3.52 \(\int (f+g x)^3 (a+b \log (c (d+e x)^n))^3 \, dx\)

Optimal. Leaf size=598 \[ \frac{2 b^2 g^2 n^2 (d+e x)^3 (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e^4}+\frac{9 b^2 g n^2 (d+e x)^2 (e f-d g)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 e^4}+\frac{3 b^2 g^3 n^2 (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{32 e^4}+\frac{6 a b^2 n^2 x (e f-d g)^3}{e^3}+\frac{g^2 (d+e x)^3 (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^4}-\frac{b g^2 n (d+e x)^3 (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^4}+\frac{3 g (d+e x)^2 (e f-d g)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 e^4}-\frac{9 b g n (d+e x)^2 (e f-d g)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^4}+\frac{(d+e x) (e f-d g)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^4}-\frac{3 b n (d+e x) (e f-d g)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^4}+\frac{g^3 (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{4 e^4}-\frac{3 b g^3 n (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{16 e^4}+\frac{6 b^3 n^2 (d+e x) (e f-d g)^3 \log \left (c (d+e x)^n\right )}{e^4}-\frac{2 b^3 g^2 n^3 (d+e x)^3 (e f-d g)}{9 e^4}-\frac{9 b^3 g n^3 (d+e x)^2 (e f-d g)^2}{8 e^4}-\frac{6 b^3 n^3 x (e f-d g)^3}{e^3}-\frac{3 b^3 g^3 n^3 (d+e x)^4}{128 e^4} \]

[Out]

(6*a*b^2*(e*f - d*g)^3*n^2*x)/e^3 - (6*b^3*(e*f - d*g)^3*n^3*x)/e^3 - (9*b^3*g*(e*f - d*g)^2*n^3*(d + e*x)^2)/
(8*e^4) - (2*b^3*g^2*(e*f - d*g)*n^3*(d + e*x)^3)/(9*e^4) - (3*b^3*g^3*n^3*(d + e*x)^4)/(128*e^4) + (6*b^3*(e*
f - d*g)^3*n^2*(d + e*x)*Log[c*(d + e*x)^n])/e^4 + (9*b^2*g*(e*f - d*g)^2*n^2*(d + e*x)^2*(a + b*Log[c*(d + e*
x)^n]))/(4*e^4) + (2*b^2*g^2*(e*f - d*g)*n^2*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n]))/(3*e^4) + (3*b^2*g^3*n^2*
(d + e*x)^4*(a + b*Log[c*(d + e*x)^n]))/(32*e^4) - (3*b*(e*f - d*g)^3*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2
)/e^4 - (9*b*g*(e*f - d*g)^2*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^2)/(4*e^4) - (b*g^2*(e*f - d*g)*n*(d + e
*x)^3*(a + b*Log[c*(d + e*x)^n])^2)/e^4 - (3*b*g^3*n*(d + e*x)^4*(a + b*Log[c*(d + e*x)^n])^2)/(16*e^4) + ((e*
f - d*g)^3*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/e^4 + (3*g*(e*f - d*g)^2*(d + e*x)^2*(a + b*Log[c*(d + e*x)
^n])^3)/(2*e^4) + (g^2*(e*f - d*g)*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n])^3)/e^4 + (g^3*(d + e*x)^4*(a + b*Log
[c*(d + e*x)^n])^3)/(4*e^4)

________________________________________________________________________________________

Rubi [A]  time = 0.552067, antiderivative size = 598, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {2401, 2389, 2296, 2295, 2390, 2305, 2304} \[ \frac{2 b^2 g^2 n^2 (d+e x)^3 (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e^4}+\frac{9 b^2 g n^2 (d+e x)^2 (e f-d g)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 e^4}+\frac{3 b^2 g^3 n^2 (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{32 e^4}+\frac{6 a b^2 n^2 x (e f-d g)^3}{e^3}+\frac{g^2 (d+e x)^3 (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^4}-\frac{b g^2 n (d+e x)^3 (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^4}+\frac{3 g (d+e x)^2 (e f-d g)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 e^4}-\frac{9 b g n (d+e x)^2 (e f-d g)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^4}+\frac{(d+e x) (e f-d g)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^4}-\frac{3 b n (d+e x) (e f-d g)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^4}+\frac{g^3 (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{4 e^4}-\frac{3 b g^3 n (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{16 e^4}+\frac{6 b^3 n^2 (d+e x) (e f-d g)^3 \log \left (c (d+e x)^n\right )}{e^4}-\frac{2 b^3 g^2 n^3 (d+e x)^3 (e f-d g)}{9 e^4}-\frac{9 b^3 g n^3 (d+e x)^2 (e f-d g)^2}{8 e^4}-\frac{6 b^3 n^3 x (e f-d g)^3}{e^3}-\frac{3 b^3 g^3 n^3 (d+e x)^4}{128 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*a*b^2*(e*f - d*g)^3*n^2*x)/e^3 - (6*b^3*(e*f - d*g)^3*n^3*x)/e^3 - (9*b^3*g*(e*f - d*g)^2*n^3*(d + e*x)^2)/
(8*e^4) - (2*b^3*g^2*(e*f - d*g)*n^3*(d + e*x)^3)/(9*e^4) - (3*b^3*g^3*n^3*(d + e*x)^4)/(128*e^4) + (6*b^3*(e*
f - d*g)^3*n^2*(d + e*x)*Log[c*(d + e*x)^n])/e^4 + (9*b^2*g*(e*f - d*g)^2*n^2*(d + e*x)^2*(a + b*Log[c*(d + e*
x)^n]))/(4*e^4) + (2*b^2*g^2*(e*f - d*g)*n^2*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n]))/(3*e^4) + (3*b^2*g^3*n^2*
(d + e*x)^4*(a + b*Log[c*(d + e*x)^n]))/(32*e^4) - (3*b*(e*f - d*g)^3*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2
)/e^4 - (9*b*g*(e*f - d*g)^2*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^2)/(4*e^4) - (b*g^2*(e*f - d*g)*n*(d + e
*x)^3*(a + b*Log[c*(d + e*x)^n])^2)/e^4 - (3*b*g^3*n*(d + e*x)^4*(a + b*Log[c*(d + e*x)^n])^2)/(16*e^4) + ((e*
f - d*g)^3*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/e^4 + (3*g*(e*f - d*g)^2*(d + e*x)^2*(a + b*Log[c*(d + e*x)
^n])^3)/(2*e^4) + (g^2*(e*f - d*g)*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n])^3)/e^4 + (g^3*(d + e*x)^4*(a + b*Log
[c*(d + e*x)^n])^3)/(4*e^4)

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 (f+g x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx &=\int \left (\frac{(e f-d g)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}+\frac{3 g (e f-d g)^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}+\frac{3 g^2 (e f-d g) (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}+\frac{g^3 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}\right ) \, dx\\ &=\frac{g^3 \int (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx}{e^3}+\frac{\left (3 g^2 (e f-d g)\right ) \int (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx}{e^3}+\frac{\left (3 g (e f-d g)^2\right ) \int (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx}{e^3}+\frac{(e f-d g)^3 \int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx}{e^3}\\ &=\frac{g^3 \operatorname{Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x\right )}{e^4}+\frac{\left (3 g^2 (e f-d g)\right ) \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x\right )}{e^4}+\frac{\left (3 g (e f-d g)^2\right ) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x\right )}{e^4}+\frac{(e f-d g)^3 \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x\right )}{e^4}\\ &=\frac{(e f-d g)^3 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^4}+\frac{3 g (e f-d g)^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 e^4}+\frac{g^2 (e f-d g) (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^4}+\frac{g^3 (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{4 e^4}-\frac{\left (3 b g^3 n\right ) \operatorname{Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{4 e^4}-\frac{\left (3 b g^2 (e f-d g) n\right ) \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e^4}-\frac{\left (9 b g (e f-d g)^2 n\right ) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{2 e^4}-\frac{\left (3 b (e f-d g)^3 n\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e^4}\\ &=-\frac{3 b (e f-d g)^3 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^4}-\frac{9 b g (e f-d g)^2 n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^4}-\frac{b g^2 (e f-d g) n (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^4}-\frac{3 b g^3 n (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{16 e^4}+\frac{(e f-d g)^3 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^4}+\frac{3 g (e f-d g)^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 e^4}+\frac{g^2 (e f-d g) (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^4}+\frac{g^3 (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{4 e^4}+\frac{\left (3 b^2 g^3 n^2\right ) \operatorname{Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{8 e^4}+\frac{\left (2 b^2 g^2 (e f-d g) n^2\right ) \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e^4}+\frac{\left (9 b^2 g (e f-d g)^2 n^2\right ) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{2 e^4}+\frac{\left (6 b^2 (e f-d g)^3 n^2\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e^4}\\ &=\frac{6 a b^2 (e f-d g)^3 n^2 x}{e^3}-\frac{9 b^3 g (e f-d g)^2 n^3 (d+e x)^2}{8 e^4}-\frac{2 b^3 g^2 (e f-d g) n^3 (d+e x)^3}{9 e^4}-\frac{3 b^3 g^3 n^3 (d+e x)^4}{128 e^4}+\frac{9 b^2 g (e f-d g)^2 n^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 e^4}+\frac{2 b^2 g^2 (e f-d g) n^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e^4}+\frac{3 b^2 g^3 n^2 (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{32 e^4}-\frac{3 b (e f-d g)^3 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^4}-\frac{9 b g (e f-d g)^2 n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^4}-\frac{b g^2 (e f-d g) n (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^4}-\frac{3 b g^3 n (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{16 e^4}+\frac{(e f-d g)^3 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^4}+\frac{3 g (e f-d g)^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 e^4}+\frac{g^2 (e f-d g) (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^4}+\frac{g^3 (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{4 e^4}+\frac{\left (6 b^3 (e f-d g)^3 n^2\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e^4}\\ &=\frac{6 a b^2 (e f-d g)^3 n^2 x}{e^3}-\frac{6 b^3 (e f-d g)^3 n^3 x}{e^3}-\frac{9 b^3 g (e f-d g)^2 n^3 (d+e x)^2}{8 e^4}-\frac{2 b^3 g^2 (e f-d g) n^3 (d+e x)^3}{9 e^4}-\frac{3 b^3 g^3 n^3 (d+e x)^4}{128 e^4}+\frac{6 b^3 (e f-d g)^3 n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e^4}+\frac{9 b^2 g (e f-d g)^2 n^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 e^4}+\frac{2 b^2 g^2 (e f-d g) n^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e^4}+\frac{3 b^2 g^3 n^2 (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{32 e^4}-\frac{3 b (e f-d g)^3 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^4}-\frac{9 b g (e f-d g)^2 n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^4}-\frac{b g^2 (e f-d g) n (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^4}-\frac{3 b g^3 n (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{16 e^4}+\frac{(e f-d g)^3 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^4}+\frac{3 g (e f-d g)^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 e^4}+\frac{g^2 (e f-d g) (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^4}+\frac{g^3 (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{4 e^4}\\ \end{align*}

Mathematica [A]  time = 0.408771, size = 475, normalized size = 0.79 \[ \frac{-128 b g^2 n (e f-d g) \left (2 b n \left (b e n x \left (3 d^2+3 d e x+e^2 x^2\right )-3 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )\right )+9 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2\right )-27 b g^3 n \left (b n \left (b e n x \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )-4 (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )\right )+8 (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2\right )+1152 g^2 (d+e x)^3 (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^3+1728 g (d+e x)^2 (e f-d g)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3+1152 (d+e x) (e f-d g)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3-3456 b n (e f-d g)^3 \left ((d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2-2 b n \left (e x (a-b n)+b (d+e x) \log \left (c (d+e x)^n\right )\right )\right )-1296 b g n (e f-d g)^2 \left (2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2+b n \left (b e n x (2 d+e x)-2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )\right )\right )+288 g^3 (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{1152 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1152*(e*f - d*g)^3*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^3 + 1728*g*(e*f - d*g)^2*(d + e*x)^2*(a + b*Log[c*(d
+ e*x)^n])^3 + 1152*g^2*(e*f - d*g)*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n])^3 + 288*g^3*(d + e*x)^4*(a + b*Log[
c*(d + e*x)^n])^3 - 3456*b*(e*f - d*g)^3*n*((d + e*x)*(a + b*Log[c*(d + e*x)^n])^2 - 2*b*n*(e*(a - b*n)*x + b*
(d + e*x)*Log[c*(d + e*x)^n])) - 1296*b*g*(e*f - d*g)^2*n*(2*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^2 + b*n*(b
*e*n*x*(2*d + e*x) - 2*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))) - 128*b*g^2*(e*f - d*g)*n*(9*(d + e*x)^3*(a +
b*Log[c*(d + e*x)^n])^2 + 2*b*n*(b*e*n*x*(3*d^2 + 3*d*e*x + e^2*x^2) - 3*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n]
))) - 27*b*g^3*n*(8*(d + e*x)^4*(a + b*Log[c*(d + e*x)^n])^2 + b*n*(b*e*n*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 +
e^3*x^3) - 4*(d + e*x)^4*(a + b*Log[c*(d + e*x)^n]))))/(1152*e^4)

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Maple [C]  time = 2.204, size = 30495, normalized size = 51. \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [B]  time = 1.55404, size = 2277, normalized size = 3.81 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b^3*g^3*x^4*log((e*x + d)^n*c)^3 + 3/4*a*b^2*g^3*x^4*log((e*x + d)^n*c)^2 + b^3*f*g^2*x^3*log((e*x + d)^n*
c)^3 + 3/4*a^2*b*g^3*x^4*log((e*x + d)^n*c) + 3*a*b^2*f*g^2*x^3*log((e*x + d)^n*c)^2 + 3/2*b^3*f^2*g*x^2*log((
e*x + d)^n*c)^3 + 1/4*a^3*g^3*x^4 + 3*a^2*b*f*g^2*x^3*log((e*x + d)^n*c) + 9/2*a*b^2*f^2*g*x^2*log((e*x + d)^n
*c)^2 + b^3*f^3*x*log((e*x + d)^n*c)^3 + a^3*f*g^2*x^3 - 3*a^2*b*e*f^3*n*(x/e - d*log(e*x + d)/e^2) - 1/16*a^2
*b*e*g^3*n*(12*d^4*log(e*x + d)/e^5 + (3*e^3*x^4 - 4*d*e^2*x^3 + 6*d^2*e*x^2 - 12*d^3*x)/e^4) + 1/2*a^2*b*e*f*
g^2*n*(6*d^3*log(e*x + d)/e^4 - (2*e^2*x^3 - 3*d*e*x^2 + 6*d^2*x)/e^3) - 9/4*a^2*b*e*f^2*g*n*(2*d^2*log(e*x +
d)/e^3 + (e*x^2 - 2*d*x)/e^2) + 9/2*a^2*b*f^2*g*x^2*log((e*x + d)^n*c) + 3*a*b^2*f^3*x*log((e*x + d)^n*c)^2 +
3/2*a^3*f^2*g*x^2 + 3*a^2*b*f^3*x*log((e*x + d)^n*c) - 3*(2*e*n*(x/e - d*log(e*x + d)/e^2)*log((e*x + d)^n*c)
+ (d*log(e*x + d)^2 - 2*e*x + 2*d*log(e*x + d))*n^2/e)*a*b^2*f^3 - (3*e*n*(x/e - d*log(e*x + d)/e^2)*log((e*x
+ d)^n*c)^2 - e*n*((d*log(e*x + d)^3 + 3*d*log(e*x + d)^2 - 6*e*x + 6*d*log(e*x + d))*n^2/e^2 - 3*(d*log(e*x +
d)^2 - 2*e*x + 2*d*log(e*x + d))*n*log((e*x + d)^n*c)/e^2))*b^3*f^3 - 9/4*(2*e*n*(2*d^2*log(e*x + d)/e^3 + (e
*x^2 - 2*d*x)/e^2)*log((e*x + d)^n*c) - (e^2*x^2 + 2*d^2*log(e*x + d)^2 - 6*d*e*x + 6*d^2*log(e*x + d))*n^2/e^
2)*a*b^2*f^2*g - 3/8*(6*e*n*(2*d^2*log(e*x + d)/e^3 + (e*x^2 - 2*d*x)/e^2)*log((e*x + d)^n*c)^2 + e*n*((4*d^2*
log(e*x + d)^3 + 3*e^2*x^2 + 18*d^2*log(e*x + d)^2 - 42*d*e*x + 42*d^2*log(e*x + d))*n^2/e^3 - 6*(e^2*x^2 + 2*
d^2*log(e*x + d)^2 - 6*d*e*x + 6*d^2*log(e*x + d))*n*log((e*x + d)^n*c)/e^3))*b^3*f^2*g + 1/6*(6*e*n*(6*d^3*lo
g(e*x + d)/e^4 - (2*e^2*x^3 - 3*d*e*x^2 + 6*d^2*x)/e^3)*log((e*x + d)^n*c) + (4*e^3*x^3 - 15*d*e^2*x^2 - 18*d^
3*log(e*x + d)^2 + 66*d^2*e*x - 66*d^3*log(e*x + d))*n^2/e^3)*a*b^2*f*g^2 + 1/36*(18*e*n*(6*d^3*log(e*x + d)/e
^4 - (2*e^2*x^3 - 3*d*e*x^2 + 6*d^2*x)/e^3)*log((e*x + d)^n*c)^2 - e*n*((8*e^3*x^3 - 36*d^3*log(e*x + d)^3 - 5
7*d*e^2*x^2 - 198*d^3*log(e*x + d)^2 + 510*d^2*e*x - 510*d^3*log(e*x + d))*n^2/e^4 - 6*(4*e^3*x^3 - 15*d*e^2*x
^2 - 18*d^3*log(e*x + d)^2 + 66*d^2*e*x - 66*d^3*log(e*x + d))*n*log((e*x + d)^n*c)/e^4))*b^3*f*g^2 - 1/96*(12
*e*n*(12*d^4*log(e*x + d)/e^5 + (3*e^3*x^4 - 4*d*e^2*x^3 + 6*d^2*e*x^2 - 12*d^3*x)/e^4)*log((e*x + d)^n*c) - (
9*e^4*x^4 - 28*d*e^3*x^3 + 78*d^2*e^2*x^2 + 72*d^4*log(e*x + d)^2 - 300*d^3*e*x + 300*d^4*log(e*x + d))*n^2/e^
4)*a*b^2*g^3 - 1/1152*(72*e*n*(12*d^4*log(e*x + d)/e^5 + (3*e^3*x^4 - 4*d*e^2*x^3 + 6*d^2*e*x^2 - 12*d^3*x)/e^
4)*log((e*x + d)^n*c)^2 + e*n*((27*e^4*x^4 - 148*d*e^3*x^3 + 288*d^4*log(e*x + d)^3 + 690*d^2*e^2*x^2 + 1800*d
^4*log(e*x + d)^2 - 4980*d^3*e*x + 4980*d^4*log(e*x + d))*n^2/e^5 - 12*(9*e^4*x^4 - 28*d*e^3*x^3 + 78*d^2*e^2*
x^2 + 72*d^4*log(e*x + d)^2 - 300*d^3*e*x + 300*d^4*log(e*x + d))*n*log((e*x + d)^n*c)/e^5))*b^3*g^3 + a^3*f^3
*x

________________________________________________________________________________________

Fricas [B]  time = 3.07314, size = 5723, normalized size = 9.57 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1152*(9*(3*b^3*e^4*g^3*n^3 - 12*a*b^2*e^4*g^3*n^2 + 24*a^2*b*e^4*g^3*n - 32*a^3*e^4*g^3)*x^4 - 4*(288*a^3*e
^4*f*g^2 - (64*b^3*e^4*f*g^2 - 37*b^3*d*e^3*g^3)*n^3 + 12*(16*a*b^2*e^4*f*g^2 - 7*a*b^2*d*e^3*g^3)*n^2 - 72*(4
*a^2*b*e^4*f*g^2 - a^2*b*d*e^3*g^3)*n)*x^3 - 288*(b^3*e^4*g^3*n^3*x^4 + 4*b^3*e^4*f*g^2*n^3*x^3 + 6*b^3*e^4*f^
2*g*n^3*x^2 + 4*b^3*e^4*f^3*n^3*x + (4*b^3*d*e^3*f^3 - 6*b^3*d^2*e^2*f^2*g + 4*b^3*d^3*e*f*g^2 - b^3*d^4*g^3)*
n^3)*log(e*x + d)^3 - 288*(b^3*e^4*g^3*x^4 + 4*b^3*e^4*f*g^2*x^3 + 6*b^3*e^4*f^2*g*x^2 + 4*b^3*e^4*f^3*x)*log(
c)^3 - 6*(288*a^3*e^4*f^2*g - (216*b^3*e^4*f^2*g - 304*b^3*d*e^3*f*g^2 + 115*b^3*d^2*e^2*g^3)*n^3 + 12*(36*a*b
^2*e^4*f^2*g - 40*a*b^2*d*e^3*f*g^2 + 13*a*b^2*d^2*e^2*g^3)*n^2 - 72*(6*a^2*b*e^4*f^2*g - 4*a^2*b*d*e^3*f*g^2
+ a^2*b*d^2*e^2*g^3)*n)*x^2 + 72*(3*(b^3*e^4*g^3*n^3 - 4*a*b^2*e^4*g^3*n^2)*x^4 + (48*b^3*d*e^3*f^3 - 108*b^3*
d^2*e^2*f^2*g + 88*b^3*d^3*e*f*g^2 - 25*b^3*d^4*g^3)*n^3 - 4*(12*a*b^2*e^4*f*g^2*n^2 - (4*b^3*e^4*f*g^2 - b^3*
d*e^3*g^3)*n^3)*x^3 - 12*(4*a*b^2*d*e^3*f^3 - 6*a*b^2*d^2*e^2*f^2*g + 4*a*b^2*d^3*e*f*g^2 - a*b^2*d^4*g^3)*n^2
- 6*(12*a*b^2*e^4*f^2*g*n^2 - (6*b^3*e^4*f^2*g - 4*b^3*d*e^3*f*g^2 + b^3*d^2*e^2*g^3)*n^3)*x^2 - 12*(4*a*b^2*
e^4*f^3*n^2 - (4*b^3*e^4*f^3 - 6*b^3*d*e^3*f^2*g + 4*b^3*d^2*e^2*f*g^2 - b^3*d^3*e*g^3)*n^3)*x - 12*(b^3*e^4*g
^3*n^2*x^4 + 4*b^3*e^4*f*g^2*n^2*x^3 + 6*b^3*e^4*f^2*g*n^2*x^2 + 4*b^3*e^4*f^3*n^2*x + (4*b^3*d*e^3*f^3 - 6*b^
3*d^2*e^2*f^2*g + 4*b^3*d^3*e*f*g^2 - b^3*d^4*g^3)*n^2)*log(c))*log(e*x + d)^2 + 72*(3*(b^3*e^4*g^3*n - 4*a*b^
2*e^4*g^3)*x^4 - 4*(12*a*b^2*e^4*f*g^2 - (4*b^3*e^4*f*g^2 - b^3*d*e^3*g^3)*n)*x^3 - 6*(12*a*b^2*e^4*f^2*g - (6
*b^3*e^4*f^2*g - 4*b^3*d*e^3*f*g^2 + b^3*d^2*e^2*g^3)*n)*x^2 - 12*(4*a*b^2*e^4*f^3 - (4*b^3*e^4*f^3 - 6*b^3*d*
e^3*f^2*g + 4*b^3*d^2*e^2*f*g^2 - b^3*d^3*e*g^3)*n)*x)*log(c)^2 - 12*(96*a^3*e^4*f^3 - (576*b^3*e^4*f^3 - 1512
*b^3*d*e^3*f^2*g + 1360*b^3*d^2*e^2*f*g^2 - 415*b^3*d^3*e*g^3)*n^3 + 12*(48*a*b^2*e^4*f^3 - 108*a*b^2*d*e^3*f^
2*g + 88*a*b^2*d^2*e^2*f*g^2 - 25*a*b^2*d^3*e*g^3)*n^2 - 72*(4*a^2*b*e^4*f^3 - 6*a^2*b*d*e^3*f^2*g + 4*a^2*b*d
^2*e^2*f*g^2 - a^2*b*d^3*e*g^3)*n)*x - 12*(9*(b^3*e^4*g^3*n^3 - 4*a*b^2*e^4*g^3*n^2 + 8*a^2*b*e^4*g^3*n)*x^4 +
(576*b^3*d*e^3*f^3 - 1512*b^3*d^2*e^2*f^2*g + 1360*b^3*d^3*e*f*g^2 - 415*b^3*d^4*g^3)*n^3 + 4*(72*a^2*b*e^4*f
*g^2*n + (16*b^3*e^4*f*g^2 - 7*b^3*d*e^3*g^3)*n^3 - 12*(4*a*b^2*e^4*f*g^2 - a*b^2*d*e^3*g^3)*n^2)*x^3 - 12*(48
*a*b^2*d*e^3*f^3 - 108*a*b^2*d^2*e^2*f^2*g + 88*a*b^2*d^3*e*f*g^2 - 25*a*b^2*d^4*g^3)*n^2 + 6*(72*a^2*b*e^4*f^
2*g*n + (36*b^3*e^4*f^2*g - 40*b^3*d*e^3*f*g^2 + 13*b^3*d^2*e^2*g^3)*n^3 - 12*(6*a*b^2*e^4*f^2*g - 4*a*b^2*d*e
^3*f*g^2 + a*b^2*d^2*e^2*g^3)*n^2)*x^2 + 72*(b^3*e^4*g^3*n*x^4 + 4*b^3*e^4*f*g^2*n*x^3 + 6*b^3*e^4*f^2*g*n*x^2
+ 4*b^3*e^4*f^3*n*x + (4*b^3*d*e^3*f^3 - 6*b^3*d^2*e^2*f^2*g + 4*b^3*d^3*e*f*g^2 - b^3*d^4*g^3)*n)*log(c)^2 +
72*(4*a^2*b*d*e^3*f^3 - 6*a^2*b*d^2*e^2*f^2*g + 4*a^2*b*d^3*e*f*g^2 - a^2*b*d^4*g^3)*n + 12*(24*a^2*b*e^4*f^3
*n + (48*b^3*e^4*f^3 - 108*b^3*d*e^3*f^2*g + 88*b^3*d^2*e^2*f*g^2 - 25*b^3*d^3*e*g^3)*n^3 - 12*(4*a*b^2*e^4*f^
3 - 6*a*b^2*d*e^3*f^2*g + 4*a*b^2*d^2*e^2*f*g^2 - a*b^2*d^3*e*g^3)*n^2)*x - 12*(3*(b^3*e^4*g^3*n^2 - 4*a*b^2*e
^4*g^3*n)*x^4 - 4*(12*a*b^2*e^4*f*g^2*n - (4*b^3*e^4*f*g^2 - b^3*d*e^3*g^3)*n^2)*x^3 + (48*b^3*d*e^3*f^3 - 108
*b^3*d^2*e^2*f^2*g + 88*b^3*d^3*e*f*g^2 - 25*b^3*d^4*g^3)*n^2 - 6*(12*a*b^2*e^4*f^2*g*n - (6*b^3*e^4*f^2*g - 4
*b^3*d*e^3*f*g^2 + b^3*d^2*e^2*g^3)*n^2)*x^2 - 12*(4*a*b^2*d*e^3*f^3 - 6*a*b^2*d^2*e^2*f^2*g + 4*a*b^2*d^3*e*f
*g^2 - a*b^2*d^4*g^3)*n - 12*(4*a*b^2*e^4*f^3*n - (4*b^3*e^4*f^3 - 6*b^3*d*e^3*f^2*g + 4*b^3*d^2*e^2*f*g^2 - b
^3*d^3*e*g^3)*n^2)*x)*log(c))*log(e*x + d) - 12*(9*(b^3*e^4*g^3*n^2 - 4*a*b^2*e^4*g^3*n + 8*a^2*b*e^4*g^3)*x^4
+ 4*(72*a^2*b*e^4*f*g^2 + (16*b^3*e^4*f*g^2 - 7*b^3*d*e^3*g^3)*n^2 - 12*(4*a*b^2*e^4*f*g^2 - a*b^2*d*e^3*g^3)
*n)*x^3 + 6*(72*a^2*b*e^4*f^2*g + (36*b^3*e^4*f^2*g - 40*b^3*d*e^3*f*g^2 + 13*b^3*d^2*e^2*g^3)*n^2 - 12*(6*a*b
^2*e^4*f^2*g - 4*a*b^2*d*e^3*f*g^2 + a*b^2*d^2*e^2*g^3)*n)*x^2 + 12*(24*a^2*b*e^4*f^3 + (48*b^3*e^4*f^3 - 108*
b^3*d*e^3*f^2*g + 88*b^3*d^2*e^2*f*g^2 - 25*b^3*d^3*e*g^3)*n^2 - 12*(4*a*b^2*e^4*f^3 - 6*a*b^2*d*e^3*f^2*g + 4
*a*b^2*d^2*e^2*f*g^2 - a*b^2*d^3*e*g^3)*n)*x)*log(c))/e^4

________________________________________________________________________________________

Sympy [A]  time = 50.5034, size = 4495, normalized size = 7.52 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)**3*(a+b*ln(c*(e*x+d)**n))**3,x)

[Out]

Piecewise((a**3*f**3*x + 3*a**3*f**2*g*x**2/2 + a**3*f*g**2*x**3 + a**3*g**3*x**4/4 - 3*a**2*b*d**4*g**3*n*log
(d + e*x)/(4*e**4) + 3*a**2*b*d**3*f*g**2*n*log(d + e*x)/e**3 + 3*a**2*b*d**3*g**3*n*x/(4*e**3) - 9*a**2*b*d**
2*f**2*g*n*log(d + e*x)/(2*e**2) - 3*a**2*b*d**2*f*g**2*n*x/e**2 - 3*a**2*b*d**2*g**3*n*x**2/(8*e**2) + 3*a**2
*b*d*f**3*n*log(d + e*x)/e + 9*a**2*b*d*f**2*g*n*x/(2*e) + 3*a**2*b*d*f*g**2*n*x**2/(2*e) + a**2*b*d*g**3*n*x*
*3/(4*e) + 3*a**2*b*f**3*n*x*log(d + e*x) - 3*a**2*b*f**3*n*x + 3*a**2*b*f**3*x*log(c) + 9*a**2*b*f**2*g*n*x**
2*log(d + e*x)/2 - 9*a**2*b*f**2*g*n*x**2/4 + 9*a**2*b*f**2*g*x**2*log(c)/2 + 3*a**2*b*f*g**2*n*x**3*log(d + e
*x) - a**2*b*f*g**2*n*x**3 + 3*a**2*b*f*g**2*x**3*log(c) + 3*a**2*b*g**3*n*x**4*log(d + e*x)/4 - 3*a**2*b*g**3
*n*x**4/16 + 3*a**2*b*g**3*x**4*log(c)/4 - 3*a*b**2*d**4*g**3*n**2*log(d + e*x)**2/(4*e**4) + 25*a*b**2*d**4*g
**3*n**2*log(d + e*x)/(8*e**4) - 3*a*b**2*d**4*g**3*n*log(c)*log(d + e*x)/(2*e**4) + 3*a*b**2*d**3*f*g**2*n**2
*log(d + e*x)**2/e**3 - 11*a*b**2*d**3*f*g**2*n**2*log(d + e*x)/e**3 + 6*a*b**2*d**3*f*g**2*n*log(c)*log(d + e
*x)/e**3 + 3*a*b**2*d**3*g**3*n**2*x*log(d + e*x)/(2*e**3) - 25*a*b**2*d**3*g**3*n**2*x/(8*e**3) + 3*a*b**2*d*
*3*g**3*n*x*log(c)/(2*e**3) - 9*a*b**2*d**2*f**2*g*n**2*log(d + e*x)**2/(2*e**2) + 27*a*b**2*d**2*f**2*g*n**2*
log(d + e*x)/(2*e**2) - 9*a*b**2*d**2*f**2*g*n*log(c)*log(d + e*x)/e**2 - 6*a*b**2*d**2*f*g**2*n**2*x*log(d +
e*x)/e**2 + 11*a*b**2*d**2*f*g**2*n**2*x/e**2 - 6*a*b**2*d**2*f*g**2*n*x*log(c)/e**2 - 3*a*b**2*d**2*g**3*n**2
*x**2*log(d + e*x)/(4*e**2) + 13*a*b**2*d**2*g**3*n**2*x**2/(16*e**2) - 3*a*b**2*d**2*g**3*n*x**2*log(c)/(4*e*
*2) + 3*a*b**2*d*f**3*n**2*log(d + e*x)**2/e - 6*a*b**2*d*f**3*n**2*log(d + e*x)/e + 6*a*b**2*d*f**3*n*log(c)*
log(d + e*x)/e + 9*a*b**2*d*f**2*g*n**2*x*log(d + e*x)/e - 27*a*b**2*d*f**2*g*n**2*x/(2*e) + 9*a*b**2*d*f**2*g
*n*x*log(c)/e + 3*a*b**2*d*f*g**2*n**2*x**2*log(d + e*x)/e - 5*a*b**2*d*f*g**2*n**2*x**2/(2*e) + 3*a*b**2*d*f*
g**2*n*x**2*log(c)/e + a*b**2*d*g**3*n**2*x**3*log(d + e*x)/(2*e) - 7*a*b**2*d*g**3*n**2*x**3/(24*e) + a*b**2*
d*g**3*n*x**3*log(c)/(2*e) + 3*a*b**2*f**3*n**2*x*log(d + e*x)**2 - 6*a*b**2*f**3*n**2*x*log(d + e*x) + 6*a*b*
*2*f**3*n**2*x + 6*a*b**2*f**3*n*x*log(c)*log(d + e*x) - 6*a*b**2*f**3*n*x*log(c) + 3*a*b**2*f**3*x*log(c)**2
+ 9*a*b**2*f**2*g*n**2*x**2*log(d + e*x)**2/2 - 9*a*b**2*f**2*g*n**2*x**2*log(d + e*x)/2 + 9*a*b**2*f**2*g*n**
2*x**2/4 + 9*a*b**2*f**2*g*n*x**2*log(c)*log(d + e*x) - 9*a*b**2*f**2*g*n*x**2*log(c)/2 + 9*a*b**2*f**2*g*x**2
*log(c)**2/2 + 3*a*b**2*f*g**2*n**2*x**3*log(d + e*x)**2 - 2*a*b**2*f*g**2*n**2*x**3*log(d + e*x) + 2*a*b**2*f
*g**2*n**2*x**3/3 + 6*a*b**2*f*g**2*n*x**3*log(c)*log(d + e*x) - 2*a*b**2*f*g**2*n*x**3*log(c) + 3*a*b**2*f*g*
*2*x**3*log(c)**2 + 3*a*b**2*g**3*n**2*x**4*log(d + e*x)**2/4 - 3*a*b**2*g**3*n**2*x**4*log(d + e*x)/8 + 3*a*b
**2*g**3*n**2*x**4/32 + 3*a*b**2*g**3*n*x**4*log(c)*log(d + e*x)/2 - 3*a*b**2*g**3*n*x**4*log(c)/8 + 3*a*b**2*
g**3*x**4*log(c)**2/4 - b**3*d**4*g**3*n**3*log(d + e*x)**3/(4*e**4) + 25*b**3*d**4*g**3*n**3*log(d + e*x)**2/
(16*e**4) - 415*b**3*d**4*g**3*n**3*log(d + e*x)/(96*e**4) - 3*b**3*d**4*g**3*n**2*log(c)*log(d + e*x)**2/(4*e
**4) + 25*b**3*d**4*g**3*n**2*log(c)*log(d + e*x)/(8*e**4) - 3*b**3*d**4*g**3*n*log(c)**2*log(d + e*x)/(4*e**4
) + b**3*d**3*f*g**2*n**3*log(d + e*x)**3/e**3 - 11*b**3*d**3*f*g**2*n**3*log(d + e*x)**2/(2*e**3) + 85*b**3*d
**3*f*g**2*n**3*log(d + e*x)/(6*e**3) + 3*b**3*d**3*f*g**2*n**2*log(c)*log(d + e*x)**2/e**3 - 11*b**3*d**3*f*g
**2*n**2*log(c)*log(d + e*x)/e**3 + 3*b**3*d**3*f*g**2*n*log(c)**2*log(d + e*x)/e**3 + 3*b**3*d**3*g**3*n**3*x
*log(d + e*x)**2/(4*e**3) - 25*b**3*d**3*g**3*n**3*x*log(d + e*x)/(8*e**3) + 415*b**3*d**3*g**3*n**3*x/(96*e**
3) + 3*b**3*d**3*g**3*n**2*x*log(c)*log(d + e*x)/(2*e**3) - 25*b**3*d**3*g**3*n**2*x*log(c)/(8*e**3) + 3*b**3*
d**3*g**3*n*x*log(c)**2/(4*e**3) - 3*b**3*d**2*f**2*g*n**3*log(d + e*x)**3/(2*e**2) + 27*b**3*d**2*f**2*g*n**3
*log(d + e*x)**2/(4*e**2) - 63*b**3*d**2*f**2*g*n**3*log(d + e*x)/(4*e**2) - 9*b**3*d**2*f**2*g*n**2*log(c)*lo
g(d + e*x)**2/(2*e**2) + 27*b**3*d**2*f**2*g*n**2*log(c)*log(d + e*x)/(2*e**2) - 9*b**3*d**2*f**2*g*n*log(c)**
2*log(d + e*x)/(2*e**2) - 3*b**3*d**2*f*g**2*n**3*x*log(d + e*x)**2/e**2 + 11*b**3*d**2*f*g**2*n**3*x*log(d +
e*x)/e**2 - 85*b**3*d**2*f*g**2*n**3*x/(6*e**2) - 6*b**3*d**2*f*g**2*n**2*x*log(c)*log(d + e*x)/e**2 + 11*b**3
*d**2*f*g**2*n**2*x*log(c)/e**2 - 3*b**3*d**2*f*g**2*n*x*log(c)**2/e**2 - 3*b**3*d**2*g**3*n**3*x**2*log(d + e
*x)**2/(8*e**2) + 13*b**3*d**2*g**3*n**3*x**2*log(d + e*x)/(16*e**2) - 115*b**3*d**2*g**3*n**3*x**2/(192*e**2)
- 3*b**3*d**2*g**3*n**2*x**2*log(c)*log(d + e*x)/(4*e**2) + 13*b**3*d**2*g**3*n**2*x**2*log(c)/(16*e**2) - 3*
b**3*d**2*g**3*n*x**2*log(c)**2/(8*e**2) + b**3*d*f**3*n**3*log(d + e*x)**3/e - 3*b**3*d*f**3*n**3*log(d + e*x
)**2/e + 6*b**3*d*f**3*n**3*log(d + e*x)/e + 3*b**3*d*f**3*n**2*log(c)*log(d + e*x)**2/e - 6*b**3*d*f**3*n**2*
log(c)*log(d + e*x)/e + 3*b**3*d*f**3*n*log(c)**2*log(d + e*x)/e + 9*b**3*d*f**2*g*n**3*x*log(d + e*x)**2/(2*e
) - 27*b**3*d*f**2*g*n**3*x*log(d + e*x)/(2*e) + 63*b**3*d*f**2*g*n**3*x/(4*e) + 9*b**3*d*f**2*g*n**2*x*log(c)
*log(d + e*x)/e - 27*b**3*d*f**2*g*n**2*x*log(c)/(2*e) + 9*b**3*d*f**2*g*n*x*log(c)**2/(2*e) + 3*b**3*d*f*g**2
*n**3*x**2*log(d + e*x)**2/(2*e) - 5*b**3*d*f*g**2*n**3*x**2*log(d + e*x)/(2*e) + 19*b**3*d*f*g**2*n**3*x**2/(
12*e) + 3*b**3*d*f*g**2*n**2*x**2*log(c)*log(d + e*x)/e - 5*b**3*d*f*g**2*n**2*x**2*log(c)/(2*e) + 3*b**3*d*f*
g**2*n*x**2*log(c)**2/(2*e) + b**3*d*g**3*n**3*x**3*log(d + e*x)**2/(4*e) - 7*b**3*d*g**3*n**3*x**3*log(d + e*
x)/(24*e) + 37*b**3*d*g**3*n**3*x**3/(288*e) + b**3*d*g**3*n**2*x**3*log(c)*log(d + e*x)/(2*e) - 7*b**3*d*g**3
*n**2*x**3*log(c)/(24*e) + b**3*d*g**3*n*x**3*log(c)**2/(4*e) + b**3*f**3*n**3*x*log(d + e*x)**3 - 3*b**3*f**3
*n**3*x*log(d + e*x)**2 + 6*b**3*f**3*n**3*x*log(d + e*x) - 6*b**3*f**3*n**3*x + 3*b**3*f**3*n**2*x*log(c)*log
(d + e*x)**2 - 6*b**3*f**3*n**2*x*log(c)*log(d + e*x) + 6*b**3*f**3*n**2*x*log(c) + 3*b**3*f**3*n*x*log(c)**2*
log(d + e*x) - 3*b**3*f**3*n*x*log(c)**2 + b**3*f**3*x*log(c)**3 + 3*b**3*f**2*g*n**3*x**2*log(d + e*x)**3/2 -
9*b**3*f**2*g*n**3*x**2*log(d + e*x)**2/4 + 9*b**3*f**2*g*n**3*x**2*log(d + e*x)/4 - 9*b**3*f**2*g*n**3*x**2/
8 + 9*b**3*f**2*g*n**2*x**2*log(c)*log(d + e*x)**2/2 - 9*b**3*f**2*g*n**2*x**2*log(c)*log(d + e*x)/2 + 9*b**3*
f**2*g*n**2*x**2*log(c)/4 + 9*b**3*f**2*g*n*x**2*log(c)**2*log(d + e*x)/2 - 9*b**3*f**2*g*n*x**2*log(c)**2/4 +
3*b**3*f**2*g*x**2*log(c)**3/2 + b**3*f*g**2*n**3*x**3*log(d + e*x)**3 - b**3*f*g**2*n**3*x**3*log(d + e*x)**
2 + 2*b**3*f*g**2*n**3*x**3*log(d + e*x)/3 - 2*b**3*f*g**2*n**3*x**3/9 + 3*b**3*f*g**2*n**2*x**3*log(c)*log(d
+ e*x)**2 - 2*b**3*f*g**2*n**2*x**3*log(c)*log(d + e*x) + 2*b**3*f*g**2*n**2*x**3*log(c)/3 + 3*b**3*f*g**2*n*x
**3*log(c)**2*log(d + e*x) - b**3*f*g**2*n*x**3*log(c)**2 + b**3*f*g**2*x**3*log(c)**3 + b**3*g**3*n**3*x**4*l
og(d + e*x)**3/4 - 3*b**3*g**3*n**3*x**4*log(d + e*x)**2/16 + 3*b**3*g**3*n**3*x**4*log(d + e*x)/32 - 3*b**3*g
**3*n**3*x**4/128 + 3*b**3*g**3*n**2*x**4*log(c)*log(d + e*x)**2/4 - 3*b**3*g**3*n**2*x**4*log(c)*log(d + e*x)
/8 + 3*b**3*g**3*n**2*x**4*log(c)/32 + 3*b**3*g**3*n*x**4*log(c)**2*log(d + e*x)/4 - 3*b**3*g**3*n*x**4*log(c)
**2/16 + b**3*g**3*x**4*log(c)**3/4, Ne(e, 0)), ((a + b*log(c*d**n))**3*(f**3*x + 3*f**2*g*x**2/2 + f*g**2*x**
3 + g**3*x**4/4), True))

________________________________________________________________________________________

Giac [B]  time = 1.51654, size = 7131, normalized size = 11.92 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(x*e + d)^4*b^3*g^3*n^3*e^(-4)*log(x*e + d)^3 - (x*e + d)^3*b^3*d*g^3*n^3*e^(-4)*log(x*e + d)^3 + 3/2*(x*e
+ d)^2*b^3*d^2*g^3*n^3*e^(-4)*log(x*e + d)^3 - (x*e + d)*b^3*d^3*g^3*n^3*e^(-4)*log(x*e + d)^3 - 3/16*(x*e +
d)^4*b^3*g^3*n^3*e^(-4)*log(x*e + d)^2 + (x*e + d)^3*b^3*d*g^3*n^3*e^(-4)*log(x*e + d)^2 - 9/4*(x*e + d)^2*b^3
*d^2*g^3*n^3*e^(-4)*log(x*e + d)^2 + 3*(x*e + d)*b^3*d^3*g^3*n^3*e^(-4)*log(x*e + d)^2 + (x*e + d)^3*b^3*f*g^2
*n^3*e^(-3)*log(x*e + d)^3 - 3*(x*e + d)^2*b^3*d*f*g^2*n^3*e^(-3)*log(x*e + d)^3 + 3*(x*e + d)*b^3*d^2*f*g^2*n
^3*e^(-3)*log(x*e + d)^3 + 3/4*(x*e + d)^4*b^3*g^3*n^2*e^(-4)*log(x*e + d)^2*log(c) - 3*(x*e + d)^3*b^3*d*g^3*
n^2*e^(-4)*log(x*e + d)^2*log(c) + 9/2*(x*e + d)^2*b^3*d^2*g^3*n^2*e^(-4)*log(x*e + d)^2*log(c) - 3*(x*e + d)*
b^3*d^3*g^3*n^2*e^(-4)*log(x*e + d)^2*log(c) + 3/32*(x*e + d)^4*b^3*g^3*n^3*e^(-4)*log(x*e + d) - 2/3*(x*e + d
)^3*b^3*d*g^3*n^3*e^(-4)*log(x*e + d) + 9/4*(x*e + d)^2*b^3*d^2*g^3*n^3*e^(-4)*log(x*e + d) - 6*(x*e + d)*b^3*
d^3*g^3*n^3*e^(-4)*log(x*e + d) - (x*e + d)^3*b^3*f*g^2*n^3*e^(-3)*log(x*e + d)^2 + 9/2*(x*e + d)^2*b^3*d*f*g^
2*n^3*e^(-3)*log(x*e + d)^2 - 9*(x*e + d)*b^3*d^2*f*g^2*n^3*e^(-3)*log(x*e + d)^2 + 3/4*(x*e + d)^4*a*b^2*g^3*
n^2*e^(-4)*log(x*e + d)^2 - 3*(x*e + d)^3*a*b^2*d*g^3*n^2*e^(-4)*log(x*e + d)^2 + 9/2*(x*e + d)^2*a*b^2*d^2*g^
3*n^2*e^(-4)*log(x*e + d)^2 - 3*(x*e + d)*a*b^2*d^3*g^3*n^2*e^(-4)*log(x*e + d)^2 + 3/2*(x*e + d)^2*b^3*f^2*g*
n^3*e^(-2)*log(x*e + d)^3 - 3*(x*e + d)*b^3*d*f^2*g*n^3*e^(-2)*log(x*e + d)^3 - 3/8*(x*e + d)^4*b^3*g^3*n^2*e^
(-4)*log(x*e + d)*log(c) + 2*(x*e + d)^3*b^3*d*g^3*n^2*e^(-4)*log(x*e + d)*log(c) - 9/2*(x*e + d)^2*b^3*d^2*g^
3*n^2*e^(-4)*log(x*e + d)*log(c) + 6*(x*e + d)*b^3*d^3*g^3*n^2*e^(-4)*log(x*e + d)*log(c) + 3*(x*e + d)^3*b^3*
f*g^2*n^2*e^(-3)*log(x*e + d)^2*log(c) - 9*(x*e + d)^2*b^3*d*f*g^2*n^2*e^(-3)*log(x*e + d)^2*log(c) + 9*(x*e +
d)*b^3*d^2*f*g^2*n^2*e^(-3)*log(x*e + d)^2*log(c) + 3/4*(x*e + d)^4*b^3*g^3*n*e^(-4)*log(x*e + d)*log(c)^2 -
3*(x*e + d)^3*b^3*d*g^3*n*e^(-4)*log(x*e + d)*log(c)^2 + 9/2*(x*e + d)^2*b^3*d^2*g^3*n*e^(-4)*log(x*e + d)*log
(c)^2 - 3*(x*e + d)*b^3*d^3*g^3*n*e^(-4)*log(x*e + d)*log(c)^2 - 3/128*(x*e + d)^4*b^3*g^3*n^3*e^(-4) + 2/9*(x
*e + d)^3*b^3*d*g^3*n^3*e^(-4) - 9/8*(x*e + d)^2*b^3*d^2*g^3*n^3*e^(-4) + 6*(x*e + d)*b^3*d^3*g^3*n^3*e^(-4) +
2/3*(x*e + d)^3*b^3*f*g^2*n^3*e^(-3)*log(x*e + d) - 9/2*(x*e + d)^2*b^3*d*f*g^2*n^3*e^(-3)*log(x*e + d) + 18*
(x*e + d)*b^3*d^2*f*g^2*n^3*e^(-3)*log(x*e + d) - 3/8*(x*e + d)^4*a*b^2*g^3*n^2*e^(-4)*log(x*e + d) + 2*(x*e +
d)^3*a*b^2*d*g^3*n^2*e^(-4)*log(x*e + d) - 9/2*(x*e + d)^2*a*b^2*d^2*g^3*n^2*e^(-4)*log(x*e + d) + 6*(x*e + d
)*a*b^2*d^3*g^3*n^2*e^(-4)*log(x*e + d) - 9/4*(x*e + d)^2*b^3*f^2*g*n^3*e^(-2)*log(x*e + d)^2 + 9*(x*e + d)*b^
3*d*f^2*g*n^3*e^(-2)*log(x*e + d)^2 + 3*(x*e + d)^3*a*b^2*f*g^2*n^2*e^(-3)*log(x*e + d)^2 - 9*(x*e + d)^2*a*b^
2*d*f*g^2*n^2*e^(-3)*log(x*e + d)^2 + 9*(x*e + d)*a*b^2*d^2*f*g^2*n^2*e^(-3)*log(x*e + d)^2 + (x*e + d)*b^3*f^
3*n^3*e^(-1)*log(x*e + d)^3 + 3/32*(x*e + d)^4*b^3*g^3*n^2*e^(-4)*log(c) - 2/3*(x*e + d)^3*b^3*d*g^3*n^2*e^(-4
)*log(c) + 9/4*(x*e + d)^2*b^3*d^2*g^3*n^2*e^(-4)*log(c) - 6*(x*e + d)*b^3*d^3*g^3*n^2*e^(-4)*log(c) - 2*(x*e
+ d)^3*b^3*f*g^2*n^2*e^(-3)*log(x*e + d)*log(c) + 9*(x*e + d)^2*b^3*d*f*g^2*n^2*e^(-3)*log(x*e + d)*log(c) - 1
8*(x*e + d)*b^3*d^2*f*g^2*n^2*e^(-3)*log(x*e + d)*log(c) + 3/2*(x*e + d)^4*a*b^2*g^3*n*e^(-4)*log(x*e + d)*log
(c) - 6*(x*e + d)^3*a*b^2*d*g^3*n*e^(-4)*log(x*e + d)*log(c) + 9*(x*e + d)^2*a*b^2*d^2*g^3*n*e^(-4)*log(x*e +
d)*log(c) - 6*(x*e + d)*a*b^2*d^3*g^3*n*e^(-4)*log(x*e + d)*log(c) + 9/2*(x*e + d)^2*b^3*f^2*g*n^2*e^(-2)*log(
x*e + d)^2*log(c) - 9*(x*e + d)*b^3*d*f^2*g*n^2*e^(-2)*log(x*e + d)^2*log(c) - 3/16*(x*e + d)^4*b^3*g^3*n*e^(-
4)*log(c)^2 + (x*e + d)^3*b^3*d*g^3*n*e^(-4)*log(c)^2 - 9/4*(x*e + d)^2*b^3*d^2*g^3*n*e^(-4)*log(c)^2 + 3*(x*e
+ d)*b^3*d^3*g^3*n*e^(-4)*log(c)^2 + 3*(x*e + d)^3*b^3*f*g^2*n*e^(-3)*log(x*e + d)*log(c)^2 - 9*(x*e + d)^2*b
^3*d*f*g^2*n*e^(-3)*log(x*e + d)*log(c)^2 + 9*(x*e + d)*b^3*d^2*f*g^2*n*e^(-3)*log(x*e + d)*log(c)^2 + 1/4*(x*
e + d)^4*b^3*g^3*e^(-4)*log(c)^3 - (x*e + d)^3*b^3*d*g^3*e^(-4)*log(c)^3 + 3/2*(x*e + d)^2*b^3*d^2*g^3*e^(-4)*
log(c)^3 - (x*e + d)*b^3*d^3*g^3*e^(-4)*log(c)^3 - 2/9*(x*e + d)^3*b^3*f*g^2*n^3*e^(-3) + 9/4*(x*e + d)^2*b^3*
d*f*g^2*n^3*e^(-3) - 18*(x*e + d)*b^3*d^2*f*g^2*n^3*e^(-3) + 3/32*(x*e + d)^4*a*b^2*g^3*n^2*e^(-4) - 2/3*(x*e
+ d)^3*a*b^2*d*g^3*n^2*e^(-4) + 9/4*(x*e + d)^2*a*b^2*d^2*g^3*n^2*e^(-4) - 6*(x*e + d)*a*b^2*d^3*g^3*n^2*e^(-4
) + 9/4*(x*e + d)^2*b^3*f^2*g*n^3*e^(-2)*log(x*e + d) - 18*(x*e + d)*b^3*d*f^2*g*n^3*e^(-2)*log(x*e + d) - 2*(
x*e + d)^3*a*b^2*f*g^2*n^2*e^(-3)*log(x*e + d) + 9*(x*e + d)^2*a*b^2*d*f*g^2*n^2*e^(-3)*log(x*e + d) - 18*(x*e
+ d)*a*b^2*d^2*f*g^2*n^2*e^(-3)*log(x*e + d) + 3/4*(x*e + d)^4*a^2*b*g^3*n*e^(-4)*log(x*e + d) - 3*(x*e + d)^
3*a^2*b*d*g^3*n*e^(-4)*log(x*e + d) + 9/2*(x*e + d)^2*a^2*b*d^2*g^3*n*e^(-4)*log(x*e + d) - 3*(x*e + d)*a^2*b*
d^3*g^3*n*e^(-4)*log(x*e + d) - 3*(x*e + d)*b^3*f^3*n^3*e^(-1)*log(x*e + d)^2 + 9/2*(x*e + d)^2*a*b^2*f^2*g*n^
2*e^(-2)*log(x*e + d)^2 - 9*(x*e + d)*a*b^2*d*f^2*g*n^2*e^(-2)*log(x*e + d)^2 + 2/3*(x*e + d)^3*b^3*f*g^2*n^2*
e^(-3)*log(c) - 9/2*(x*e + d)^2*b^3*d*f*g^2*n^2*e^(-3)*log(c) + 18*(x*e + d)*b^3*d^2*f*g^2*n^2*e^(-3)*log(c) -
3/8*(x*e + d)^4*a*b^2*g^3*n*e^(-4)*log(c) + 2*(x*e + d)^3*a*b^2*d*g^3*n*e^(-4)*log(c) - 9/2*(x*e + d)^2*a*b^2
*d^2*g^3*n*e^(-4)*log(c) + 6*(x*e + d)*a*b^2*d^3*g^3*n*e^(-4)*log(c) - 9/2*(x*e + d)^2*b^3*f^2*g*n^2*e^(-2)*lo
g(x*e + d)*log(c) + 18*(x*e + d)*b^3*d*f^2*g*n^2*e^(-2)*log(x*e + d)*log(c) + 6*(x*e + d)^3*a*b^2*f*g^2*n*e^(-
3)*log(x*e + d)*log(c) - 18*(x*e + d)^2*a*b^2*d*f*g^2*n*e^(-3)*log(x*e + d)*log(c) + 18*(x*e + d)*a*b^2*d^2*f*
g^2*n*e^(-3)*log(x*e + d)*log(c) + 3*(x*e + d)*b^3*f^3*n^2*e^(-1)*log(x*e + d)^2*log(c) - (x*e + d)^3*b^3*f*g^
2*n*e^(-3)*log(c)^2 + 9/2*(x*e + d)^2*b^3*d*f*g^2*n*e^(-3)*log(c)^2 - 9*(x*e + d)*b^3*d^2*f*g^2*n*e^(-3)*log(c
)^2 + 3/4*(x*e + d)^4*a*b^2*g^3*e^(-4)*log(c)^2 - 3*(x*e + d)^3*a*b^2*d*g^3*e^(-4)*log(c)^2 + 9/2*(x*e + d)^2*
a*b^2*d^2*g^3*e^(-4)*log(c)^2 - 3*(x*e + d)*a*b^2*d^3*g^3*e^(-4)*log(c)^2 + 9/2*(x*e + d)^2*b^3*f^2*g*n*e^(-2)
*log(x*e + d)*log(c)^2 - 9*(x*e + d)*b^3*d*f^2*g*n*e^(-2)*log(x*e + d)*log(c)^2 + (x*e + d)^3*b^3*f*g^2*e^(-3)
*log(c)^3 - 3*(x*e + d)^2*b^3*d*f*g^2*e^(-3)*log(c)^3 + 3*(x*e + d)*b^3*d^2*f*g^2*e^(-3)*log(c)^3 - 9/8*(x*e +
d)^2*b^3*f^2*g*n^3*e^(-2) + 18*(x*e + d)*b^3*d*f^2*g*n^3*e^(-2) + 2/3*(x*e + d)^3*a*b^2*f*g^2*n^2*e^(-3) - 9/
2*(x*e + d)^2*a*b^2*d*f*g^2*n^2*e^(-3) + 18*(x*e + d)*a*b^2*d^2*f*g^2*n^2*e^(-3) - 3/16*(x*e + d)^4*a^2*b*g^3*
n*e^(-4) + (x*e + d)^3*a^2*b*d*g^3*n*e^(-4) - 9/4*(x*e + d)^2*a^2*b*d^2*g^3*n*e^(-4) + 3*(x*e + d)*a^2*b*d^3*g
^3*n*e^(-4) + 6*(x*e + d)*b^3*f^3*n^3*e^(-1)*log(x*e + d) - 9/2*(x*e + d)^2*a*b^2*f^2*g*n^2*e^(-2)*log(x*e + d
) + 18*(x*e + d)*a*b^2*d*f^2*g*n^2*e^(-2)*log(x*e + d) + 3*(x*e + d)^3*a^2*b*f*g^2*n*e^(-3)*log(x*e + d) - 9*(
x*e + d)^2*a^2*b*d*f*g^2*n*e^(-3)*log(x*e + d) + 9*(x*e + d)*a^2*b*d^2*f*g^2*n*e^(-3)*log(x*e + d) + 3*(x*e +
d)*a*b^2*f^3*n^2*e^(-1)*log(x*e + d)^2 + 9/4*(x*e + d)^2*b^3*f^2*g*n^2*e^(-2)*log(c) - 18*(x*e + d)*b^3*d*f^2*
g*n^2*e^(-2)*log(c) - 2*(x*e + d)^3*a*b^2*f*g^2*n*e^(-3)*log(c) + 9*(x*e + d)^2*a*b^2*d*f*g^2*n*e^(-3)*log(c)
- 18*(x*e + d)*a*b^2*d^2*f*g^2*n*e^(-3)*log(c) + 3/4*(x*e + d)^4*a^2*b*g^3*e^(-4)*log(c) - 3*(x*e + d)^3*a^2*b
*d*g^3*e^(-4)*log(c) + 9/2*(x*e + d)^2*a^2*b*d^2*g^3*e^(-4)*log(c) - 3*(x*e + d)*a^2*b*d^3*g^3*e^(-4)*log(c) -
6*(x*e + d)*b^3*f^3*n^2*e^(-1)*log(x*e + d)*log(c) + 9*(x*e + d)^2*a*b^2*f^2*g*n*e^(-2)*log(x*e + d)*log(c) -
18*(x*e + d)*a*b^2*d*f^2*g*n*e^(-2)*log(x*e + d)*log(c) - 9/4*(x*e + d)^2*b^3*f^2*g*n*e^(-2)*log(c)^2 + 9*(x*
e + d)*b^3*d*f^2*g*n*e^(-2)*log(c)^2 + 3*(x*e + d)^3*a*b^2*f*g^2*e^(-3)*log(c)^2 - 9*(x*e + d)^2*a*b^2*d*f*g^2
*e^(-3)*log(c)^2 + 9*(x*e + d)*a*b^2*d^2*f*g^2*e^(-3)*log(c)^2 + 3*(x*e + d)*b^3*f^3*n*e^(-1)*log(x*e + d)*log
(c)^2 + 3/2*(x*e + d)^2*b^3*f^2*g*e^(-2)*log(c)^3 - 3*(x*e + d)*b^3*d*f^2*g*e^(-2)*log(c)^3 - 6*(x*e + d)*b^3*
f^3*n^3*e^(-1) + 9/4*(x*e + d)^2*a*b^2*f^2*g*n^2*e^(-2) - 18*(x*e + d)*a*b^2*d*f^2*g*n^2*e^(-2) - (x*e + d)^3*
a^2*b*f*g^2*n*e^(-3) + 9/2*(x*e + d)^2*a^2*b*d*f*g^2*n*e^(-3) - 9*(x*e + d)*a^2*b*d^2*f*g^2*n*e^(-3) + 1/4*(x*
e + d)^4*a^3*g^3*e^(-4) - (x*e + d)^3*a^3*d*g^3*e^(-4) + 3/2*(x*e + d)^2*a^3*d^2*g^3*e^(-4) - (x*e + d)*a^3*d^
3*g^3*e^(-4) - 6*(x*e + d)*a*b^2*f^3*n^2*e^(-1)*log(x*e + d) + 9/2*(x*e + d)^2*a^2*b*f^2*g*n*e^(-2)*log(x*e +
d) - 9*(x*e + d)*a^2*b*d*f^2*g*n*e^(-2)*log(x*e + d) + 6*(x*e + d)*b^3*f^3*n^2*e^(-1)*log(c) - 9/2*(x*e + d)^2
*a*b^2*f^2*g*n*e^(-2)*log(c) + 18*(x*e + d)*a*b^2*d*f^2*g*n*e^(-2)*log(c) + 3*(x*e + d)^3*a^2*b*f*g^2*e^(-3)*l
og(c) - 9*(x*e + d)^2*a^2*b*d*f*g^2*e^(-3)*log(c) + 9*(x*e + d)*a^2*b*d^2*f*g^2*e^(-3)*log(c) + 6*(x*e + d)*a*
b^2*f^3*n*e^(-1)*log(x*e + d)*log(c) - 3*(x*e + d)*b^3*f^3*n*e^(-1)*log(c)^2 + 9/2*(x*e + d)^2*a*b^2*f^2*g*e^(
-2)*log(c)^2 - 9*(x*e + d)*a*b^2*d*f^2*g*e^(-2)*log(c)^2 + (x*e + d)*b^3*f^3*e^(-1)*log(c)^3 + 6*(x*e + d)*a*b
^2*f^3*n^2*e^(-1) - 9/4*(x*e + d)^2*a^2*b*f^2*g*n*e^(-2) + 9*(x*e + d)*a^2*b*d*f^2*g*n*e^(-2) + (x*e + d)^3*a^
3*f*g^2*e^(-3) - 3*(x*e + d)^2*a^3*d*f*g^2*e^(-3) + 3*(x*e + d)*a^3*d^2*f*g^2*e^(-3) + 3*(x*e + d)*a^2*b*f^3*n
*e^(-1)*log(x*e + d) - 6*(x*e + d)*a*b^2*f^3*n*e^(-1)*log(c) + 9/2*(x*e + d)^2*a^2*b*f^2*g*e^(-2)*log(c) - 9*(
x*e + d)*a^2*b*d*f^2*g*e^(-2)*log(c) + 3*(x*e + d)*a*b^2*f^3*e^(-1)*log(c)^2 - 3*(x*e + d)*a^2*b*f^3*n*e^(-1)
+ 3/2*(x*e + d)^2*a^3*f^2*g*e^(-2) - 3*(x*e + d)*a^3*d*f^2*g*e^(-2) + 3*(x*e + d)*a^2*b*f^3*e^(-1)*log(c) + (x
*e + d)*a^3*f^3*e^(-1)