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

Optimal. Leaf size=267 \[ \frac{2 b d^3 n \log \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^3}-\frac{6 b d^2 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^3}+\frac{3 b d n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^3}-\frac{2 b n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^3}+x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2+\frac{6 b^2 d^2 n^2 \sqrt [3]{x}}{e^2}-\frac{b^2 d^3 n^2 \log ^2\left (d+e \sqrt [3]{x}\right )}{e^3}-\frac{3 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^2}{2 e^3}+\frac{2 b^2 n^2 \left (d+e \sqrt [3]{x}\right )^3}{9 e^3} \]

[Out]

(-3*b^2*d*n^2*(d + e*x^(1/3))^2)/(2*e^3) + (2*b^2*n^2*(d + e*x^(1/3))^3)/(9*e^3) + (6*b^2*d^2*n^2*x^(1/3))/e^2
- (b^2*d^3*n^2*Log[d + e*x^(1/3)]^2)/e^3 - (6*b*d^2*n*(d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n]))/e^3 +
(3*b*d*n*(d + e*x^(1/3))^2*(a + b*Log[c*(d + e*x^(1/3))^n]))/e^3 - (2*b*n*(d + e*x^(1/3))^3*(a + b*Log[c*(d +
e*x^(1/3))^n]))/(3*e^3) + (2*b*d^3*n*Log[d + e*x^(1/3)]*(a + b*Log[c*(d + e*x^(1/3))^n]))/e^3 + x*(a + b*Log[
c*(d + e*x^(1/3))^n])^2

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Rubi [A]  time = 0.289783, antiderivative size = 210, normalized size of antiderivative = 0.79, number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {2451, 2398, 2411, 43, 2334, 12, 14, 2301} \[ -\frac{1}{3} b n \left (\frac{18 d^2 \left (d+e \sqrt [3]{x}\right )}{e^3}-\frac{6 d^3 \log \left (d+e \sqrt [3]{x}\right )}{e^3}-\frac{9 d \left (d+e \sqrt [3]{x}\right )^2}{e^3}+\frac{2 \left (d+e \sqrt [3]{x}\right )^3}{e^3}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )+x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2+\frac{6 b^2 d^2 n^2 \sqrt [3]{x}}{e^2}-\frac{b^2 d^3 n^2 \log ^2\left (d+e \sqrt [3]{x}\right )}{e^3}-\frac{3 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^2}{2 e^3}+\frac{2 b^2 n^2 \left (d+e \sqrt [3]{x}\right )^3}{9 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*b^2*d*n^2*(d + e*x^(1/3))^2)/(2*e^3) + (2*b^2*n^2*(d + e*x^(1/3))^3)/(9*e^3) + (6*b^2*d^2*n^2*x^(1/3))/e^2
- (b^2*d^3*n^2*Log[d + e*x^(1/3)]^2)/e^3 - (b*n*((18*d^2*(d + e*x^(1/3)))/e^3 - (9*d*(d + e*x^(1/3))^2)/e^3 +
(2*(d + e*x^(1/3))^3)/e^3 - (6*d^3*Log[d + e*x^(1/3)])/e^3)*(a + b*Log[c*(d + e*x^(1/3))^n]))/3 + x*(a + b*Lo
g[c*(d + e*x^(1/3))^n])^2

Rule 2451

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_), x_Symbol] :> With[{k = Denominator[n]}, Di
st[k, Subst[Int[x^(k - 1)*(a + b*Log[c*(d + e*x^(k*n))^p])^q, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p,
q}, x] && FractionQ[n]

Rule 2398

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

Rule 2411

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2334

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rubi steps

\begin{align*} \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \, dx &=3 \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )\\ &=x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2-(2 b e n) \operatorname{Subst}\left (\int \frac{x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx,x,\sqrt [3]{x}\right )\\ &=x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2-(2 b n) \operatorname{Subst}\left (\int \frac{\left (-\frac{d}{e}+\frac{x}{e}\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+e \sqrt [3]{x}\right )\\ &=-\frac{1}{3} b n \left (\frac{18 d^2 \left (d+e \sqrt [3]{x}\right )}{e^3}-\frac{9 d \left (d+e \sqrt [3]{x}\right )^2}{e^3}+\frac{2 \left (d+e \sqrt [3]{x}\right )^3}{e^3}-\frac{6 d^3 \log \left (d+e \sqrt [3]{x}\right )}{e^3}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )+x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2+\left (2 b^2 n^2\right ) \operatorname{Subst}\left (\int \frac{18 d^2 x-9 d x^2+2 x^3-6 d^3 \log (x)}{6 e^3 x} \, dx,x,d+e \sqrt [3]{x}\right )\\ &=-\frac{1}{3} b n \left (\frac{18 d^2 \left (d+e \sqrt [3]{x}\right )}{e^3}-\frac{9 d \left (d+e \sqrt [3]{x}\right )^2}{e^3}+\frac{2 \left (d+e \sqrt [3]{x}\right )^3}{e^3}-\frac{6 d^3 \log \left (d+e \sqrt [3]{x}\right )}{e^3}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )+x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2+\frac{\left (b^2 n^2\right ) \operatorname{Subst}\left (\int \frac{18 d^2 x-9 d x^2+2 x^3-6 d^3 \log (x)}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{3 e^3}\\ &=-\frac{1}{3} b n \left (\frac{18 d^2 \left (d+e \sqrt [3]{x}\right )}{e^3}-\frac{9 d \left (d+e \sqrt [3]{x}\right )^2}{e^3}+\frac{2 \left (d+e \sqrt [3]{x}\right )^3}{e^3}-\frac{6 d^3 \log \left (d+e \sqrt [3]{x}\right )}{e^3}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )+x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2+\frac{\left (b^2 n^2\right ) \operatorname{Subst}\left (\int \left (18 d^2-9 d x+2 x^2-\frac{6 d^3 \log (x)}{x}\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{3 e^3}\\ &=-\frac{3 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^2}{2 e^3}+\frac{2 b^2 n^2 \left (d+e \sqrt [3]{x}\right )^3}{9 e^3}+\frac{6 b^2 d^2 n^2 \sqrt [3]{x}}{e^2}-\frac{1}{3} b n \left (\frac{18 d^2 \left (d+e \sqrt [3]{x}\right )}{e^3}-\frac{9 d \left (d+e \sqrt [3]{x}\right )^2}{e^3}+\frac{2 \left (d+e \sqrt [3]{x}\right )^3}{e^3}-\frac{6 d^3 \log \left (d+e \sqrt [3]{x}\right )}{e^3}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )+x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2-\frac{\left (2 b^2 d^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{e^3}\\ &=-\frac{3 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^2}{2 e^3}+\frac{2 b^2 n^2 \left (d+e \sqrt [3]{x}\right )^3}{9 e^3}+\frac{6 b^2 d^2 n^2 \sqrt [3]{x}}{e^2}-\frac{b^2 d^3 n^2 \log ^2\left (d+e \sqrt [3]{x}\right )}{e^3}-\frac{1}{3} b n \left (\frac{18 d^2 \left (d+e \sqrt [3]{x}\right )}{e^3}-\frac{9 d \left (d+e \sqrt [3]{x}\right )^2}{e^3}+\frac{2 \left (d+e \sqrt [3]{x}\right )^3}{e^3}-\frac{6 d^3 \log \left (d+e \sqrt [3]{x}\right )}{e^3}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )+x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2\\ \end{align*}

Mathematica [A]  time = 0.126614, size = 197, normalized size = 0.74 \[ \frac{18 a^2 \left (d^3+e^3 x\right )+6 b \left (6 a \left (d^3+e^3 x\right )-b n \left (6 d^2 e \sqrt [3]{x}+11 d^3-3 d e^2 x^{2/3}+2 e^3 x\right )\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )+6 a b n \left (-6 d^2 e \sqrt [3]{x}+7 d^3+3 d e^2 x^{2/3}-2 e^3 x\right )+18 b^2 \left (d^3+e^3 x\right ) \log ^2\left (c \left (d+e \sqrt [3]{x}\right )^n\right )+b^2 e n^2 \sqrt [3]{x} \left (66 d^2-15 d e \sqrt [3]{x}+4 e^2 x^{2/3}\right )}{18 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b^2*e*n^2*(66*d^2 - 15*d*e*x^(1/3) + 4*e^2*x^(2/3))*x^(1/3) + 6*a*b*n*(7*d^3 - 6*d^2*e*x^(1/3) + 3*d*e^2*x^(2
/3) - 2*e^3*x) + 18*a^2*(d^3 + e^3*x) + 6*b*(6*a*(d^3 + e^3*x) - b*n*(11*d^3 + 6*d^2*e*x^(1/3) - 3*d*e^2*x^(2/
3) + 2*e^3*x))*Log[c*(d + e*x^(1/3))^n] + 18*b^2*(d^3 + e^3*x)*Log[c*(d + e*x^(1/3))^n]^2)/(18*e^3)

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Maple [F]  time = 0.093, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c \left ( d+e\sqrt [3]{x} \right ) ^{n} \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.05169, size = 293, normalized size = 1.1 \begin{align*} \frac{1}{3} \,{\left (e n{\left (\frac{6 \, d^{3} \log \left (e x^{\frac{1}{3}} + d\right )}{e^{4}} - \frac{2 \, e^{2} x - 3 \, d e x^{\frac{2}{3}} + 6 \, d^{2} x^{\frac{1}{3}}}{e^{3}}\right )} + 6 \, x \log \left ({\left (e x^{\frac{1}{3}} + d\right )}^{n} c\right )\right )} a b + \frac{1}{18} \,{\left (6 \, e n{\left (\frac{6 \, d^{3} \log \left (e x^{\frac{1}{3}} + d\right )}{e^{4}} - \frac{2 \, e^{2} x - 3 \, d e x^{\frac{2}{3}} + 6 \, d^{2} x^{\frac{1}{3}}}{e^{3}}\right )} \log \left ({\left (e x^{\frac{1}{3}} + d\right )}^{n} c\right ) + 18 \, x \log \left ({\left (e x^{\frac{1}{3}} + d\right )}^{n} c\right )^{2} - \frac{{\left (18 \, d^{3} \log \left (e x^{\frac{1}{3}} + d\right )^{2} - 4 \, e^{3} x + 66 \, d^{3} \log \left (e x^{\frac{1}{3}} + d\right ) + 15 \, d e^{2} x^{\frac{2}{3}} - 66 \, d^{2} e x^{\frac{1}{3}}\right )} n^{2}}{e^{3}}\right )} b^{2} + a^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(e*n*(6*d^3*log(e*x^(1/3) + d)/e^4 - (2*e^2*x - 3*d*e*x^(2/3) + 6*d^2*x^(1/3))/e^3) + 6*x*log((e*x^(1/3) +
d)^n*c))*a*b + 1/18*(6*e*n*(6*d^3*log(e*x^(1/3) + d)/e^4 - (2*e^2*x - 3*d*e*x^(2/3) + 6*d^2*x^(1/3))/e^3)*log
((e*x^(1/3) + d)^n*c) + 18*x*log((e*x^(1/3) + d)^n*c)^2 - (18*d^3*log(e*x^(1/3) + d)^2 - 4*e^3*x + 66*d^3*log(
e*x^(1/3) + d) + 15*d*e^2*x^(2/3) - 66*d^2*e*x^(1/3))*n^2/e^3)*b^2 + a^2*x

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Fricas [A]  time = 1.97478, size = 664, normalized size = 2.49 \begin{align*} \frac{18 \, b^{2} e^{3} x \log \left (c\right )^{2} + 18 \,{\left (b^{2} e^{3} n^{2} x + b^{2} d^{3} n^{2}\right )} \log \left (e x^{\frac{1}{3}} + d\right )^{2} - 12 \,{\left (b^{2} e^{3} n - 3 \, a b e^{3}\right )} x \log \left (c\right ) + 2 \,{\left (2 \, b^{2} e^{3} n^{2} - 6 \, a b e^{3} n + 9 \, a^{2} e^{3}\right )} x + 6 \,{\left (3 \, b^{2} d e^{2} n^{2} x^{\frac{2}{3}} - 6 \, b^{2} d^{2} e n^{2} x^{\frac{1}{3}} - 11 \, b^{2} d^{3} n^{2} + 6 \, a b d^{3} n - 2 \,{\left (b^{2} e^{3} n^{2} - 3 \, a b e^{3} n\right )} x + 6 \,{\left (b^{2} e^{3} n x + b^{2} d^{3} n\right )} \log \left (c\right )\right )} \log \left (e x^{\frac{1}{3}} + d\right ) - 3 \,{\left (5 \, b^{2} d e^{2} n^{2} - 6 \, b^{2} d e^{2} n \log \left (c\right ) - 6 \, a b d e^{2} n\right )} x^{\frac{2}{3}} + 6 \,{\left (11 \, b^{2} d^{2} e n^{2} - 6 \, b^{2} d^{2} e n \log \left (c\right ) - 6 \, a b d^{2} e n\right )} x^{\frac{1}{3}}}{18 \, e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*(18*b^2*e^3*x*log(c)^2 + 18*(b^2*e^3*n^2*x + b^2*d^3*n^2)*log(e*x^(1/3) + d)^2 - 12*(b^2*e^3*n - 3*a*b*e^
3)*x*log(c) + 2*(2*b^2*e^3*n^2 - 6*a*b*e^3*n + 9*a^2*e^3)*x + 6*(3*b^2*d*e^2*n^2*x^(2/3) - 6*b^2*d^2*e*n^2*x^(
1/3) - 11*b^2*d^3*n^2 + 6*a*b*d^3*n - 2*(b^2*e^3*n^2 - 3*a*b*e^3*n)*x + 6*(b^2*e^3*n*x + b^2*d^3*n)*log(c))*lo
g(e*x^(1/3) + d) - 3*(5*b^2*d*e^2*n^2 - 6*b^2*d*e^2*n*log(c) - 6*a*b*d*e^2*n)*x^(2/3) + 6*(11*b^2*d^2*e*n^2 -
6*b^2*d^2*e*n*log(c) - 6*a*b*d^2*e*n)*x^(1/3))/e^3

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \log{\left (c \left (d + e \sqrt [3]{x}\right )^{n} \right )}\right )^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.37276, size = 647, normalized size = 2.42 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*(18*b^2*x*e*log(c)^2 + (18*(x^(1/3)*e + d)^3*e^(-2)*log(x^(1/3)*e + d)^2 - 54*(x^(1/3)*e + d)^2*d*e^(-2)*
log(x^(1/3)*e + d)^2 + 54*(x^(1/3)*e + d)*d^2*e^(-2)*log(x^(1/3)*e + d)^2 - 12*(x^(1/3)*e + d)^3*e^(-2)*log(x^
(1/3)*e + d) + 54*(x^(1/3)*e + d)^2*d*e^(-2)*log(x^(1/3)*e + d) - 108*(x^(1/3)*e + d)*d^2*e^(-2)*log(x^(1/3)*e
+ d) + 4*(x^(1/3)*e + d)^3*e^(-2) - 27*(x^(1/3)*e + d)^2*d*e^(-2) + 108*(x^(1/3)*e + d)*d^2*e^(-2))*b^2*n^2 +
6*(6*(x^(1/3)*e + d)^3*e^(-2)*log(x^(1/3)*e + d) - 18*(x^(1/3)*e + d)^2*d*e^(-2)*log(x^(1/3)*e + d) + 18*(x^(
1/3)*e + d)*d^2*e^(-2)*log(x^(1/3)*e + d) - 2*(x^(1/3)*e + d)^3*e^(-2) + 9*(x^(1/3)*e + d)^2*d*e^(-2) - 18*(x^
(1/3)*e + d)*d^2*e^(-2))*b^2*n*log(c) + 36*a*b*x*e*log(c) + 6*(6*(x^(1/3)*e + d)^3*e^(-2)*log(x^(1/3)*e + d) -
18*(x^(1/3)*e + d)^2*d*e^(-2)*log(x^(1/3)*e + d) + 18*(x^(1/3)*e + d)*d^2*e^(-2)*log(x^(1/3)*e + d) - 2*(x^(1
/3)*e + d)^3*e^(-2) + 9*(x^(1/3)*e + d)^2*d*e^(-2) - 18*(x^(1/3)*e + d)*d^2*e^(-2))*a*b*n + 18*a^2*x*e)*e^(-1)