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

Optimal. Leaf size=907 \[ \text{result too large to display} \]

[Out]

(-45*b^3*d^4*n^3*(d + e*x^(1/3))^2)/(8*e^6) + (20*b^3*d^3*n^3*(d + e*x^(1/3))^3)/(9*e^6) - (45*b^3*d^2*n^3*(d
+ e*x^(1/3))^4)/(64*e^6) + (18*b^3*d*n^3*(d + e*x^(1/3))^5)/(125*e^6) - (b^3*n^3*(d + e*x^(1/3))^6)/(72*e^6) -
(18*a*b^2*d^5*n^2*x^(1/3))/e^5 + (18*b^3*d^5*n^3*x^(1/3))/e^5 - (18*b^3*d^5*n^2*(d + e*x^(1/3))*Log[c*(d + e*
x^(1/3))^n])/e^6 + (45*b^2*d^4*n^2*(d + e*x^(1/3))^2*(a + b*Log[c*(d + e*x^(1/3))^n]))/(4*e^6) - (20*b^2*d^3*n
^2*(d + e*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^n]))/(3*e^6) + (45*b^2*d^2*n^2*(d + e*x^(1/3))^4*(a + b*Log[
c*(d + e*x^(1/3))^n]))/(16*e^6) - (18*b^2*d*n^2*(d + e*x^(1/3))^5*(a + b*Log[c*(d + e*x^(1/3))^n]))/(25*e^6) +
(b^2*n^2*(d + e*x^(1/3))^6*(a + b*Log[c*(d + e*x^(1/3))^n]))/(12*e^6) + (9*b*d^5*n*(d + e*x^(1/3))*(a + b*Log
[c*(d + e*x^(1/3))^n])^2)/e^6 - (45*b*d^4*n*(d + e*x^(1/3))^2*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(4*e^6) + (1
0*b*d^3*n*(d + e*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/e^6 - (45*b*d^2*n*(d + e*x^(1/3))^4*(a + b*Log
[c*(d + e*x^(1/3))^n])^2)/(8*e^6) + (9*b*d*n*(d + e*x^(1/3))^5*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(5*e^6) - (
b*n*(d + e*x^(1/3))^6*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(4*e^6) - (3*d^5*(d + e*x^(1/3))*(a + b*Log[c*(d + e
*x^(1/3))^n])^3)/e^6 + (15*d^4*(d + e*x^(1/3))^2*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/(2*e^6) - (10*d^3*(d + e*
x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^6 + (15*d^2*(d + e*x^(1/3))^4*(a + b*Log[c*(d + e*x^(1/3))^n]
)^3)/(2*e^6) - (3*d*(d + e*x^(1/3))^5*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^6 + ((d + e*x^(1/3))^6*(a + b*Log[
c*(d + e*x^(1/3))^n])^3)/(2*e^6)

________________________________________________________________________________________

Rubi [A]  time = 0.979217, antiderivative size = 907, normalized size of antiderivative = 1., number of steps used = 28, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {2454, 2401, 2389, 2296, 2295, 2390, 2305, 2304} \[ -\frac{b^3 n^3 \left (d+e \sqrt [3]{x}\right )^6}{72 e^6}+\frac{\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \left (d+e \sqrt [3]{x}\right )^6}{2 e^6}-\frac{b n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \left (d+e \sqrt [3]{x}\right )^6}{4 e^6}+\frac{b^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \left (d+e \sqrt [3]{x}\right )^6}{12 e^6}+\frac{18 b^3 d n^3 \left (d+e \sqrt [3]{x}\right )^5}{125 e^6}-\frac{3 d \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \left (d+e \sqrt [3]{x}\right )^5}{e^6}+\frac{9 b d n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \left (d+e \sqrt [3]{x}\right )^5}{5 e^6}-\frac{18 b^2 d n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \left (d+e \sqrt [3]{x}\right )^5}{25 e^6}-\frac{45 b^3 d^2 n^3 \left (d+e \sqrt [3]{x}\right )^4}{64 e^6}+\frac{15 d^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \left (d+e \sqrt [3]{x}\right )^4}{2 e^6}-\frac{45 b d^2 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \left (d+e \sqrt [3]{x}\right )^4}{8 e^6}+\frac{45 b^2 d^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \left (d+e \sqrt [3]{x}\right )^4}{16 e^6}+\frac{20 b^3 d^3 n^3 \left (d+e \sqrt [3]{x}\right )^3}{9 e^6}-\frac{10 d^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \left (d+e \sqrt [3]{x}\right )^3}{e^6}+\frac{10 b d^3 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \left (d+e \sqrt [3]{x}\right )^3}{e^6}-\frac{20 b^2 d^3 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \left (d+e \sqrt [3]{x}\right )^3}{3 e^6}-\frac{45 b^3 d^4 n^3 \left (d+e \sqrt [3]{x}\right )^2}{8 e^6}+\frac{15 d^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \left (d+e \sqrt [3]{x}\right )^2}{2 e^6}-\frac{45 b d^4 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \left (d+e \sqrt [3]{x}\right )^2}{4 e^6}+\frac{45 b^2 d^4 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \left (d+e \sqrt [3]{x}\right )^2}{4 e^6}-\frac{3 d^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \left (d+e \sqrt [3]{x}\right )}{e^6}+\frac{9 b d^5 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \left (d+e \sqrt [3]{x}\right )}{e^6}-\frac{18 b^3 d^5 n^2 \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right ) \left (d+e \sqrt [3]{x}\right )}{e^6}+\frac{18 b^3 d^5 n^3 \sqrt [3]{x}}{e^5}-\frac{18 a b^2 d^5 n^2 \sqrt [3]{x}}{e^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-45*b^3*d^4*n^3*(d + e*x^(1/3))^2)/(8*e^6) + (20*b^3*d^3*n^3*(d + e*x^(1/3))^3)/(9*e^6) - (45*b^3*d^2*n^3*(d
+ e*x^(1/3))^4)/(64*e^6) + (18*b^3*d*n^3*(d + e*x^(1/3))^5)/(125*e^6) - (b^3*n^3*(d + e*x^(1/3))^6)/(72*e^6) -
(18*a*b^2*d^5*n^2*x^(1/3))/e^5 + (18*b^3*d^5*n^3*x^(1/3))/e^5 - (18*b^3*d^5*n^2*(d + e*x^(1/3))*Log[c*(d + e*
x^(1/3))^n])/e^6 + (45*b^2*d^4*n^2*(d + e*x^(1/3))^2*(a + b*Log[c*(d + e*x^(1/3))^n]))/(4*e^6) - (20*b^2*d^3*n
^2*(d + e*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^n]))/(3*e^6) + (45*b^2*d^2*n^2*(d + e*x^(1/3))^4*(a + b*Log[
c*(d + e*x^(1/3))^n]))/(16*e^6) - (18*b^2*d*n^2*(d + e*x^(1/3))^5*(a + b*Log[c*(d + e*x^(1/3))^n]))/(25*e^6) +
(b^2*n^2*(d + e*x^(1/3))^6*(a + b*Log[c*(d + e*x^(1/3))^n]))/(12*e^6) + (9*b*d^5*n*(d + e*x^(1/3))*(a + b*Log
[c*(d + e*x^(1/3))^n])^2)/e^6 - (45*b*d^4*n*(d + e*x^(1/3))^2*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(4*e^6) + (1
0*b*d^3*n*(d + e*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/e^6 - (45*b*d^2*n*(d + e*x^(1/3))^4*(a + b*Log
[c*(d + e*x^(1/3))^n])^2)/(8*e^6) + (9*b*d*n*(d + e*x^(1/3))^5*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(5*e^6) - (
b*n*(d + e*x^(1/3))^6*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(4*e^6) - (3*d^5*(d + e*x^(1/3))*(a + b*Log[c*(d + e
*x^(1/3))^n])^3)/e^6 + (15*d^4*(d + e*x^(1/3))^2*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/(2*e^6) - (10*d^3*(d + e*
x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^6 + (15*d^2*(d + e*x^(1/3))^4*(a + b*Log[c*(d + e*x^(1/3))^n]
)^3)/(2*e^6) - (3*d*(d + e*x^(1/3))^5*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^6 + ((d + e*x^(1/3))^6*(a + b*Log[
c*(d + e*x^(1/3))^n])^3)/(2*e^6)

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 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx &=3 \operatorname{Subst}\left (\int x^5 \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^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^5}+\frac{5 d^4 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^5}-\frac{10 d^3 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^5}+\frac{10 d^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^5}-\frac{5 d (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^5}+\frac{(d+e x)^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^5}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{3 \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^5}-\frac{(15 d) \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^5}+\frac{\left (30 d^2\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^5}-\frac{\left (30 d^3\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^5}+\frac{\left (15 d^4\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^5}-\frac{\left (3 d^5\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^5}\\ &=\frac{3 \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^6}-\frac{(15 d) \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^6}+\frac{\left (30 d^2\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^6}-\frac{\left (30 d^3\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^6}+\frac{\left (15 d^4\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^6}-\frac{\left (3 d^5\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^6}\\ &=-\frac{3 d^5 \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^6}+\frac{15 d^4 \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^6}-\frac{10 d^3 \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^6}+\frac{15 d^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 )^3}{2 e^6}-\frac{3 d \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^6}+\frac{\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}{2 e^6}-\frac{(3 b n) \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 )}{2 e^6}+\frac{(9 b d n) \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^6}-\frac{\left (45 b d^2 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 )}{2 e^6}+\frac{\left (30 b d^3 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^6}-\frac{\left (45 b d^4 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^6}+\frac{\left (9 b d^5 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^6}\\ &=\frac{9 b d^5 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^6}-\frac{45 b d^4 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^6}+\frac{10 b d^3 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^6}-\frac{45 b d^2 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}{8 e^6}+\frac{9 b d 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^6}-\frac{b 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}{4 e^6}-\frac{3 d^5 \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^6}+\frac{15 d^4 \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^6}-\frac{10 d^3 \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^6}+\frac{15 d^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 )^3}{2 e^6}-\frac{3 d \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^6}+\frac{\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}{2 e^6}+\frac{\left (b^2 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 )}{2 e^6}-\frac{\left (18 b^2 d 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^6}+\frac{\left (45 b^2 d^2 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 )}{4 e^6}-\frac{\left (20 b^2 d^3 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^6}+\frac{\left (45 b^2 d^4 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^6}-\frac{\left (18 b^2 d^5 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^6}\\ &=-\frac{45 b^3 d^4 n^3 \left (d+e \sqrt [3]{x}\right )^2}{8 e^6}+\frac{20 b^3 d^3 n^3 \left (d+e \sqrt [3]{x}\right )^3}{9 e^6}-\frac{45 b^3 d^2 n^3 \left (d+e \sqrt [3]{x}\right )^4}{64 e^6}+\frac{18 b^3 d n^3 \left (d+e \sqrt [3]{x}\right )^5}{125 e^6}-\frac{b^3 n^3 \left (d+e \sqrt [3]{x}\right )^6}{72 e^6}-\frac{18 a b^2 d^5 n^2 \sqrt [3]{x}}{e^5}+\frac{45 b^2 d^4 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^6}-\frac{20 b^2 d^3 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^6}+\frac{45 b^2 d^2 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 )}{16 e^6}-\frac{18 b^2 d 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^6}+\frac{b^2 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 )}{12 e^6}+\frac{9 b d^5 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^6}-\frac{45 b d^4 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^6}+\frac{10 b d^3 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^6}-\frac{45 b d^2 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}{8 e^6}+\frac{9 b d 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^6}-\frac{b 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}{4 e^6}-\frac{3 d^5 \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^6}+\frac{15 d^4 \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^6}-\frac{10 d^3 \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^6}+\frac{15 d^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 )^3}{2 e^6}-\frac{3 d \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^6}+\frac{\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}{2 e^6}-\frac{\left (18 b^3 d^5 n^2\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{e^6}\\ &=-\frac{45 b^3 d^4 n^3 \left (d+e \sqrt [3]{x}\right )^2}{8 e^6}+\frac{20 b^3 d^3 n^3 \left (d+e \sqrt [3]{x}\right )^3}{9 e^6}-\frac{45 b^3 d^2 n^3 \left (d+e \sqrt [3]{x}\right )^4}{64 e^6}+\frac{18 b^3 d n^3 \left (d+e \sqrt [3]{x}\right )^5}{125 e^6}-\frac{b^3 n^3 \left (d+e \sqrt [3]{x}\right )^6}{72 e^6}-\frac{18 a b^2 d^5 n^2 \sqrt [3]{x}}{e^5}+\frac{18 b^3 d^5 n^3 \sqrt [3]{x}}{e^5}-\frac{18 b^3 d^5 n^2 \left (d+e \sqrt [3]{x}\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{e^6}+\frac{45 b^2 d^4 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^6}-\frac{20 b^2 d^3 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^6}+\frac{45 b^2 d^2 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 )}{16 e^6}-\frac{18 b^2 d 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^6}+\frac{b^2 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 )}{12 e^6}+\frac{9 b d^5 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^6}-\frac{45 b d^4 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^6}+\frac{10 b d^3 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^6}-\frac{45 b d^2 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}{8 e^6}+\frac{9 b d 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^6}-\frac{b 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}{4 e^6}-\frac{3 d^5 \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^6}+\frac{15 d^4 \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^6}-\frac{10 d^3 \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^6}+\frac{15 d^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 )^3}{2 e^6}-\frac{3 d \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^6}+\frac{\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}{2 e^6}\\ \end{align*}

Mathematica [A]  time = 0.466504, size = 589, normalized size = 0.65 \[ \frac{-60 b \left (1800 a^2 \left (d^6-e^6 x^2\right )-60 a b n \left (-30 d^4 e^2 x^{2/3}-15 d^2 e^4 x^{4/3}+20 d^3 e^3 x+60 d^5 e \sqrt [3]{x}+147 d^6+12 d e^5 x^{5/3}-10 e^6 x^2\right )+b^2 n^2 \left (-2610 d^4 e^2 x^{2/3}-555 d^2 e^4 x^{4/3}+1140 d^3 e^3 x+8820 d^5 e \sqrt [3]{x}+13489 d^6+264 d e^5 x^{5/3}-100 e^6 x^2\right )\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )+1800 a^2 b n \left (-30 d^4 e^2 x^{2/3}-15 d^2 e^4 x^{4/3}+20 d^3 e^3 x+60 d^5 e \sqrt [3]{x}+147 d^6+12 d e^5 x^{5/3}-10 e^6 x^2\right )-36000 a^3 \left (d^6-e^6 x^2\right )-1800 b^2 \left (60 a \left (d^6-e^6 x^2\right )+b n \left (30 d^4 e^2 x^{2/3}+15 d^2 e^4 x^{4/3}-20 d^3 e^3 x-60 d^5 e \sqrt [3]{x}-147 d^6-12 d e^5 x^{5/3}+10 e^6 x^2\right )\right ) \log ^2\left (c \left (d+e \sqrt [3]{x}\right )^n\right )+60 a b^2 n^2 \left (2610 d^4 e^2 x^{2/3}+555 d^2 e^4 x^{4/3}-1140 d^3 e^3 x-8820 d^5 e \sqrt [3]{x}+8111 d^6-264 d e^5 x^{5/3}+100 e^6 x^2\right )-36000 b^3 \left (d^6-e^6 x^2\right ) \log ^3\left (c \left (d+e \sqrt [3]{x}\right )^n\right )+b^3 e n^3 \sqrt [3]{x} \left (41180 d^3 e^2 x^{2/3}-13785 d^2 e^3 x-140070 d^4 e \sqrt [3]{x}+809340 d^5+4368 d e^4 x^{4/3}-1000 e^5 x^{5/3}\right )}{72000 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b^3*e*n^3*x^(1/3)*(809340*d^5 - 140070*d^4*e*x^(1/3) + 41180*d^3*e^2*x^(2/3) - 13785*d^2*e^3*x + 4368*d*e^4*x
^(4/3) - 1000*e^5*x^(5/3)) + 1800*a^2*b*n*(147*d^6 + 60*d^5*e*x^(1/3) - 30*d^4*e^2*x^(2/3) + 20*d^3*e^3*x - 15
*d^2*e^4*x^(4/3) + 12*d*e^5*x^(5/3) - 10*e^6*x^2) - 36000*a^3*(d^6 - e^6*x^2) + 60*a*b^2*n^2*(8111*d^6 - 8820*
d^5*e*x^(1/3) + 2610*d^4*e^2*x^(2/3) - 1140*d^3*e^3*x + 555*d^2*e^4*x^(4/3) - 264*d*e^5*x^(5/3) + 100*e^6*x^2)
- 60*b*(b^2*n^2*(13489*d^6 + 8820*d^5*e*x^(1/3) - 2610*d^4*e^2*x^(2/3) + 1140*d^3*e^3*x - 555*d^2*e^4*x^(4/3)
+ 264*d*e^5*x^(5/3) - 100*e^6*x^2) - 60*a*b*n*(147*d^6 + 60*d^5*e*x^(1/3) - 30*d^4*e^2*x^(2/3) + 20*d^3*e^3*x
- 15*d^2*e^4*x^(4/3) + 12*d*e^5*x^(5/3) - 10*e^6*x^2) + 1800*a^2*(d^6 - e^6*x^2))*Log[c*(d + e*x^(1/3))^n] -
1800*b^2*(60*a*(d^6 - e^6*x^2) + b*n*(-147*d^6 - 60*d^5*e*x^(1/3) + 30*d^4*e^2*x^(2/3) - 20*d^3*e^3*x + 15*d^2
*e^4*x^(4/3) - 12*d*e^5*x^(5/3) + 10*e^6*x^2))*Log[c*(d + e*x^(1/3))^n]^2 - 36000*b^3*(d^6 - e^6*x^2)*Log[c*(d
+ e*x^(1/3))^n]^3)/(72000*e^6)

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Maple [F]  time = 0.098, size = 0, normalized size = 0. \begin{align*} \int x \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*(a+b*ln(c*(d+e*x^(1/3))^n))^3,x)

[Out]

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

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Maxima [A]  time = 1.09962, size = 902, normalized size = 0.99 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b^3*x^2*log((e*x^(1/3) + d)^n*c)^3 + 3/2*a*b^2*x^2*log((e*x^(1/3) + d)^n*c)^2 - 1/40*a^2*b*e*n*(60*d^6*log
(e*x^(1/3) + d)/e^7 + (10*e^5*x^2 - 12*d*e^4*x^(5/3) + 15*d^2*e^3*x^(4/3) - 20*d^3*e^2*x + 30*d^4*e*x^(2/3) -
60*d^5*x^(1/3))/e^6) + 3/2*a^2*b*x^2*log((e*x^(1/3) + d)^n*c) + 1/2*a^3*x^2 - 1/1200*(60*e*n*(60*d^6*log(e*x^(
1/3) + d)/e^7 + (10*e^5*x^2 - 12*d*e^4*x^(5/3) + 15*d^2*e^3*x^(4/3) - 20*d^3*e^2*x + 30*d^4*e*x^(2/3) - 60*d^5
*x^(1/3))/e^6)*log((e*x^(1/3) + d)^n*c) - (100*e^6*x^2 + 1800*d^6*log(e*x^(1/3) + d)^2 - 264*d*e^5*x^(5/3) + 5
55*d^2*e^4*x^(4/3) - 1140*d^3*e^3*x + 8820*d^6*log(e*x^(1/3) + d) + 2610*d^4*e^2*x^(2/3) - 8820*d^5*e*x^(1/3))
*n^2/e^6)*a*b^2 - 1/72000*(1800*e*n*(60*d^6*log(e*x^(1/3) + d)/e^7 + (10*e^5*x^2 - 12*d*e^4*x^(5/3) + 15*d^2*e
^3*x^(4/3) - 20*d^3*e^2*x + 30*d^4*e*x^(2/3) - 60*d^5*x^(1/3))/e^6)*log((e*x^(1/3) + d)^n*c)^2 + e*n*((36000*d
^6*log(e*x^(1/3) + d)^3 + 1000*e^6*x^2 + 264600*d^6*log(e*x^(1/3) + d)^2 - 4368*d*e^5*x^(5/3) + 13785*d^2*e^4*
x^(4/3) - 41180*d^3*e^3*x + 809340*d^6*log(e*x^(1/3) + d) + 140070*d^4*e^2*x^(2/3) - 809340*d^5*e*x^(1/3))*n^2
/e^7 - 60*(100*e^6*x^2 + 1800*d^6*log(e*x^(1/3) + d)^2 - 264*d*e^5*x^(5/3) + 555*d^2*e^4*x^(4/3) - 1140*d^3*e^
3*x + 8820*d^6*log(e*x^(1/3) + d) + 2610*d^4*e^2*x^(2/3) - 8820*d^5*e*x^(1/3))*n*log((e*x^(1/3) + d)^n*c)/e^7)
)*b^3

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Fricas [A]  time = 2.58221, size = 2668, normalized size = 2.94 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/72000*(36000*b^3*e^6*x^2*log(c)^3 + 36000*(b^3*e^6*n^3*x^2 - b^3*d^6*n^3)*log(e*x^(1/3) + d)^3 - 1000*(b^3*e
^6*n^3 - 6*a*b^2*e^6*n^2 + 18*a^2*b*e^6*n - 36*a^3*e^6)*x^2 + 1800*(20*b^3*d^3*e^3*n^3*x + 147*b^3*d^6*n^3 - 6
0*a*b^2*d^6*n^2 - 10*(b^3*e^6*n^3 - 6*a*b^2*e^6*n^2)*x^2 + 60*(b^3*e^6*n^2*x^2 - b^3*d^6*n^2)*log(c) + 6*(2*b^
3*d*e^5*n^3*x - 5*b^3*d^4*e^2*n^3)*x^(2/3) - 15*(b^3*d^2*e^4*n^3*x - 4*b^3*d^5*e*n^3)*x^(1/3))*log(e*x^(1/3) +
d)^2 + 18000*(2*b^3*d^3*e^3*n*x - (b^3*e^6*n - 6*a*b^2*e^6)*x^2)*log(c)^2 + 20*(2059*b^3*d^3*e^3*n^3 - 3420*a
*b^2*d^3*e^3*n^2 + 1800*a^2*b*d^3*e^3*n)*x - 60*(13489*b^3*d^6*n^3 - 8820*a*b^2*d^6*n^2 + 1800*a^2*b*d^6*n - 1
00*(b^3*e^6*n^3 - 6*a*b^2*e^6*n^2 + 18*a^2*b*e^6*n)*x^2 - 1800*(b^3*e^6*n*x^2 - b^3*d^6*n)*log(c)^2 + 60*(19*b
^3*d^3*e^3*n^3 - 20*a*b^2*d^3*e^3*n^2)*x - 60*(20*b^3*d^3*e^3*n^2*x + 147*b^3*d^6*n^2 - 60*a*b^2*d^6*n - 10*(b
^3*e^6*n^2 - 6*a*b^2*e^6*n)*x^2)*log(c) - 6*(435*b^3*d^4*e^2*n^3 - 300*a*b^2*d^4*e^2*n^2 - 4*(11*b^3*d*e^5*n^3
- 30*a*b^2*d*e^5*n^2)*x + 60*(2*b^3*d*e^5*n^2*x - 5*b^3*d^4*e^2*n^2)*log(c))*x^(2/3) + 15*(588*b^3*d^5*e*n^3
- 240*a*b^2*d^5*e*n^2 - (37*b^3*d^2*e^4*n^3 - 60*a*b^2*d^2*e^4*n^2)*x + 60*(b^3*d^2*e^4*n^2*x - 4*b^3*d^5*e*n^
2)*log(c))*x^(1/3))*log(e*x^(1/3) + d) + 1200*(5*(b^3*e^6*n^2 - 6*a*b^2*e^6*n + 18*a^2*b*e^6)*x^2 - 3*(19*b^3*
d^3*e^3*n^2 - 20*a*b^2*d^3*e^3*n)*x)*log(c) - 6*(23345*b^3*d^4*e^2*n^3 - 26100*a*b^2*d^4*e^2*n^2 + 9000*a^2*b*
d^4*e^2*n - 1800*(2*b^3*d*e^5*n*x - 5*b^3*d^4*e^2*n)*log(c)^2 - 8*(91*b^3*d*e^5*n^3 - 330*a*b^2*d*e^5*n^2 + 45
0*a^2*b*d*e^5*n)*x - 60*(435*b^3*d^4*e^2*n^2 - 300*a*b^2*d^4*e^2*n - 4*(11*b^3*d*e^5*n^2 - 30*a*b^2*d*e^5*n)*x
)*log(c))*x^(2/3) + 15*(53956*b^3*d^5*e*n^3 - 35280*a*b^2*d^5*e*n^2 + 7200*a^2*b*d^5*e*n - 1800*(b^3*d^2*e^4*n
*x - 4*b^3*d^5*e*n)*log(c)^2 - (919*b^3*d^2*e^4*n^3 - 2220*a*b^2*d^2*e^4*n^2 + 1800*a^2*b*d^2*e^4*n)*x - 60*(5
88*b^3*d^5*e*n^2 - 240*a*b^2*d^5*e*n - (37*b^3*d^2*e^4*n^2 - 60*a*b^2*d^2*e^4*n)*x)*log(c))*x^(1/3))/e^6

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

[Out]

Timed out

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Giac [B]  time = 1.406, size = 3001, normalized size = 3.31 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/72000*(36000*b^3*x^2*e*log(c)^3 + 108000*a*b^2*x^2*e*log(c)^2 + (36000*(x^(1/3)*e + d)^6*e^(-5)*log(x^(1/3)*
e + d)^3 - 216000*(x^(1/3)*e + d)^5*d*e^(-5)*log(x^(1/3)*e + d)^3 + 540000*(x^(1/3)*e + d)^4*d^2*e^(-5)*log(x^
(1/3)*e + d)^3 - 720000*(x^(1/3)*e + d)^3*d^3*e^(-5)*log(x^(1/3)*e + d)^3 + 540000*(x^(1/3)*e + d)^2*d^4*e^(-5
)*log(x^(1/3)*e + d)^3 - 216000*(x^(1/3)*e + d)*d^5*e^(-5)*log(x^(1/3)*e + d)^3 - 18000*(x^(1/3)*e + d)^6*e^(-
5)*log(x^(1/3)*e + d)^2 + 129600*(x^(1/3)*e + d)^5*d*e^(-5)*log(x^(1/3)*e + d)^2 - 405000*(x^(1/3)*e + d)^4*d^
2*e^(-5)*log(x^(1/3)*e + d)^2 + 720000*(x^(1/3)*e + d)^3*d^3*e^(-5)*log(x^(1/3)*e + d)^2 - 810000*(x^(1/3)*e +
d)^2*d^4*e^(-5)*log(x^(1/3)*e + d)^2 + 648000*(x^(1/3)*e + d)*d^5*e^(-5)*log(x^(1/3)*e + d)^2 + 6000*(x^(1/3)
*e + d)^6*e^(-5)*log(x^(1/3)*e + d) - 51840*(x^(1/3)*e + d)^5*d*e^(-5)*log(x^(1/3)*e + d) + 202500*(x^(1/3)*e
+ d)^4*d^2*e^(-5)*log(x^(1/3)*e + d) - 480000*(x^(1/3)*e + d)^3*d^3*e^(-5)*log(x^(1/3)*e + d) + 810000*(x^(1/3
)*e + d)^2*d^4*e^(-5)*log(x^(1/3)*e + d) - 1296000*(x^(1/3)*e + d)*d^5*e^(-5)*log(x^(1/3)*e + d) - 1000*(x^(1/
3)*e + d)^6*e^(-5) + 10368*(x^(1/3)*e + d)^5*d*e^(-5) - 50625*(x^(1/3)*e + d)^4*d^2*e^(-5) + 160000*(x^(1/3)*e
+ d)^3*d^3*e^(-5) - 405000*(x^(1/3)*e + d)^2*d^4*e^(-5) + 1296000*(x^(1/3)*e + d)*d^5*e^(-5))*b^3*n^3 + 60*(1
800*(x^(1/3)*e + d)^6*e^(-5)*log(x^(1/3)*e + d)^2 - 10800*(x^(1/3)*e + d)^5*d*e^(-5)*log(x^(1/3)*e + d)^2 + 27
000*(x^(1/3)*e + d)^4*d^2*e^(-5)*log(x^(1/3)*e + d)^2 - 36000*(x^(1/3)*e + d)^3*d^3*e^(-5)*log(x^(1/3)*e + d)^
2 + 27000*(x^(1/3)*e + d)^2*d^4*e^(-5)*log(x^(1/3)*e + d)^2 - 10800*(x^(1/3)*e + d)*d^5*e^(-5)*log(x^(1/3)*e +
d)^2 - 600*(x^(1/3)*e + d)^6*e^(-5)*log(x^(1/3)*e + d) + 4320*(x^(1/3)*e + d)^5*d*e^(-5)*log(x^(1/3)*e + d) -
13500*(x^(1/3)*e + d)^4*d^2*e^(-5)*log(x^(1/3)*e + d) + 24000*(x^(1/3)*e + d)^3*d^3*e^(-5)*log(x^(1/3)*e + d)
- 27000*(x^(1/3)*e + d)^2*d^4*e^(-5)*log(x^(1/3)*e + d) + 21600*(x^(1/3)*e + d)*d^5*e^(-5)*log(x^(1/3)*e + d)
+ 100*(x^(1/3)*e + d)^6*e^(-5) - 864*(x^(1/3)*e + d)^5*d*e^(-5) + 3375*(x^(1/3)*e + d)^4*d^2*e^(-5) - 8000*(x
^(1/3)*e + d)^3*d^3*e^(-5) + 13500*(x^(1/3)*e + d)^2*d^4*e^(-5) - 21600*(x^(1/3)*e + d)*d^5*e^(-5))*b^3*n^2*lo
g(c) + 108000*a^2*b*x^2*e*log(c) + 1800*(60*(x^(1/3)*e + d)^6*e^(-5)*log(x^(1/3)*e + d) - 360*(x^(1/3)*e + d)^
5*d*e^(-5)*log(x^(1/3)*e + d) + 900*(x^(1/3)*e + d)^4*d^2*e^(-5)*log(x^(1/3)*e + d) - 1200*(x^(1/3)*e + d)^3*d
^3*e^(-5)*log(x^(1/3)*e + d) + 900*(x^(1/3)*e + d)^2*d^4*e^(-5)*log(x^(1/3)*e + d) - 360*(x^(1/3)*e + d)*d^5*e
^(-5)*log(x^(1/3)*e + d) - 10*(x^(1/3)*e + d)^6*e^(-5) + 72*(x^(1/3)*e + d)^5*d*e^(-5) - 225*(x^(1/3)*e + d)^4
*d^2*e^(-5) + 400*(x^(1/3)*e + d)^3*d^3*e^(-5) - 450*(x^(1/3)*e + d)^2*d^4*e^(-5) + 360*(x^(1/3)*e + d)*d^5*e^
(-5))*b^3*n*log(c)^2 + 60*(1800*(x^(1/3)*e + d)^6*e^(-5)*log(x^(1/3)*e + d)^2 - 10800*(x^(1/3)*e + d)^5*d*e^(-
5)*log(x^(1/3)*e + d)^2 + 27000*(x^(1/3)*e + d)^4*d^2*e^(-5)*log(x^(1/3)*e + d)^2 - 36000*(x^(1/3)*e + d)^3*d^
3*e^(-5)*log(x^(1/3)*e + d)^2 + 27000*(x^(1/3)*e + d)^2*d^4*e^(-5)*log(x^(1/3)*e + d)^2 - 10800*(x^(1/3)*e + d
)*d^5*e^(-5)*log(x^(1/3)*e + d)^2 - 600*(x^(1/3)*e + d)^6*e^(-5)*log(x^(1/3)*e + d) + 4320*(x^(1/3)*e + d)^5*d
*e^(-5)*log(x^(1/3)*e + d) - 13500*(x^(1/3)*e + d)^4*d^2*e^(-5)*log(x^(1/3)*e + d) + 24000*(x^(1/3)*e + d)^3*d
^3*e^(-5)*log(x^(1/3)*e + d) - 27000*(x^(1/3)*e + d)^2*d^4*e^(-5)*log(x^(1/3)*e + d) + 21600*(x^(1/3)*e + d)*d
^5*e^(-5)*log(x^(1/3)*e + d) + 100*(x^(1/3)*e + d)^6*e^(-5) - 864*(x^(1/3)*e + d)^5*d*e^(-5) + 3375*(x^(1/3)*e
+ d)^4*d^2*e^(-5) - 8000*(x^(1/3)*e + d)^3*d^3*e^(-5) + 13500*(x^(1/3)*e + d)^2*d^4*e^(-5) - 21600*(x^(1/3)*e
+ d)*d^5*e^(-5))*a*b^2*n^2 + 36000*a^3*x^2*e + 3600*(60*(x^(1/3)*e + d)^6*e^(-5)*log(x^(1/3)*e + d) - 360*(x^
(1/3)*e + d)^5*d*e^(-5)*log(x^(1/3)*e + d) + 900*(x^(1/3)*e + d)^4*d^2*e^(-5)*log(x^(1/3)*e + d) - 1200*(x^(1/
3)*e + d)^3*d^3*e^(-5)*log(x^(1/3)*e + d) + 900*(x^(1/3)*e + d)^2*d^4*e^(-5)*log(x^(1/3)*e + d) - 360*(x^(1/3)
*e + d)*d^5*e^(-5)*log(x^(1/3)*e + d) - 10*(x^(1/3)*e + d)^6*e^(-5) + 72*(x^(1/3)*e + d)^5*d*e^(-5) - 225*(x^(
1/3)*e + d)^4*d^2*e^(-5) + 400*(x^(1/3)*e + d)^3*d^3*e^(-5) - 450*(x^(1/3)*e + d)^2*d^4*e^(-5) + 360*(x^(1/3)*
e + d)*d^5*e^(-5))*a*b^2*n*log(c) + 1800*(60*(x^(1/3)*e + d)^6*e^(-5)*log(x^(1/3)*e + d) - 360*(x^(1/3)*e + d)
^5*d*e^(-5)*log(x^(1/3)*e + d) + 900*(x^(1/3)*e + d)^4*d^2*e^(-5)*log(x^(1/3)*e + d) - 1200*(x^(1/3)*e + d)^3*
d^3*e^(-5)*log(x^(1/3)*e + d) + 900*(x^(1/3)*e + d)^2*d^4*e^(-5)*log(x^(1/3)*e + d) - 360*(x^(1/3)*e + d)*d^5*
e^(-5)*log(x^(1/3)*e + d) - 10*(x^(1/3)*e + d)^6*e^(-5) + 72*(x^(1/3)*e + d)^5*d*e^(-5) - 225*(x^(1/3)*e + d)^
4*d^2*e^(-5) + 400*(x^(1/3)*e + d)^3*d^3*e^(-5) - 450*(x^(1/3)*e + d)^2*d^4*e^(-5) + 360*(x^(1/3)*e + d)*d^5*e
^(-5))*a^2*b*n)*e^(-1)