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

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

[Out]

(-15*b^3*d^4*n^3*(d + e*Sqrt[x])^2)/(4*e^6) + (40*b^3*d^3*n^3*(d + e*Sqrt[x])^3)/(27*e^6) - (15*b^3*d^2*n^3*(d
+ e*Sqrt[x])^4)/(32*e^6) + (12*b^3*d*n^3*(d + e*Sqrt[x])^5)/(125*e^6) - (b^3*n^3*(d + e*Sqrt[x])^6)/(108*e^6)
- (12*a*b^2*d^5*n^2*Sqrt[x])/e^5 + (12*b^3*d^5*n^3*Sqrt[x])/e^5 - (12*b^3*d^5*n^2*(d + e*Sqrt[x])*Log[c*(d +
e*Sqrt[x])^n])/e^6 + (15*b^2*d^4*n^2*(d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(2*e^6) - (40*b^2*d^3
*n^2*(d + e*Sqrt[x])^3*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(9*e^6) + (15*b^2*d^2*n^2*(d + e*Sqrt[x])^4*(a + b*Lo
g[c*(d + e*Sqrt[x])^n]))/(8*e^6) - (12*b^2*d*n^2*(d + e*Sqrt[x])^5*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(25*e^6)
+ (b^2*n^2*(d + e*Sqrt[x])^6*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(18*e^6) + (6*b*d^5*n*(d + e*Sqrt[x])*(a + b*Lo
g[c*(d + e*Sqrt[x])^n])^2)/e^6 - (15*b*d^4*n*(d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/(2*e^6) + (
20*b*d^3*n*(d + e*Sqrt[x])^3*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/(3*e^6) - (15*b*d^2*n*(d + e*Sqrt[x])^4*(a +
b*Log[c*(d + e*Sqrt[x])^n])^2)/(4*e^6) + (6*b*d*n*(d + e*Sqrt[x])^5*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/(5*e^6
) - (b*n*(d + e*Sqrt[x])^6*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/(6*e^6) - (2*d^5*(d + e*Sqrt[x])*(a + b*Log[c*(
d + e*Sqrt[x])^n])^3)/e^6 + (5*d^4*(d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n])^3)/e^6 - (20*d^3*(d + e*
Sqrt[x])^3*(a + b*Log[c*(d + e*Sqrt[x])^n])^3)/(3*e^6) + (5*d^2*(d + e*Sqrt[x])^4*(a + b*Log[c*(d + e*Sqrt[x])
^n])^3)/e^6 - (2*d*(d + e*Sqrt[x])^5*(a + b*Log[c*(d + e*Sqrt[x])^n])^3)/e^6 + ((d + e*Sqrt[x])^6*(a + b*Log[c
*(d + e*Sqrt[x])^n])^3)/(3*e^6)

________________________________________________________________________________________

Rubi [A]  time = 1.00708, antiderivative size = 907, normalized size of antiderivative = 1., number of steps used = 28, 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} \[ -\frac{b^3 n^3 \left (d+e \sqrt{x}\right )^6}{108 e^6}+\frac{\left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3 \left (d+e \sqrt{x}\right )^6}{3 e^6}-\frac{b n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2 \left (d+e \sqrt{x}\right )^6}{6 e^6}+\frac{b^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right ) \left (d+e \sqrt{x}\right )^6}{18 e^6}+\frac{12 b^3 d n^3 \left (d+e \sqrt{x}\right )^5}{125 e^6}-\frac{2 d \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3 \left (d+e \sqrt{x}\right )^5}{e^6}+\frac{6 b d n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2 \left (d+e \sqrt{x}\right )^5}{5 e^6}-\frac{12 b^2 d n^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right ) \left (d+e \sqrt{x}\right )^5}{25 e^6}-\frac{15 b^3 d^2 n^3 \left (d+e \sqrt{x}\right )^4}{32 e^6}+\frac{5 d^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3 \left (d+e \sqrt{x}\right )^4}{e^6}-\frac{15 b d^2 n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2 \left (d+e \sqrt{x}\right )^4}{4 e^6}+\frac{15 b^2 d^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right ) \left (d+e \sqrt{x}\right )^4}{8 e^6}+\frac{40 b^3 d^3 n^3 \left (d+e \sqrt{x}\right )^3}{27 e^6}-\frac{20 d^3 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3 \left (d+e \sqrt{x}\right )^3}{3 e^6}+\frac{20 b d^3 n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2 \left (d+e \sqrt{x}\right )^3}{3 e^6}-\frac{40 b^2 d^3 n^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right ) \left (d+e \sqrt{x}\right )^3}{9 e^6}-\frac{15 b^3 d^4 n^3 \left (d+e \sqrt{x}\right )^2}{4 e^6}+\frac{5 d^4 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3 \left (d+e \sqrt{x}\right )^2}{e^6}-\frac{15 b d^4 n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2 \left (d+e \sqrt{x}\right )^2}{2 e^6}+\frac{15 b^2 d^4 n^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right ) \left (d+e \sqrt{x}\right )^2}{2 e^6}-\frac{2 d^5 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3 \left (d+e \sqrt{x}\right )}{e^6}+\frac{6 b d^5 n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2 \left (d+e \sqrt{x}\right )}{e^6}-\frac{12 b^3 d^5 n^2 \log \left (c \left (d+e \sqrt{x}\right )^n\right ) \left (d+e \sqrt{x}\right )}{e^6}+\frac{12 b^3 d^5 n^3 \sqrt{x}}{e^5}-\frac{12 a b^2 d^5 n^2 \sqrt{x}}{e^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-15*b^3*d^4*n^3*(d + e*Sqrt[x])^2)/(4*e^6) + (40*b^3*d^3*n^3*(d + e*Sqrt[x])^3)/(27*e^6) - (15*b^3*d^2*n^3*(d
+ e*Sqrt[x])^4)/(32*e^6) + (12*b^3*d*n^3*(d + e*Sqrt[x])^5)/(125*e^6) - (b^3*n^3*(d + e*Sqrt[x])^6)/(108*e^6)
- (12*a*b^2*d^5*n^2*Sqrt[x])/e^5 + (12*b^3*d^5*n^3*Sqrt[x])/e^5 - (12*b^3*d^5*n^2*(d + e*Sqrt[x])*Log[c*(d +
e*Sqrt[x])^n])/e^6 + (15*b^2*d^4*n^2*(d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(2*e^6) - (40*b^2*d^3
*n^2*(d + e*Sqrt[x])^3*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(9*e^6) + (15*b^2*d^2*n^2*(d + e*Sqrt[x])^4*(a + b*Lo
g[c*(d + e*Sqrt[x])^n]))/(8*e^6) - (12*b^2*d*n^2*(d + e*Sqrt[x])^5*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(25*e^6)
+ (b^2*n^2*(d + e*Sqrt[x])^6*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(18*e^6) + (6*b*d^5*n*(d + e*Sqrt[x])*(a + b*Lo
g[c*(d + e*Sqrt[x])^n])^2)/e^6 - (15*b*d^4*n*(d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/(2*e^6) + (
20*b*d^3*n*(d + e*Sqrt[x])^3*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/(3*e^6) - (15*b*d^2*n*(d + e*Sqrt[x])^4*(a +
b*Log[c*(d + e*Sqrt[x])^n])^2)/(4*e^6) + (6*b*d*n*(d + e*Sqrt[x])^5*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/(5*e^6
) - (b*n*(d + e*Sqrt[x])^6*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/(6*e^6) - (2*d^5*(d + e*Sqrt[x])*(a + b*Log[c*(
d + e*Sqrt[x])^n])^3)/e^6 + (5*d^4*(d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n])^3)/e^6 - (20*d^3*(d + e*
Sqrt[x])^3*(a + b*Log[c*(d + e*Sqrt[x])^n])^3)/(3*e^6) + (5*d^2*(d + e*Sqrt[x])^4*(a + b*Log[c*(d + e*Sqrt[x])
^n])^3)/e^6 - (2*d*(d + e*Sqrt[x])^5*(a + b*Log[c*(d + e*Sqrt[x])^n])^3)/e^6 + ((d + e*Sqrt[x])^6*(a + b*Log[c
*(d + e*Sqrt[x])^n])^3)/(3*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^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3 \, dx &=2 \operatorname{Subst}\left (\int x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt{x}\right )\\ &=2 \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{x}\right )\\ &=\frac{2 \operatorname{Subst}\left (\int (d+e x)^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt{x}\right )}{e^5}-\frac{(10 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{x}\right )}{e^5}+\frac{\left (20 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{x}\right )}{e^5}-\frac{\left (20 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{x}\right )}{e^5}+\frac{\left (10 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{x}\right )}{e^5}-\frac{\left (2 d^5\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt{x}\right )}{e^5}\\ &=\frac{2 \operatorname{Subst}\left (\int x^5 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt{x}\right )}{e^6}-\frac{(10 d) \operatorname{Subst}\left (\int x^4 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt{x}\right )}{e^6}+\frac{\left (20 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{x}\right )}{e^6}-\frac{\left (20 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{x}\right )}{e^6}+\frac{\left (10 d^4\right ) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt{x}\right )}{e^6}-\frac{\left (2 d^5\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt{x}\right )}{e^6}\\ &=-\frac{2 d^5 \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{e^6}+\frac{5 d^4 \left (d+e \sqrt{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{e^6}-\frac{20 d^3 \left (d+e \sqrt{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{3 e^6}+\frac{5 d^2 \left (d+e \sqrt{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{e^6}-\frac{2 d \left (d+e \sqrt{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{e^6}+\frac{\left (d+e \sqrt{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{3 e^6}-\frac{(b n) \operatorname{Subst}\left (\int x^5 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt{x}\right )}{e^6}+\frac{(6 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{x}\right )}{e^6}-\frac{\left (15 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{x}\right )}{e^6}+\frac{\left (20 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{x}\right )}{e^6}-\frac{\left (15 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{x}\right )}{e^6}+\frac{\left (6 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{x}\right )}{e^6}\\ &=\frac{6 b d^5 n \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{e^6}-\frac{15 b d^4 n \left (d+e \sqrt{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{2 e^6}+\frac{20 b d^3 n \left (d+e \sqrt{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{3 e^6}-\frac{15 b d^2 n \left (d+e \sqrt{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{4 e^6}+\frac{6 b d n \left (d+e \sqrt{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{5 e^6}-\frac{b n \left (d+e \sqrt{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{6 e^6}-\frac{2 d^5 \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{e^6}+\frac{5 d^4 \left (d+e \sqrt{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{e^6}-\frac{20 d^3 \left (d+e \sqrt{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{3 e^6}+\frac{5 d^2 \left (d+e \sqrt{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{e^6}-\frac{2 d \left (d+e \sqrt{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{e^6}+\frac{\left (d+e \sqrt{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{3 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{x}\right )}{3 e^6}-\frac{\left (12 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{x}\right )}{5 e^6}+\frac{\left (15 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{x}\right )}{2 e^6}-\frac{\left (40 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{x}\right )}{3 e^6}+\frac{\left (15 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{x}\right )}{e^6}-\frac{\left (12 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{x}\right )}{e^6}\\ &=-\frac{15 b^3 d^4 n^3 \left (d+e \sqrt{x}\right )^2}{4 e^6}+\frac{40 b^3 d^3 n^3 \left (d+e \sqrt{x}\right )^3}{27 e^6}-\frac{15 b^3 d^2 n^3 \left (d+e \sqrt{x}\right )^4}{32 e^6}+\frac{12 b^3 d n^3 \left (d+e \sqrt{x}\right )^5}{125 e^6}-\frac{b^3 n^3 \left (d+e \sqrt{x}\right )^6}{108 e^6}-\frac{12 a b^2 d^5 n^2 \sqrt{x}}{e^5}+\frac{15 b^2 d^4 n^2 \left (d+e \sqrt{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{2 e^6}-\frac{40 b^2 d^3 n^2 \left (d+e \sqrt{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{9 e^6}+\frac{15 b^2 d^2 n^2 \left (d+e \sqrt{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{8 e^6}-\frac{12 b^2 d n^2 \left (d+e \sqrt{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{25 e^6}+\frac{b^2 n^2 \left (d+e \sqrt{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{18 e^6}+\frac{6 b d^5 n \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{e^6}-\frac{15 b d^4 n \left (d+e \sqrt{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{2 e^6}+\frac{20 b d^3 n \left (d+e \sqrt{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{3 e^6}-\frac{15 b d^2 n \left (d+e \sqrt{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{4 e^6}+\frac{6 b d n \left (d+e \sqrt{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{5 e^6}-\frac{b n \left (d+e \sqrt{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{6 e^6}-\frac{2 d^5 \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{e^6}+\frac{5 d^4 \left (d+e \sqrt{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{e^6}-\frac{20 d^3 \left (d+e \sqrt{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{3 e^6}+\frac{5 d^2 \left (d+e \sqrt{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{e^6}-\frac{2 d \left (d+e \sqrt{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{e^6}+\frac{\left (d+e \sqrt{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{3 e^6}-\frac{\left (12 b^3 d^5 n^2\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e \sqrt{x}\right )}{e^6}\\ &=-\frac{15 b^3 d^4 n^3 \left (d+e \sqrt{x}\right )^2}{4 e^6}+\frac{40 b^3 d^3 n^3 \left (d+e \sqrt{x}\right )^3}{27 e^6}-\frac{15 b^3 d^2 n^3 \left (d+e \sqrt{x}\right )^4}{32 e^6}+\frac{12 b^3 d n^3 \left (d+e \sqrt{x}\right )^5}{125 e^6}-\frac{b^3 n^3 \left (d+e \sqrt{x}\right )^6}{108 e^6}-\frac{12 a b^2 d^5 n^2 \sqrt{x}}{e^5}+\frac{12 b^3 d^5 n^3 \sqrt{x}}{e^5}-\frac{12 b^3 d^5 n^2 \left (d+e \sqrt{x}\right ) \log \left (c \left (d+e \sqrt{x}\right )^n\right )}{e^6}+\frac{15 b^2 d^4 n^2 \left (d+e \sqrt{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{2 e^6}-\frac{40 b^2 d^3 n^2 \left (d+e \sqrt{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{9 e^6}+\frac{15 b^2 d^2 n^2 \left (d+e \sqrt{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{8 e^6}-\frac{12 b^2 d n^2 \left (d+e \sqrt{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{25 e^6}+\frac{b^2 n^2 \left (d+e \sqrt{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{18 e^6}+\frac{6 b d^5 n \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{e^6}-\frac{15 b d^4 n \left (d+e \sqrt{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{2 e^6}+\frac{20 b d^3 n \left (d+e \sqrt{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{3 e^6}-\frac{15 b d^2 n \left (d+e \sqrt{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{4 e^6}+\frac{6 b d n \left (d+e \sqrt{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{5 e^6}-\frac{b n \left (d+e \sqrt{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{6 e^6}-\frac{2 d^5 \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{e^6}+\frac{5 d^4 \left (d+e \sqrt{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{e^6}-\frac{20 d^3 \left (d+e \sqrt{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{3 e^6}+\frac{5 d^2 \left (d+e \sqrt{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{e^6}-\frac{2 d \left (d+e \sqrt{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{e^6}+\frac{\left (d+e \sqrt{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{3 e^6}\\ \end{align*}

Mathematica [A]  time = 0.47715, size = 577, normalized size = 0.64 \[ \frac{-60 b \left (1800 a^2 \left (d^6-e^6 x^3\right )-60 a b n \left (20 d^3 e^3 x^{3/2}-15 d^2 e^4 x^2-30 d^4 e^2 x+60 d^5 e \sqrt{x}+147 d^6+12 d e^5 x^{5/2}-10 e^6 x^3\right )+b^2 n^2 \left (1140 d^3 e^3 x^{3/2}-555 d^2 e^4 x^2-2610 d^4 e^2 x+8820 d^5 e \sqrt{x}+13489 d^6+264 d e^5 x^{5/2}-100 e^6 x^3\right )\right ) \log \left (c \left (d+e \sqrt{x}\right )^n\right )+1800 a^2 b n \left (20 d^3 e^3 x^{3/2}-15 d^2 e^4 x^2-30 d^4 e^2 x+60 d^5 e \sqrt{x}+147 d^6+12 d e^5 x^{5/2}-10 e^6 x^3\right )-36000 a^3 \left (d^6-e^6 x^3\right )-1800 b^2 \left (60 a \left (d^6-e^6 x^3\right )+b n \left (-20 d^3 e^3 x^{3/2}+15 d^2 e^4 x^2+30 d^4 e^2 x-60 d^5 e \sqrt{x}-147 d^6-12 d e^5 x^{5/2}+10 e^6 x^3\right )\right ) \log ^2\left (c \left (d+e \sqrt{x}\right )^n\right )+60 a b^2 n^2 \left (-1140 d^3 e^3 x^{3/2}+555 d^2 e^4 x^2+2610 d^4 e^2 x-8820 d^5 e \sqrt{x}+8111 d^6-264 d e^5 x^{5/2}+100 e^6 x^3\right )-36000 b^3 \left (d^6-e^6 x^3\right ) \log ^3\left (c \left (d+e \sqrt{x}\right )^n\right )+b^3 e n^3 \sqrt{x} \left (-13785 d^2 e^3 x^{3/2}+41180 d^3 e^2 x-140070 d^4 e \sqrt{x}+809340 d^5+4368 d e^4 x^2-1000 e^5 x^{5/2}\right )}{108000 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b^3*e*n^3*Sqrt[x]*(809340*d^5 - 140070*d^4*e*Sqrt[x] + 41180*d^3*e^2*x - 13785*d^2*e^3*x^(3/2) + 4368*d*e^4*x
^2 - 1000*e^5*x^(5/2)) + 1800*a^2*b*n*(147*d^6 + 60*d^5*e*Sqrt[x] - 30*d^4*e^2*x + 20*d^3*e^3*x^(3/2) - 15*d^2
*e^4*x^2 + 12*d*e^5*x^(5/2) - 10*e^6*x^3) - 36000*a^3*(d^6 - e^6*x^3) + 60*a*b^2*n^2*(8111*d^6 - 8820*d^5*e*Sq
rt[x] + 2610*d^4*e^2*x - 1140*d^3*e^3*x^(3/2) + 555*d^2*e^4*x^2 - 264*d*e^5*x^(5/2) + 100*e^6*x^3) - 60*b*(b^2
*n^2*(13489*d^6 + 8820*d^5*e*Sqrt[x] - 2610*d^4*e^2*x + 1140*d^3*e^3*x^(3/2) - 555*d^2*e^4*x^2 + 264*d*e^5*x^(
5/2) - 100*e^6*x^3) - 60*a*b*n*(147*d^6 + 60*d^5*e*Sqrt[x] - 30*d^4*e^2*x + 20*d^3*e^3*x^(3/2) - 15*d^2*e^4*x^
2 + 12*d*e^5*x^(5/2) - 10*e^6*x^3) + 1800*a^2*(d^6 - e^6*x^3))*Log[c*(d + e*Sqrt[x])^n] - 1800*b^2*(60*a*(d^6
- e^6*x^3) + b*n*(-147*d^6 - 60*d^5*e*Sqrt[x] + 30*d^4*e^2*x - 20*d^3*e^3*x^(3/2) + 15*d^2*e^4*x^2 - 12*d*e^5*
x^(5/2) + 10*e^6*x^3))*Log[c*(d + e*Sqrt[x])^n]^2 - 36000*b^3*(d^6 - e^6*x^3)*Log[c*(d + e*Sqrt[x])^n]^3)/(108
000*e^6)

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Maple [F]  time = 0.098, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( a+b\ln \left ( c \left ( d+e\sqrt{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^(1/2))^n))^3,x)

[Out]

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

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Maxima [A]  time = 1.09476, size = 899, 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^2*(a+b*log(c*(d+e*x^(1/2))^n))^3,x, algorithm="maxima")

[Out]

1/3*b^3*x^3*log((e*sqrt(x) + d)^n*c)^3 + a*b^2*x^3*log((e*sqrt(x) + d)^n*c)^2 + a^2*b*x^3*log((e*sqrt(x) + d)^
n*c) + 1/3*a^3*x^3 - 1/60*a^2*b*e*n*(60*d^6*log(e*sqrt(x) + d)/e^7 + (10*e^5*x^3 - 12*d*e^4*x^(5/2) + 15*d^2*e
^3*x^2 - 20*d^3*e^2*x^(3/2) + 30*d^4*e*x - 60*d^5*sqrt(x))/e^6) - 1/1800*(60*e*n*(60*d^6*log(e*sqrt(x) + d)/e^
7 + (10*e^5*x^3 - 12*d*e^4*x^(5/2) + 15*d^2*e^3*x^2 - 20*d^3*e^2*x^(3/2) + 30*d^4*e*x - 60*d^5*sqrt(x))/e^6)*l
og((e*sqrt(x) + d)^n*c) - (100*e^6*x^3 - 264*d*e^5*x^(5/2) + 555*d^2*e^4*x^2 + 1800*d^6*log(e*sqrt(x) + d)^2 -
1140*d^3*e^3*x^(3/2) + 2610*d^4*e^2*x + 8820*d^6*log(e*sqrt(x) + d) - 8820*d^5*e*sqrt(x))*n^2/e^6)*a*b^2 - 1/
108000*(1800*e*n*(60*d^6*log(e*sqrt(x) + d)/e^7 + (10*e^5*x^3 - 12*d*e^4*x^(5/2) + 15*d^2*e^3*x^2 - 20*d^3*e^2
*x^(3/2) + 30*d^4*e*x - 60*d^5*sqrt(x))/e^6)*log((e*sqrt(x) + d)^n*c)^2 + e*n*((1000*e^6*x^3 + 36000*d^6*log(e
*sqrt(x) + d)^3 - 4368*d*e^5*x^(5/2) + 13785*d^2*e^4*x^2 + 264600*d^6*log(e*sqrt(x) + d)^2 - 41180*d^3*e^3*x^(
3/2) + 140070*d^4*e^2*x + 809340*d^6*log(e*sqrt(x) + d) - 809340*d^5*e*sqrt(x))*n^2/e^7 - 60*(100*e^6*x^3 - 26
4*d*e^5*x^(5/2) + 555*d^2*e^4*x^2 + 1800*d^6*log(e*sqrt(x) + d)^2 - 1140*d^3*e^3*x^(3/2) + 2610*d^4*e^2*x + 88
20*d^6*log(e*sqrt(x) + d) - 8820*d^5*e*sqrt(x))*n*log((e*sqrt(x) + d)^n*c)/e^7))*b^3

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Fricas [A]  time = 2.47634, size = 2655, normalized size = 2.93 \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^(1/2))^n))^3,x, algorithm="fricas")

[Out]

1/108000*(36000*b^3*e^6*x^3*log(c)^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^3
+ 36000*(b^3*e^6*n^3*x^3 - b^3*d^6*n^3)*log(e*sqrt(x) + d)^3 - 15*(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^2 - 1800*(15*b^3*d^2*e^4*n^3*x^2 + 30*b^3*d^4*e^2*n^3*x - 147*b^3*d^6*n^3 + 60*a*b
^2*d^6*n^2 + 10*(b^3*e^6*n^3 - 6*a*b^2*e^6*n^2)*x^3 - 60*(b^3*e^6*n^2*x^3 - b^3*d^6*n^2)*log(c) - 4*(3*b^3*d*e
^5*n^3*x^2 + 5*b^3*d^3*e^3*n^3*x + 15*b^3*d^5*e*n^3)*sqrt(x))*log(e*sqrt(x) + d)^2 - 9000*(3*b^3*d^2*e^4*n*x^2
+ 6*b^3*d^4*e^2*n*x + 2*(b^3*e^6*n - 6*a*b^2*e^6)*x^3)*log(c)^2 - 30*(4669*b^3*d^4*e^2*n^3 - 5220*a*b^2*d^4*e
^2*n^2 + 1800*a^2*b*d^4*e^2*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 - 100*(b^3*e^
6*n^3 - 6*a*b^2*e^6*n^2 + 18*a^2*b*e^6*n)*x^3 - 15*(37*b^3*d^2*e^4*n^3 - 60*a*b^2*d^2*e^4*n^2)*x^2 - 1800*(b^3
*e^6*n*x^3 - b^3*d^6*n)*log(c)^2 - 90*(29*b^3*d^4*e^2*n^3 - 20*a*b^2*d^4*e^2*n^2)*x + 60*(15*b^3*d^2*e^4*n^2*x
^2 + 30*b^3*d^4*e^2*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^3)*log(c) +
12*(735*b^3*d^5*e*n^3 - 300*a*b^2*d^5*e*n^2 + 2*(11*b^3*d*e^5*n^3 - 30*a*b^2*d*e^5*n^2)*x^2 + 5*(19*b^3*d^3*e^
3*n^3 - 20*a*b^2*d^3*e^3*n^2)*x - 20*(3*b^3*d*e^5*n^2*x^2 + 5*b^3*d^3*e^3*n^2*x + 15*b^3*d^5*e*n^2)*log(c))*sq
rt(x))*log(e*sqrt(x) + d) + 300*(20*(b^3*e^6*n^2 - 6*a*b^2*e^6*n + 18*a^2*b*e^6)*x^3 + 3*(37*b^3*d^2*e^4*n^2 -
60*a*b^2*d^2*e^4*n)*x^2 + 18*(29*b^3*d^4*e^2*n^2 - 20*a*b^2*d^4*e^2*n)*x)*log(c) + 4*(202335*b^3*d^5*e*n^3 -
132300*a*b^2*d^5*e*n^2 + 27000*a^2*b*d^5*e*n + 12*(91*b^3*d*e^5*n^3 - 330*a*b^2*d*e^5*n^2 + 450*a^2*b*d*e^5*n)
*x^2 + 1800*(3*b^3*d*e^5*n*x^2 + 5*b^3*d^3*e^3*n*x + 15*b^3*d^5*e*n)*log(c)^2 + 5*(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 - 180*(735*b^3*d^5*e*n^2 - 300*a*b^2*d^5*e*n + 2*(11*b^3*d*e^5*n^
2 - 30*a*b^2*d*e^5*n)*x^2 + 5*(19*b^3*d^3*e^3*n^2 - 20*a*b^2*d^3*e^3*n)*x)*log(c))*sqrt(x))/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**2*(a+b*ln(c*(d+e*x**(1/2))**n))**3,x)

[Out]

Timed out

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Giac [B]  time = 1.4142, size = 3444, normalized size = 3.8 \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^(1/2))^n))^3,x, algorithm="giac")

[Out]

1/108000*((36000*(sqrt(x)*e + d)^6*e^(-4)*log(sqrt(x)*e + d)^3 - 216000*(sqrt(x)*e + d)^5*d*e^(-4)*log(sqrt(x)
*e + d)^3 + 540000*(sqrt(x)*e + d)^4*d^2*e^(-4)*log(sqrt(x)*e + d)^3 - 720000*(sqrt(x)*e + d)^3*d^3*e^(-4)*log
(sqrt(x)*e + d)^3 + 540000*(sqrt(x)*e + d)^2*d^4*e^(-4)*log(sqrt(x)*e + d)^3 - 216000*(sqrt(x)*e + d)*d^5*e^(-
4)*log(sqrt(x)*e + d)^3 - 18000*(sqrt(x)*e + d)^6*e^(-4)*log(sqrt(x)*e + d)^2 + 129600*(sqrt(x)*e + d)^5*d*e^(
-4)*log(sqrt(x)*e + d)^2 - 405000*(sqrt(x)*e + d)^4*d^2*e^(-4)*log(sqrt(x)*e + d)^2 + 720000*(sqrt(x)*e + d)^3
*d^3*e^(-4)*log(sqrt(x)*e + d)^2 - 810000*(sqrt(x)*e + d)^2*d^4*e^(-4)*log(sqrt(x)*e + d)^2 + 648000*(sqrt(x)*
e + d)*d^5*e^(-4)*log(sqrt(x)*e + d)^2 + 6000*(sqrt(x)*e + d)^6*e^(-4)*log(sqrt(x)*e + d) - 51840*(sqrt(x)*e +
d)^5*d*e^(-4)*log(sqrt(x)*e + d) + 202500*(sqrt(x)*e + d)^4*d^2*e^(-4)*log(sqrt(x)*e + d) - 480000*(sqrt(x)*e
+ d)^3*d^3*e^(-4)*log(sqrt(x)*e + d) + 810000*(sqrt(x)*e + d)^2*d^4*e^(-4)*log(sqrt(x)*e + d) - 1296000*(sqrt
(x)*e + d)*d^5*e^(-4)*log(sqrt(x)*e + d) - 1000*(sqrt(x)*e + d)^6*e^(-4) + 10368*(sqrt(x)*e + d)^5*d*e^(-4) -
50625*(sqrt(x)*e + d)^4*d^2*e^(-4) + 160000*(sqrt(x)*e + d)^3*d^3*e^(-4) - 405000*(sqrt(x)*e + d)^2*d^4*e^(-4)
+ 1296000*(sqrt(x)*e + d)*d^5*e^(-4))*b^3*n^3*e^(-1) + 60*(1800*(sqrt(x)*e + d)^6*e^(-4)*log(sqrt(x)*e + d)^2
- 10800*(sqrt(x)*e + d)^5*d*e^(-4)*log(sqrt(x)*e + d)^2 + 27000*(sqrt(x)*e + d)^4*d^2*e^(-4)*log(sqrt(x)*e +
d)^2 - 36000*(sqrt(x)*e + d)^3*d^3*e^(-4)*log(sqrt(x)*e + d)^2 + 27000*(sqrt(x)*e + d)^2*d^4*e^(-4)*log(sqrt(x
)*e + d)^2 - 10800*(sqrt(x)*e + d)*d^5*e^(-4)*log(sqrt(x)*e + d)^2 - 600*(sqrt(x)*e + d)^6*e^(-4)*log(sqrt(x)*
e + d) + 4320*(sqrt(x)*e + d)^5*d*e^(-4)*log(sqrt(x)*e + d) - 13500*(sqrt(x)*e + d)^4*d^2*e^(-4)*log(sqrt(x)*e
+ d) + 24000*(sqrt(x)*e + d)^3*d^3*e^(-4)*log(sqrt(x)*e + d) - 27000*(sqrt(x)*e + d)^2*d^4*e^(-4)*log(sqrt(x)
*e + d) + 21600*(sqrt(x)*e + d)*d^5*e^(-4)*log(sqrt(x)*e + d) + 100*(sqrt(x)*e + d)^6*e^(-4) - 864*(sqrt(x)*e
+ d)^5*d*e^(-4) + 3375*(sqrt(x)*e + d)^4*d^2*e^(-4) - 8000*(sqrt(x)*e + d)^3*d^3*e^(-4) + 13500*(sqrt(x)*e + d
)^2*d^4*e^(-4) - 21600*(sqrt(x)*e + d)*d^5*e^(-4))*b^3*n^2*e^(-1)*log(c) + 1800*(60*(sqrt(x)*e + d)^6*e^(-4)*l
og(sqrt(x)*e + d) - 360*(sqrt(x)*e + d)^5*d*e^(-4)*log(sqrt(x)*e + d) + 900*(sqrt(x)*e + d)^4*d^2*e^(-4)*log(s
qrt(x)*e + d) - 1200*(sqrt(x)*e + d)^3*d^3*e^(-4)*log(sqrt(x)*e + d) + 900*(sqrt(x)*e + d)^2*d^4*e^(-4)*log(sq
rt(x)*e + d) - 360*(sqrt(x)*e + d)*d^5*e^(-4)*log(sqrt(x)*e + d) - 10*(sqrt(x)*e + d)^6*e^(-4) + 72*(sqrt(x)*e
+ d)^5*d*e^(-4) - 225*(sqrt(x)*e + d)^4*d^2*e^(-4) + 400*(sqrt(x)*e + d)^3*d^3*e^(-4) - 450*(sqrt(x)*e + d)^2
*d^4*e^(-4) + 360*(sqrt(x)*e + d)*d^5*e^(-4))*b^3*n*e^(-1)*log(c)^2 + 36000*((sqrt(x)*e + d)^6 - 6*(sqrt(x)*e
+ d)^5*d + 15*(sqrt(x)*e + d)^4*d^2 - 20*(sqrt(x)*e + d)^3*d^3 + 15*(sqrt(x)*e + d)^2*d^4 - 6*(sqrt(x)*e + d)*
d^5)*b^3*e^(-5)*log(c)^3 + 60*(1800*(sqrt(x)*e + d)^6*e^(-4)*log(sqrt(x)*e + d)^2 - 10800*(sqrt(x)*e + d)^5*d*
e^(-4)*log(sqrt(x)*e + d)^2 + 27000*(sqrt(x)*e + d)^4*d^2*e^(-4)*log(sqrt(x)*e + d)^2 - 36000*(sqrt(x)*e + d)^
3*d^3*e^(-4)*log(sqrt(x)*e + d)^2 + 27000*(sqrt(x)*e + d)^2*d^4*e^(-4)*log(sqrt(x)*e + d)^2 - 10800*(sqrt(x)*e
+ d)*d^5*e^(-4)*log(sqrt(x)*e + d)^2 - 600*(sqrt(x)*e + d)^6*e^(-4)*log(sqrt(x)*e + d) + 4320*(sqrt(x)*e + d)
^5*d*e^(-4)*log(sqrt(x)*e + d) - 13500*(sqrt(x)*e + d)^4*d^2*e^(-4)*log(sqrt(x)*e + d) + 24000*(sqrt(x)*e + d)
^3*d^3*e^(-4)*log(sqrt(x)*e + d) - 27000*(sqrt(x)*e + d)^2*d^4*e^(-4)*log(sqrt(x)*e + d) + 21600*(sqrt(x)*e +
d)*d^5*e^(-4)*log(sqrt(x)*e + d) + 100*(sqrt(x)*e + d)^6*e^(-4) - 864*(sqrt(x)*e + d)^5*d*e^(-4) + 3375*(sqrt(
x)*e + d)^4*d^2*e^(-4) - 8000*(sqrt(x)*e + d)^3*d^3*e^(-4) + 13500*(sqrt(x)*e + d)^2*d^4*e^(-4) - 21600*(sqrt(
x)*e + d)*d^5*e^(-4))*a*b^2*n^2*e^(-1) + 3600*(60*(sqrt(x)*e + d)^6*e^(-4)*log(sqrt(x)*e + d) - 360*(sqrt(x)*e
+ d)^5*d*e^(-4)*log(sqrt(x)*e + d) + 900*(sqrt(x)*e + d)^4*d^2*e^(-4)*log(sqrt(x)*e + d) - 1200*(sqrt(x)*e +
d)^3*d^3*e^(-4)*log(sqrt(x)*e + d) + 900*(sqrt(x)*e + d)^2*d^4*e^(-4)*log(sqrt(x)*e + d) - 360*(sqrt(x)*e + d)
*d^5*e^(-4)*log(sqrt(x)*e + d) - 10*(sqrt(x)*e + d)^6*e^(-4) + 72*(sqrt(x)*e + d)^5*d*e^(-4) - 225*(sqrt(x)*e
+ d)^4*d^2*e^(-4) + 400*(sqrt(x)*e + d)^3*d^3*e^(-4) - 450*(sqrt(x)*e + d)^2*d^4*e^(-4) + 360*(sqrt(x)*e + d)*
d^5*e^(-4))*a*b^2*n*e^(-1)*log(c) + 108000*((sqrt(x)*e + d)^6 - 6*(sqrt(x)*e + d)^5*d + 15*(sqrt(x)*e + d)^4*d
^2 - 20*(sqrt(x)*e + d)^3*d^3 + 15*(sqrt(x)*e + d)^2*d^4 - 6*(sqrt(x)*e + d)*d^5)*a*b^2*e^(-5)*log(c)^2 + 1800
*(60*(sqrt(x)*e + d)^6*e^(-4)*log(sqrt(x)*e + d) - 360*(sqrt(x)*e + d)^5*d*e^(-4)*log(sqrt(x)*e + d) + 900*(sq
rt(x)*e + d)^4*d^2*e^(-4)*log(sqrt(x)*e + d) - 1200*(sqrt(x)*e + d)^3*d^3*e^(-4)*log(sqrt(x)*e + d) + 900*(sqr
t(x)*e + d)^2*d^4*e^(-4)*log(sqrt(x)*e + d) - 360*(sqrt(x)*e + d)*d^5*e^(-4)*log(sqrt(x)*e + d) - 10*(sqrt(x)*
e + d)^6*e^(-4) + 72*(sqrt(x)*e + d)^5*d*e^(-4) - 225*(sqrt(x)*e + d)^4*d^2*e^(-4) + 400*(sqrt(x)*e + d)^3*d^3
*e^(-4) - 450*(sqrt(x)*e + d)^2*d^4*e^(-4) + 360*(sqrt(x)*e + d)*d^5*e^(-4))*a^2*b*n*e^(-1) + 108000*((sqrt(x)
*e + d)^6 - 6*(sqrt(x)*e + d)^5*d + 15*(sqrt(x)*e + d)^4*d^2 - 20*(sqrt(x)*e + d)^3*d^3 + 15*(sqrt(x)*e + d)^2
*d^4 - 6*(sqrt(x)*e + d)*d^5)*a^2*b*e^(-5)*log(c) + 36000*((sqrt(x)*e + d)^6 - 6*(sqrt(x)*e + d)^5*d + 15*(sqr
t(x)*e + d)^4*d^2 - 20*(sqrt(x)*e + d)^3*d^3 + 15*(sqrt(x)*e + d)^2*d^4 - 6*(sqrt(x)*e + d)*d^5)*a^3*e^(-5))*e
^(-1)