*d^3*e^5*x^(5/3) - 219465000*d^2*e^6*x^2 + 83734875*d*e^7*x^(7/
3) - 21952000*e^8*x^(8/3)) - 2520*a*b^2*n^2*(26853209*d^9 - 17965080*d^8*e*x^(1/3) + 5807340*d^7*e^2*x^(2/3) -
2813160*d^6*e^3*x + 1580670*d^5*e^4*x^(4/3) - 947016*d^4*e^5*x^(5/3) + 577500*d^3*e^6*x^2 - 343800*d^2*e^7*x^
(7/3) + 187425*d*e^8*x^(8/3) - 78400*e^9*x^3) + 2667168000*a^3*(d^9 + e^9*x^3) - 3175200*a^2*b*n*(7129*d^9 + 2
520*d^8*e*x^(1/3) - 1260*d^7*e^2*x^(2/3) + 840*d^6*e^3*x - 630*d^5*e^4*x^(4/3) + 504*d^4*e^5*x^(5/3) - 420*d^3
*e^6*x^2 + 360*d^2*e^7*x^(7/3) - 315*d*e^8*x^(8/3) + 280*e^9*x^3) + 2520*b*(3175200*a^2*(d^9 + e^9*x^3) - 2520
*a*b*n*(7129*d^9 + 2520*d^8*e*x^(1/3) - 1260*d^7*e^2*x^(2/3) + 840*d^6*e^3*x - 630*d^5*e^4*x^(4/3) + 504*d^4*e
^5*x^(5/3) - 420*d^3*e^6*x^2 + 360*d^2*e^7*x^(7/3) - 315*d*e^8*x^(8/3) + 280*e^9*x^3) + b^2*n^2*(30300391*d^9
+ 17965080*d^8*e*x^(1/3) - 5807340*d^7*e^2*x^(2/3) + 2813160*d^6*e^3*x - 1580670*d^5*e^4*x^(4/3) + 947016*d^4*
e^5*x^(5/3) - 577500*d^3*e^6*x^2 + 343800*d^2*e^7*x^(7/3) - 187425*d*e^8*x^(8/3) + 78400*e^9*x^3))*Log[c*(d +
e*x^(1/3))^n] + 3175200*b^2*(2520*a*(d^9 + e^9*x^3) - b*n*(7129*d^9 + 2520*d^8*e*x^(1/3) - 1260*d^7*e^2*x^(2/3
) + 840*d^6*e^3*x - 630*d^5*e^4*x^(4/3) + 504*d^4*e^5*x^(5/3) - 420*d^3*e^6*x^2 + 360*d^2*e^7*x^(7/3) - 315*d*
e^8*x^(8/3) + 280*e^9*x^3))*Log[c*(d + e*x^(1/3))^n]^2 + 2667168000*b^3*(d^9 + e^9*x^3)*Log[c*(d + e*x^(1/3))^
n]^3)/(8001504000*e^9)

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Maple [F]  time = 0.112, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \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^2*(a+b*ln(c*(d+e*x^(1/3))^n))^3,x)

[Out]

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

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Maxima [A]  time = 1.12994, size = 1170, normalized size = 0.86 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b^3*x^3*log((e*x^(1/3) + d)^n*c)^3 + a*b^2*x^3*log((e*x^(1/3) + d)^n*c)^2 + a^2*b*x^3*log((e*x^(1/3) + d)^
n*c) + 1/3*a^3*x^3 + 1/2520*a^2*b*e*n*(2520*d^9*log(e*x^(1/3) + d)/e^10 - (280*e^8*x^3 - 315*d*e^7*x^(8/3) + 3
60*d^2*e^6*x^(7/3) - 420*d^3*e^5*x^2 + 504*d^4*e^4*x^(5/3) - 630*d^5*e^3*x^(4/3) + 840*d^6*e^2*x - 1260*d^7*e*
x^(2/3) + 2520*d^8*x^(1/3))/e^9) + 1/3175200*(2520*e*n*(2520*d^9*log(e*x^(1/3) + d)/e^10 - (280*e^8*x^3 - 315*
d*e^7*x^(8/3) + 360*d^2*e^6*x^(7/3) - 420*d^3*e^5*x^2 + 504*d^4*e^4*x^(5/3) - 630*d^5*e^3*x^(4/3) + 840*d^6*e^
2*x - 1260*d^7*e*x^(2/3) + 2520*d^8*x^(1/3))/e^9)*log((e*x^(1/3) + d)^n*c) + (78400*e^9*x^3 - 187425*d*e^8*x^(
8/3) + 343800*d^2*e^7*x^(7/3) - 577500*d^3*e^6*x^2 - 3175200*d^9*log(e*x^(1/3) + d)^2 + 947016*d^4*e^5*x^(5/3)
- 1580670*d^5*e^4*x^(4/3) + 2813160*d^6*e^3*x - 17965080*d^9*log(e*x^(1/3) + d) - 5807340*d^7*e^2*x^(2/3) + 1
7965080*d^8*e*x^(1/3))*n^2/e^9)*a*b^2 + 1/8001504000*(3175200*e*n*(2520*d^9*log(e*x^(1/3) + d)/e^10 - (280*e^8
*x^3 - 315*d*e^7*x^(8/3) + 360*d^2*e^6*x^(7/3) - 420*d^3*e^5*x^2 + 504*d^4*e^4*x^(5/3) - 630*d^5*e^3*x^(4/3) +
840*d^6*e^2*x - 1260*d^7*e*x^(2/3) + 2520*d^8*x^(1/3))/e^9)*log((e*x^(1/3) + d)^n*c)^2 - e*n*((21952000*e^9*x
^3 - 2667168000*d^9*log(e*x^(1/3) + d)^3 - 83734875*d*e^8*x^(8/3) + 219465000*d^2*e^7*x^(7/3) - 498592500*d^3*
e^6*x^2 - 22636000800*d^9*log(e*x^(1/3) + d)^2 + 1075607064*d^4*e^5*x^(5/3) - 2340330930*d^5*e^4*x^(4/3) + 548
3495640*d^6*e^3*x - 76356985320*d^9*log(e*x^(1/3) + d) - 15542491860*d^7*e^2*x^(2/3) + 76356985320*d^8*e*x^(1/
3))*n^2/e^10 - 2520*(78400*e^9*x^3 - 187425*d*e^8*x^(8/3) + 343800*d^2*e^7*x^(7/3) - 577500*d^3*e^6*x^2 - 3175
200*d^9*log(e*x^(1/3) + d)^2 + 947016*d^4*e^5*x^(5/3) - 1580670*d^5*e^4*x^(4/3) + 2813160*d^6*e^3*x - 17965080
*d^9*log(e*x^(1/3) + d) - 5807340*d^7*e^2*x^(2/3) + 17965080*d^8*e*x^(1/3))*n*log((e*x^(1/3) + d)^n*c)/e^10))*
b^3

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Fricas [A]  time = 3.95246, size = 4018, normalized size = 2.96 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8001504000*(2667168000*b^3*e^9*x^3*log(c)^3 - 10976000*(2*b^3*e^9*n^3 - 18*a*b^2*e^9*n^2 + 81*a^2*b*e^9*n -
243*a^3*e^9)*x^3 + 2667168000*(b^3*e^9*n^3*x^3 + b^3*d^9*n^3)*log(e*x^(1/3) + d)^3 + 10500*(47485*b^3*d^3*e^6*
n^3 - 138600*a*b^2*d^3*e^6*n^2 + 127008*a^2*b*d^3*e^6*n)*x^2 + 3175200*(420*b^3*d^3*e^6*n^3*x^2 - 840*b^3*d^6*
e^3*n^3*x - 7129*b^3*d^9*n^3 + 2520*a*b^2*d^9*n^2 - 280*(b^3*e^9*n^3 - 9*a*b^2*e^9*n^2)*x^3 + 2520*(b^3*e^9*n^
2*x^3 + b^3*d^9*n^2)*log(c) + 63*(5*b^3*d*e^8*n^3*x^2 - 8*b^3*d^4*e^5*n^3*x + 20*b^3*d^7*e^2*n^3)*x^(2/3) - 90
*(4*b^3*d^2*e^7*n^3*x^2 - 7*b^3*d^5*e^4*n^3*x + 28*b^3*d^8*e*n^3)*x^(1/3))*log(e*x^(1/3) + d)^2 + 444528000*(3
*b^3*d^3*e^6*n*x^2 - 6*b^3*d^6*e^3*n*x - 2*(b^3*e^9*n - 9*a*b^2*e^9)*x^3)*log(c)^2 - 840*(6527971*b^3*d^6*e^3*
n^3 - 8439480*a*b^2*d^6*e^3*n^2 + 3175200*a^2*b*d^6*e^3*n)*x + 2520*(30300391*b^3*d^9*n^3 - 17965080*a*b^2*d^9
*n^2 + 3175200*a^2*b*d^9*n + 39200*(2*b^3*e^9*n^3 - 18*a*b^2*e^9*n^2 + 81*a^2*b*e^9*n)*x^3 - 2100*(275*b^3*d^3
*e^6*n^3 - 504*a*b^2*d^3*e^6*n^2)*x^2 + 3175200*(b^3*e^9*n*x^3 + b^3*d^9*n)*log(c)^2 + 840*(3349*b^3*d^6*e^3*n
^3 - 2520*a*b^2*d^6*e^3*n^2)*x + 2520*(420*b^3*d^3*e^6*n^2*x^2 - 840*b^3*d^6*e^3*n^2*x - 7129*b^3*d^9*n^2 + 25
20*a*b^2*d^9*n - 280*(b^3*e^9*n^2 - 9*a*b^2*e^9*n)*x^3)*log(c) - 63*(92180*b^3*d^7*e^2*n^3 - 50400*a*b^2*d^7*e
^2*n^2 + 175*(17*b^3*d*e^8*n^3 - 72*a*b^2*d*e^8*n^2)*x^2 - 8*(1879*b^3*d^4*e^5*n^3 - 2520*a*b^2*d^4*e^5*n^2)*x
- 2520*(5*b^3*d*e^8*n^2*x^2 - 8*b^3*d^4*e^5*n^2*x + 20*b^3*d^7*e^2*n^2)*log(c))*x^(2/3) + 90*(199612*b^3*d^8*
e*n^3 - 70560*a*b^2*d^8*e*n^2 + 20*(191*b^3*d^2*e^7*n^3 - 504*a*b^2*d^2*e^7*n^2)*x^2 - 7*(2509*b^3*d^5*e^4*n^3
- 2520*a*b^2*d^5*e^4*n^2)*x - 2520*(4*b^3*d^2*e^7*n^2*x^2 - 7*b^3*d^5*e^4*n^2*x + 28*b^3*d^8*e*n^2)*log(c))*x
^(1/3))*log(e*x^(1/3) + d) + 352800*(280*(2*b^3*e^9*n^2 - 18*a*b^2*e^9*n + 81*a^2*b*e^9)*x^3 - 15*(275*b^3*d^3
*e^6*n^2 - 504*a*b^2*d^3*e^6*n)*x^2 + 6*(3349*b^3*d^6*e^3*n^2 - 2520*a*b^2*d^6*e^3*n)*x)*log(c) + 63*(24670622
0*b^3*d^7*e^2*n^3 - 232293600*a*b^2*d^7*e^2*n^2 + 63504000*a^2*b*d^7*e^2*n + 6125*(217*b^3*d*e^8*n^3 - 1224*a*
b^2*d*e^8*n^2 + 2592*a^2*b*d*e^8*n)*x^2 + 3175200*(5*b^3*d*e^8*n*x^2 - 8*b^3*d^4*e^5*n*x + 20*b^3*d^7*e^2*n)*l
og(c)^2 - 8*(2134141*b^3*d^4*e^5*n^3 - 4735080*a*b^2*d^4*e^5*n^2 + 3175200*a^2*b*d^4*e^5*n)*x - 2520*(92180*b^
3*d^7*e^2*n^2 - 50400*a*b^2*d^7*e^2*n + 175*(17*b^3*d*e^8*n^2 - 72*a*b^2*d*e^8*n)*x^2 - 8*(1879*b^3*d^4*e^5*n^
2 - 2520*a*b^2*d^4*e^5*n)*x)*log(c))*x^(2/3) - 90*(848410948*b^3*d^8*e*n^3 - 503022240*a*b^2*d^8*e*n^2 + 88905
600*a^2*b*d^8*e*n + 100*(24385*b^3*d^2*e^7*n^3 - 96264*a*b^2*d^2*e^7*n^2 + 127008*a^2*b*d^2*e^7*n)*x^2 + 31752
00*(4*b^3*d^2*e^7*n*x^2 - 7*b^3*d^5*e^4*n*x + 28*b^3*d^8*e*n)*log(c)^2 - 7*(3714811*b^3*d^5*e^4*n^3 - 6322680*
a*b^2*d^5*e^4*n^2 + 3175200*a^2*b*d^5*e^4*n)*x - 2520*(199612*b^3*d^8*e*n^2 - 70560*a*b^2*d^8*e*n + 20*(191*b^
3*d^2*e^7*n^2 - 504*a*b^2*d^2*e^7*n)*x^2 - 7*(2509*b^3*d^5*e^4*n^2 - 2520*a*b^2*d^5*e^4*n)*x)*log(c))*x^(1/3))
/e^9

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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**(1/3))**n))**3,x)

[Out]

Timed out

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Giac [B]  time = 1.47896, size = 4500, normalized size = 3.32 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8001504000*(2667168000*b^3*x^3*e*log(c)^3 + 8001504000*a*b^2*x^3*e*log(c)^2 + 8001504000*a^2*b*x^3*e*log(c)
+ (2667168000*(x^(1/3)*e + d)^9*e^(-8)*log(x^(1/3)*e + d)^3 - 24004512000*(x^(1/3)*e + d)^8*d*e^(-8)*log(x^(1/
3)*e + d)^3 + 96018048000*(x^(1/3)*e + d)^7*d^2*e^(-8)*log(x^(1/3)*e + d)^3 - 224042112000*(x^(1/3)*e + d)^6*d
^3*e^(-8)*log(x^(1/3)*e + d)^3 + 336063168000*(x^(1/3)*e + d)^5*d^4*e^(-8)*log(x^(1/3)*e + d)^3 - 336063168000
*(x^(1/3)*e + d)^4*d^5*e^(-8)*log(x^(1/3)*e + d)^3 + 224042112000*(x^(1/3)*e + d)^3*d^6*e^(-8)*log(x^(1/3)*e +
d)^3 - 96018048000*(x^(1/3)*e + d)^2*d^7*e^(-8)*log(x^(1/3)*e + d)^3 + 24004512000*(x^(1/3)*e + d)*d^8*e^(-8)
*log(x^(1/3)*e + d)^3 - 889056000*(x^(1/3)*e + d)^9*e^(-8)*log(x^(1/3)*e + d)^2 + 9001692000*(x^(1/3)*e + d)^8
*d*e^(-8)*log(x^(1/3)*e + d)^2 - 41150592000*(x^(1/3)*e + d)^7*d^2*e^(-8)*log(x^(1/3)*e + d)^2 + 112021056000*
(x^(1/3)*e + d)^6*d^3*e^(-8)*log(x^(1/3)*e + d)^2 - 201637900800*(x^(1/3)*e + d)^5*d^4*e^(-8)*log(x^(1/3)*e +
d)^2 + 252047376000*(x^(1/3)*e + d)^4*d^5*e^(-8)*log(x^(1/3)*e + d)^2 - 224042112000*(x^(1/3)*e + d)^3*d^6*e^(
-8)*log(x^(1/3)*e + d)^2 + 144027072000*(x^(1/3)*e + d)^2*d^7*e^(-8)*log(x^(1/3)*e + d)^2 - 72013536000*(x^(1/
3)*e + d)*d^8*e^(-8)*log(x^(1/3)*e + d)^2 + 197568000*(x^(1/3)*e + d)^9*e^(-8)*log(x^(1/3)*e + d) - 2250423000
*(x^(1/3)*e + d)^8*d*e^(-8)*log(x^(1/3)*e + d) + 11757312000*(x^(1/3)*e + d)^7*d^2*e^(-8)*log(x^(1/3)*e + d) -
37340352000*(x^(1/3)*e + d)^6*d^3*e^(-8)*log(x^(1/3)*e + d) + 80655160320*(x^(1/3)*e + d)^5*d^4*e^(-8)*log(x^
(1/3)*e + d) - 126023688000*(x^(1/3)*e + d)^4*d^5*e^(-8)*log(x^(1/3)*e + d) + 149361408000*(x^(1/3)*e + d)^3*d
^6*e^(-8)*log(x^(1/3)*e + d) - 144027072000*(x^(1/3)*e + d)^2*d^7*e^(-8)*log(x^(1/3)*e + d) + 144027072000*(x^
(1/3)*e + d)*d^8*e^(-8)*log(x^(1/3)*e + d) - 21952000*(x^(1/3)*e + d)^9*e^(-8) + 281302875*(x^(1/3)*e + d)^8*d
*e^(-8) - 1679616000*(x^(1/3)*e + d)^7*d^2*e^(-8) + 6223392000*(x^(1/3)*e + d)^6*d^3*e^(-8) - 16131032064*(x^(
1/3)*e + d)^5*d^4*e^(-8) + 31505922000*(x^(1/3)*e + d)^4*d^5*e^(-8) - 49787136000*(x^(1/3)*e + d)^3*d^6*e^(-8)
+ 72013536000*(x^(1/3)*e + d)^2*d^7*e^(-8) - 144027072000*(x^(1/3)*e + d)*d^8*e^(-8))*b^3*n^3 + 2667168000*a^
3*x^3*e + 2520*(3175200*(x^(1/3)*e + d)^9*e^(-8)*log(x^(1/3)*e + d)^2 - 28576800*(x^(1/3)*e + d)^8*d*e^(-8)*lo
g(x^(1/3)*e + d)^2 + 114307200*(x^(1/3)*e + d)^7*d^2*e^(-8)*log(x^(1/3)*e + d)^2 - 266716800*(x^(1/3)*e + d)^6
*d^3*e^(-8)*log(x^(1/3)*e + d)^2 + 400075200*(x^(1/3)*e + d)^5*d^4*e^(-8)*log(x^(1/3)*e + d)^2 - 400075200*(x^
(1/3)*e + d)^4*d^5*e^(-8)*log(x^(1/3)*e + d)^2 + 266716800*(x^(1/3)*e + d)^3*d^6*e^(-8)*log(x^(1/3)*e + d)^2 -
114307200*(x^(1/3)*e + d)^2*d^7*e^(-8)*log(x^(1/3)*e + d)^2 + 28576800*(x^(1/3)*e + d)*d^8*e^(-8)*log(x^(1/3)
*e + d)^2 - 705600*(x^(1/3)*e + d)^9*e^(-8)*log(x^(1/3)*e + d) + 7144200*(x^(1/3)*e + d)^8*d*e^(-8)*log(x^(1/3
)*e + d) - 32659200*(x^(1/3)*e + d)^7*d^2*e^(-8)*log(x^(1/3)*e + d) + 88905600*(x^(1/3)*e + d)^6*d^3*e^(-8)*lo
g(x^(1/3)*e + d) - 160030080*(x^(1/3)*e + d)^5*d^4*e^(-8)*log(x^(1/3)*e + d) + 200037600*(x^(1/3)*e + d)^4*d^5
*e^(-8)*log(x^(1/3)*e + d) - 177811200*(x^(1/3)*e + d)^3*d^6*e^(-8)*log(x^(1/3)*e + d) + 114307200*(x^(1/3)*e
+ d)^2*d^7*e^(-8)*log(x^(1/3)*e + d) - 57153600*(x^(1/3)*e + d)*d^8*e^(-8)*log(x^(1/3)*e + d) + 78400*(x^(1/3)
*e + d)^9*e^(-8) - 893025*(x^(1/3)*e + d)^8*d*e^(-8) + 4665600*(x^(1/3)*e + d)^7*d^2*e^(-8) - 14817600*(x^(1/3
)*e + d)^6*d^3*e^(-8) + 32006016*(x^(1/3)*e + d)^5*d^4*e^(-8) - 50009400*(x^(1/3)*e + d)^4*d^5*e^(-8) + 592704
00*(x^(1/3)*e + d)^3*d^6*e^(-8) - 57153600*(x^(1/3)*e + d)^2*d^7*e^(-8) + 57153600*(x^(1/3)*e + d)*d^8*e^(-8))
*b^3*n^2*log(c) + 3175200*(2520*(x^(1/3)*e + d)^9*e^(-8)*log(x^(1/3)*e + d) - 22680*(x^(1/3)*e + d)^8*d*e^(-8)
*log(x^(1/3)*e + d) + 90720*(x^(1/3)*e + d)^7*d^2*e^(-8)*log(x^(1/3)*e + d) - 211680*(x^(1/3)*e + d)^6*d^3*e^(
-8)*log(x^(1/3)*e + d) + 317520*(x^(1/3)*e + d)^5*d^4*e^(-8)*log(x^(1/3)*e + d) - 317520*(x^(1/3)*e + d)^4*d^5
*e^(-8)*log(x^(1/3)*e + d) + 211680*(x^(1/3)*e + d)^3*d^6*e^(-8)*log(x^(1/3)*e + d) - 90720*(x^(1/3)*e + d)^2*
d^7*e^(-8)*log(x^(1/3)*e + d) + 22680*(x^(1/3)*e + d)*d^8*e^(-8)*log(x^(1/3)*e + d) - 280*(x^(1/3)*e + d)^9*e^
(-8) + 2835*(x^(1/3)*e + d)^8*d*e^(-8) - 12960*(x^(1/3)*e + d)^7*d^2*e^(-8) + 35280*(x^(1/3)*e + d)^6*d^3*e^(-
8) - 63504*(x^(1/3)*e + d)^5*d^4*e^(-8) + 79380*(x^(1/3)*e + d)^4*d^5*e^(-8) - 70560*(x^(1/3)*e + d)^3*d^6*e^(
-8) + 45360*(x^(1/3)*e + d)^2*d^7*e^(-8) - 22680*(x^(1/3)*e + d)*d^8*e^(-8))*b^3*n*log(c)^2 + 2520*(3175200*(x
^(1/3)*e + d)^9*e^(-8)*log(x^(1/3)*e + d)^2 - 28576800*(x^(1/3)*e + d)^8*d*e^(-8)*log(x^(1/3)*e + d)^2 + 11430
7200*(x^(1/3)*e + d)^7*d^2*e^(-8)*log(x^(1/3)*e + d)^2 - 266716800*(x^(1/3)*e + d)^6*d^3*e^(-8)*log(x^(1/3)*e
+ d)^2 + 400075200*(x^(1/3)*e + d)^5*d^4*e^(-8)*log(x^(1/3)*e + d)^2 - 400075200*(x^(1/3)*e + d)^4*d^5*e^(-8)*
log(x^(1/3)*e + d)^2 + 266716800*(x^(1/3)*e + d)^3*d^6*e^(-8)*log(x^(1/3)*e + d)^2 - 114307200*(x^(1/3)*e + d)
^2*d^7*e^(-8)*log(x^(1/3)*e + d)^2 + 28576800*(x^(1/3)*e + d)*d^8*e^(-8)*log(x^(1/3)*e + d)^2 - 705600*(x^(1/3
)*e + d)^9*e^(-8)*log(x^(1/3)*e + d) + 7144200*(x^(1/3)*e + d)^8*d*e^(-8)*log(x^(1/3)*e + d) - 32659200*(x^(1/
3)*e + d)^7*d^2*e^(-8)*log(x^(1/3)*e + d) + 88905600*(x^(1/3)*e + d)^6*d^3*e^(-8)*log(x^(1/3)*e + d) - 1600300
80*(x^(1/3)*e + d)^5*d^4*e^(-8)*log(x^(1/3)*e + d) + 200037600*(x^(1/3)*e + d)^4*d^5*e^(-8)*log(x^(1/3)*e + d)
- 177811200*(x^(1/3)*e + d)^3*d^6*e^(-8)*log(x^(1/3)*e + d) + 114307200*(x^(1/3)*e + d)^2*d^7*e^(-8)*log(x^(1
/3)*e + d) - 57153600*(x^(1/3)*e + d)*d^8*e^(-8)*log(x^(1/3)*e + d) + 78400*(x^(1/3)*e + d)^9*e^(-8) - 893025*
(x^(1/3)*e + d)^8*d*e^(-8) + 4665600*(x^(1/3)*e + d)^7*d^2*e^(-8) - 14817600*(x^(1/3)*e + d)^6*d^3*e^(-8) + 32
006016*(x^(1/3)*e + d)^5*d^4*e^(-8) - 50009400*(x^(1/3)*e + d)^4*d^5*e^(-8) + 59270400*(x^(1/3)*e + d)^3*d^6*e
^(-8) - 57153600*(x^(1/3)*e + d)^2*d^7*e^(-8) + 57153600*(x^(1/3)*e + d)*d^8*e^(-8))*a*b^2*n^2 + 6350400*(2520
*(x^(1/3)*e + d)^9*e^(-8)*log(x^(1/3)*e + d) - 22680*(x^(1/3)*e + d)^8*d*e^(-8)*log(x^(1/3)*e + d) + 90720*(x^
(1/3)*e + d)^7*d^2*e^(-8)*log(x^(1/3)*e + d) - 211680*(x^(1/3)*e + d)^6*d^3*e^(-8)*log(x^(1/3)*e + d) + 317520
*(x^(1/3)*e + d)^5*d^4*e^(-8)*log(x^(1/3)*e + d) - 317520*(x^(1/3)*e + d)^4*d^5*e^(-8)*log(x^(1/3)*e + d) + 21
1680*(x^(1/3)*e + d)^3*d^6*e^(-8)*log(x^(1/3)*e + d) - 90720*(x^(1/3)*e + d)^2*d^7*e^(-8)*log(x^(1/3)*e + d) +
22680*(x^(1/3)*e + d)*d^8*e^(-8)*log(x^(1/3)*e + d) - 280*(x^(1/3)*e + d)^9*e^(-8) + 2835*(x^(1/3)*e + d)^8*d
*e^(-8) - 12960*(x^(1/3)*e + d)^7*d^2*e^(-8) + 35280*(x^(1/3)*e + d)^6*d^3*e^(-8) - 63504*(x^(1/3)*e + d)^5*d^
4*e^(-8) + 79380*(x^(1/3)*e + d)^4*d^5*e^(-8) - 70560*(x^(1/3)*e + d)^3*d^6*e^(-8) + 45360*(x^(1/3)*e + d)^2*d
^7*e^(-8) - 22680*(x^(1/3)*e + d)*d^8*e^(-8))*a*b^2*n*log(c) + 3175200*(2520*(x^(1/3)*e + d)^9*e^(-8)*log(x^(1
/3)*e + d) - 22680*(x^(1/3)*e + d)^8*d*e^(-8)*log(x^(1/3)*e + d) + 90720*(x^(1/3)*e + d)^7*d^2*e^(-8)*log(x^(1
/3)*e + d) - 211680*(x^(1/3)*e + d)^6*d^3*e^(-8)*log(x^(1/3)*e + d) + 317520*(x^(1/3)*e + d)^5*d^4*e^(-8)*log(
x^(1/3)*e + d) - 317520*(x^(1/3)*e + d)^4*d^5*e^(-8)*log(x^(1/3)*e + d) + 211680*(x^(1/3)*e + d)^3*d^6*e^(-8)*
log(x^(1/3)*e + d) - 90720*(x^(1/3)*e + d)^2*d^7*e^(-8)*log(x^(1/3)*e + d) + 22680*(x^(1/3)*e + d)*d^8*e^(-8)*
log(x^(1/3)*e + d) - 280*(x^(1/3)*e + d)^9*e^(-8) + 2835*(x^(1/3)*e + d)^8*d*e^(-8) - 12960*(x^(1/3)*e + d)^7*
d^2*e^(-8) + 35280*(x^(1/3)*e + d)^6*d^3*e^(-8) - 63504*(x^(1/3)*e + d)^5*d^4*e^(-8) + 79380*(x^(1/3)*e + d)^4
*d^5*e^(-8) - 70560*(x^(1/3)*e + d)^3*d^6*e^(-8) + 45360*(x^(1/3)*e + d)^2*d^7*e^(-8) - 22680*(x^(1/3)*e + d)*
d^8*e^(-8))*a^2*b*n)*e^(-1)