3.53 \(\int (f+g x)^2 (a+b \log (c (d+e x)^n))^3 \, dx\)

Optimal. Leaf size=432 \[ \frac{3 b^2 g n^2 (d+e x)^2 (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^3}+\frac{2 b^2 g^2 n^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 e^3}+\frac{6 a b^2 n^2 x (e f-d g)^2}{e^2}-\frac{3 b g n (d+e x)^2 (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^3}-\frac{3 b n (d+e x) (e f-d g)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^3}+\frac{g (d+e x)^2 (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}+\frac{(d+e x) (e f-d g)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}-\frac{b g^2 n (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 e^3}+\frac{g^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 e^3}+\frac{6 b^3 n^2 (d+e x) (e f-d g)^2 \log \left (c (d+e x)^n\right )}{e^3}-\frac{3 b^3 g n^3 (d+e x)^2 (e f-d g)}{4 e^3}-\frac{6 b^3 n^3 x (e f-d g)^2}{e^2}-\frac{2 b^3 g^2 n^3 (d+e x)^3}{27 e^3} \]

[Out]

(6*a*b^2*(e*f - d*g)^2*n^2*x)/e^2 - (6*b^3*(e*f - d*g)^2*n^3*x)/e^2 - (3*b^3*g*(e*f - d*g)*n^3*(d + e*x)^2)/(4
*e^3) - (2*b^3*g^2*n^3*(d + e*x)^3)/(27*e^3) + (6*b^3*(e*f - d*g)^2*n^2*(d + e*x)*Log[c*(d + e*x)^n])/e^3 + (3
*b^2*g*(e*f - d*g)*n^2*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(2*e^3) + (2*b^2*g^2*n^2*(d + e*x)^3*(a + b*Log
[c*(d + e*x)^n]))/(9*e^3) - (3*b*(e*f - d*g)^2*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/e^3 - (3*b*g*(e*f - d
*g)*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^2)/(2*e^3) - (b*g^2*n*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n])^2)/(
3*e^3) + ((e*f - d*g)^2*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/e^3 + (g*(e*f - d*g)*(d + e*x)^2*(a + b*Log[c*
(d + e*x)^n])^3)/e^3 + (g^2*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n])^3)/(3*e^3)

________________________________________________________________________________________

Rubi [A]  time = 0.384478, antiderivative size = 432, normalized size of antiderivative = 1., number of steps used = 15, 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} \[ \frac{3 b^2 g n^2 (d+e x)^2 (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^3}+\frac{2 b^2 g^2 n^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 e^3}+\frac{6 a b^2 n^2 x (e f-d g)^2}{e^2}-\frac{3 b g n (d+e x)^2 (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^3}-\frac{3 b n (d+e x) (e f-d g)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^3}+\frac{g (d+e x)^2 (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}+\frac{(d+e x) (e f-d g)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}-\frac{b g^2 n (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 e^3}+\frac{g^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 e^3}+\frac{6 b^3 n^2 (d+e x) (e f-d g)^2 \log \left (c (d+e x)^n\right )}{e^3}-\frac{3 b^3 g n^3 (d+e x)^2 (e f-d g)}{4 e^3}-\frac{6 b^3 n^3 x (e f-d g)^2}{e^2}-\frac{2 b^3 g^2 n^3 (d+e x)^3}{27 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*a*b^2*(e*f - d*g)^2*n^2*x)/e^2 - (6*b^3*(e*f - d*g)^2*n^3*x)/e^2 - (3*b^3*g*(e*f - d*g)*n^3*(d + e*x)^2)/(4
*e^3) - (2*b^3*g^2*n^3*(d + e*x)^3)/(27*e^3) + (6*b^3*(e*f - d*g)^2*n^2*(d + e*x)*Log[c*(d + e*x)^n])/e^3 + (3
*b^2*g*(e*f - d*g)*n^2*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(2*e^3) + (2*b^2*g^2*n^2*(d + e*x)^3*(a + b*Log
[c*(d + e*x)^n]))/(9*e^3) - (3*b*(e*f - d*g)^2*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/e^3 - (3*b*g*(e*f - d
*g)*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^2)/(2*e^3) - (b*g^2*n*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n])^2)/(
3*e^3) + ((e*f - d*g)^2*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/e^3 + (g*(e*f - d*g)*(d + e*x)^2*(a + b*Log[c*
(d + e*x)^n])^3)/e^3 + (g^2*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n])^3)/(3*e^3)

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)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx &=\int \left (\frac{(e f-d g)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2}+\frac{2 g (e f-d g) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2}+\frac{g^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2}\right ) \, dx\\ &=\frac{g^2 \int (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx}{e^2}+\frac{(2 g (e f-d g)) \int (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx}{e^2}+\frac{(e f-d g)^2 \int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx}{e^2}\\ &=\frac{g^2 \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x\right )}{e^3}+\frac{(2 g (e f-d g)) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x\right )}{e^3}+\frac{(e f-d g)^2 \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x\right )}{e^3}\\ &=\frac{(e f-d g)^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}+\frac{g (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^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 e^3}-\frac{\left (b g^2 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^3}-\frac{(3 b g (e f-d g) n) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e^3}-\frac{\left (3 b (e f-d g)^2 n\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e^3}\\ &=-\frac{3 b (e f-d g)^2 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^3}-\frac{3 b g (e f-d g) n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^3}-\frac{b g^2 n (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 e^3}+\frac{(e f-d g)^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}+\frac{g (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^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 e^3}+\frac{\left (2 b^2 g^2 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 )}{3 e^3}+\frac{\left (3 b^2 g (e f-d g) n^2\right ) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e^3}+\frac{\left (6 b^2 (e f-d g)^2 n^2\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e^3}\\ &=\frac{6 a b^2 (e f-d g)^2 n^2 x}{e^2}-\frac{3 b^3 g (e f-d g) n^3 (d+e x)^2}{4 e^3}-\frac{2 b^3 g^2 n^3 (d+e x)^3}{27 e^3}+\frac{3 b^2 g (e f-d g) n^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^3}+\frac{2 b^2 g^2 n^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 e^3}-\frac{3 b (e f-d g)^2 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^3}-\frac{3 b g (e f-d g) n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^3}-\frac{b g^2 n (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 e^3}+\frac{(e f-d g)^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}+\frac{g (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^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 e^3}+\frac{\left (6 b^3 (e f-d g)^2 n^2\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e^3}\\ &=\frac{6 a b^2 (e f-d g)^2 n^2 x}{e^2}-\frac{6 b^3 (e f-d g)^2 n^3 x}{e^2}-\frac{3 b^3 g (e f-d g) n^3 (d+e x)^2}{4 e^3}-\frac{2 b^3 g^2 n^3 (d+e x)^3}{27 e^3}+\frac{6 b^3 (e f-d g)^2 n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e^3}+\frac{3 b^2 g (e f-d g) n^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^3}+\frac{2 b^2 g^2 n^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 e^3}-\frac{3 b (e f-d g)^2 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^3}-\frac{3 b g (e f-d g) n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^3}-\frac{b g^2 n (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 e^3}+\frac{(e f-d g)^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}+\frac{g (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^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 e^3}\\ \end{align*}

Mathematica [A]  time = 0.213052, size = 333, normalized size = 0.77 \[ \frac{-4 b g^2 n \left (2 b n \left (b e n x \left (3 d^2+3 d e x+e^2 x^2\right )-3 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )\right )+9 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2\right )+108 g (d+e x)^2 (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^3+108 (d+e x) (e f-d g)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3-324 b n (e f-d g)^2 \left ((d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2-2 b n \left (e x (a-b n)+b (d+e x) \log \left (c (d+e x)^n\right )\right )\right )-81 b g n (e f-d g) \left (2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2+b n \left (b e n x (2 d+e x)-2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )\right )\right )+36 g^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{108 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(108*(e*f - d*g)^2*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^3 + 108*g*(e*f - d*g)*(d + e*x)^2*(a + b*Log[c*(d + e*
x)^n])^3 + 36*g^2*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n])^3 - 324*b*(e*f - d*g)^2*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])) - 81*b*g*(e*f - d*g)*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]))) - 4*b*g^2*
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]))))/(108*e^3)

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Maple [C]  time = 1.743, size = 20417, normalized size = 47.3 \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)^2*(a+b*ln(c*(e*x+d)^n))^3,x)

[Out]

result too large to display

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Maxima [B]  time = 1.8809, size = 1539, normalized size = 3.56 \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)^2*(a+b*log(c*(e*x+d)^n))^3,x, algorithm="maxima")

[Out]

1/3*b^3*g^2*x^3*log((e*x + d)^n*c)^3 + a*b^2*g^2*x^3*log((e*x + d)^n*c)^2 + b^3*f*g*x^2*log((e*x + d)^n*c)^3 +
a^2*b*g^2*x^3*log((e*x + d)^n*c) + 3*a*b^2*f*g*x^2*log((e*x + d)^n*c)^2 + b^3*f^2*x*log((e*x + d)^n*c)^3 + 1/
3*a^3*g^2*x^3 - 3*a^2*b*e*f^2*n*(x/e - d*log(e*x + d)/e^2) + 1/6*a^2*b*e*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) - 3/2*a^2*b*e*f*g*n*(2*d^2*log(e*x + d)/e^3 + (e*x^2 - 2*d*x)/e^2) + 3*a^2*b
*f*g*x^2*log((e*x + d)^n*c) + 3*a*b^2*f^2*x*log((e*x + d)^n*c)^2 + a^3*f*g*x^2 + 3*a^2*b*f^2*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^2 - (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^2 - 3/2*(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*g - 1/4*(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*g + 1/18*(6*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) + (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*g^2 + 1/108*(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 - 57*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*g^2 + a^3*f^2*x

________________________________________________________________________________________

Fricas [B]  time = 2.56195, size = 3644, normalized size = 8.44 \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)^2*(a+b*log(c*(e*x+d)^n))^3,x, algorithm="fricas")

[Out]

-1/108*(4*(2*b^3*e^3*g^2*n^3 - 6*a*b^2*e^3*g^2*n^2 + 9*a^2*b*e^3*g^2*n - 9*a^3*e^3*g^2)*x^3 - 36*(b^3*e^3*g^2*
n^3*x^3 + 3*b^3*e^3*f*g*n^3*x^2 + 3*b^3*e^3*f^2*n^3*x + (3*b^3*d*e^2*f^2 - 3*b^3*d^2*e*f*g + b^3*d^3*g^2)*n^3)
*log(e*x + d)^3 - 36*(b^3*e^3*g^2*x^3 + 3*b^3*e^3*f*g*x^2 + 3*b^3*e^3*f^2*x)*log(c)^3 - 3*(36*a^3*e^3*f*g - (2
7*b^3*e^3*f*g - 19*b^3*d*e^2*g^2)*n^3 + 6*(9*a*b^2*e^3*f*g - 5*a*b^2*d*e^2*g^2)*n^2 - 18*(3*a^2*b*e^3*f*g - a^
2*b*d*e^2*g^2)*n)*x^2 + 18*((18*b^3*d*e^2*f^2 - 27*b^3*d^2*e*f*g + 11*b^3*d^3*g^2)*n^3 + 2*(b^3*e^3*g^2*n^3 -
3*a*b^2*e^3*g^2*n^2)*x^3 - 6*(3*a*b^2*d*e^2*f^2 - 3*a*b^2*d^2*e*f*g + a*b^2*d^3*g^2)*n^2 - 3*(6*a*b^2*e^3*f*g*
n^2 - (3*b^3*e^3*f*g - b^3*d*e^2*g^2)*n^3)*x^2 - 6*(3*a*b^2*e^3*f^2*n^2 - (3*b^3*e^3*f^2 - 3*b^3*d*e^2*f*g + b
^3*d^2*e*g^2)*n^3)*x - 6*(b^3*e^3*g^2*n^2*x^3 + 3*b^3*e^3*f*g*n^2*x^2 + 3*b^3*e^3*f^2*n^2*x + (3*b^3*d*e^2*f^2
- 3*b^3*d^2*e*f*g + b^3*d^3*g^2)*n^2)*log(c))*log(e*x + d)^2 + 18*(2*(b^3*e^3*g^2*n - 3*a*b^2*e^3*g^2)*x^3 -
3*(6*a*b^2*e^3*f*g - (3*b^3*e^3*f*g - b^3*d*e^2*g^2)*n)*x^2 - 6*(3*a*b^2*e^3*f^2 - (3*b^3*e^3*f^2 - 3*b^3*d*e^
2*f*g + b^3*d^2*e*g^2)*n)*x)*log(c)^2 - 6*(18*a^3*e^3*f^2 - (108*b^3*e^3*f^2 - 189*b^3*d*e^2*f*g + 85*b^3*d^2*
e*g^2)*n^3 + 6*(18*a*b^2*e^3*f^2 - 27*a*b^2*d*e^2*f*g + 11*a*b^2*d^2*e*g^2)*n^2 - 18*(3*a^2*b*e^3*f^2 - 3*a^2*
b*d*e^2*f*g + a^2*b*d^2*e*g^2)*n)*x - 6*((108*b^3*d*e^2*f^2 - 189*b^3*d^2*e*f*g + 85*b^3*d^3*g^2)*n^3 + 2*(2*b
^3*e^3*g^2*n^3 - 6*a*b^2*e^3*g^2*n^2 + 9*a^2*b*e^3*g^2*n)*x^3 - 6*(18*a*b^2*d*e^2*f^2 - 27*a*b^2*d^2*e*f*g + 1
1*a*b^2*d^3*g^2)*n^2 + 3*(18*a^2*b*e^3*f*g*n + (9*b^3*e^3*f*g - 5*b^3*d*e^2*g^2)*n^3 - 6*(3*a*b^2*e^3*f*g - a*
b^2*d*e^2*g^2)*n^2)*x^2 + 18*(b^3*e^3*g^2*n*x^3 + 3*b^3*e^3*f*g*n*x^2 + 3*b^3*e^3*f^2*n*x + (3*b^3*d*e^2*f^2 -
3*b^3*d^2*e*f*g + b^3*d^3*g^2)*n)*log(c)^2 + 18*(3*a^2*b*d*e^2*f^2 - 3*a^2*b*d^2*e*f*g + a^2*b*d^3*g^2)*n + 6
*(9*a^2*b*e^3*f^2*n + (18*b^3*e^3*f^2 - 27*b^3*d*e^2*f*g + 11*b^3*d^2*e*g^2)*n^3 - 6*(3*a*b^2*e^3*f^2 - 3*a*b^
2*d*e^2*f*g + a*b^2*d^2*e*g^2)*n^2)*x - 6*(2*(b^3*e^3*g^2*n^2 - 3*a*b^2*e^3*g^2*n)*x^3 + (18*b^3*d*e^2*f^2 - 2
7*b^3*d^2*e*f*g + 11*b^3*d^3*g^2)*n^2 - 3*(6*a*b^2*e^3*f*g*n - (3*b^3*e^3*f*g - b^3*d*e^2*g^2)*n^2)*x^2 - 6*(3
*a*b^2*d*e^2*f^2 - 3*a*b^2*d^2*e*f*g + a*b^2*d^3*g^2)*n - 6*(3*a*b^2*e^3*f^2*n - (3*b^3*e^3*f^2 - 3*b^3*d*e^2*
f*g + b^3*d^2*e*g^2)*n^2)*x)*log(c))*log(e*x + d) - 6*(2*(2*b^3*e^3*g^2*n^2 - 6*a*b^2*e^3*g^2*n + 9*a^2*b*e^3*
g^2)*x^3 + 3*(18*a^2*b*e^3*f*g + (9*b^3*e^3*f*g - 5*b^3*d*e^2*g^2)*n^2 - 6*(3*a*b^2*e^3*f*g - a*b^2*d*e^2*g^2)
*n)*x^2 + 6*(9*a^2*b*e^3*f^2 + (18*b^3*e^3*f^2 - 27*b^3*d*e^2*f*g + 11*b^3*d^2*e*g^2)*n^2 - 6*(3*a*b^2*e^3*f^2
- 3*a*b^2*d*e^2*f*g + a*b^2*d^2*e*g^2)*n)*x)*log(c))/e^3

________________________________________________________________________________________

Sympy [A]  time = 23.6911, size = 2746, normalized size = 6.36 \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)**2*(a+b*ln(c*(e*x+d)**n))**3,x)

[Out]

Piecewise((a**3*f**2*x + a**3*f*g*x**2 + a**3*g**2*x**3/3 + a**2*b*d**3*g**2*n*log(d + e*x)/e**3 - 3*a**2*b*d*
*2*f*g*n*log(d + e*x)/e**2 - a**2*b*d**2*g**2*n*x/e**2 + 3*a**2*b*d*f**2*n*log(d + e*x)/e + 3*a**2*b*d*f*g*n*x
/e + a**2*b*d*g**2*n*x**2/(2*e) + 3*a**2*b*f**2*n*x*log(d + e*x) - 3*a**2*b*f**2*n*x + 3*a**2*b*f**2*x*log(c)
+ 3*a**2*b*f*g*n*x**2*log(d + e*x) - 3*a**2*b*f*g*n*x**2/2 + 3*a**2*b*f*g*x**2*log(c) + a**2*b*g**2*n*x**3*log
(d + e*x) - a**2*b*g**2*n*x**3/3 + a**2*b*g**2*x**3*log(c) + a*b**2*d**3*g**2*n**2*log(d + e*x)**2/e**3 - 11*a
*b**2*d**3*g**2*n**2*log(d + e*x)/(3*e**3) + 2*a*b**2*d**3*g**2*n*log(c)*log(d + e*x)/e**3 - 3*a*b**2*d**2*f*g
*n**2*log(d + e*x)**2/e**2 + 9*a*b**2*d**2*f*g*n**2*log(d + e*x)/e**2 - 6*a*b**2*d**2*f*g*n*log(c)*log(d + e*x
)/e**2 - 2*a*b**2*d**2*g**2*n**2*x*log(d + e*x)/e**2 + 11*a*b**2*d**2*g**2*n**2*x/(3*e**2) - 2*a*b**2*d**2*g**
2*n*x*log(c)/e**2 + 3*a*b**2*d*f**2*n**2*log(d + e*x)**2/e - 6*a*b**2*d*f**2*n**2*log(d + e*x)/e + 6*a*b**2*d*
f**2*n*log(c)*log(d + e*x)/e + 6*a*b**2*d*f*g*n**2*x*log(d + e*x)/e - 9*a*b**2*d*f*g*n**2*x/e + 6*a*b**2*d*f*g
*n*x*log(c)/e + a*b**2*d*g**2*n**2*x**2*log(d + e*x)/e - 5*a*b**2*d*g**2*n**2*x**2/(6*e) + a*b**2*d*g**2*n*x**
2*log(c)/e + 3*a*b**2*f**2*n**2*x*log(d + e*x)**2 - 6*a*b**2*f**2*n**2*x*log(d + e*x) + 6*a*b**2*f**2*n**2*x +
6*a*b**2*f**2*n*x*log(c)*log(d + e*x) - 6*a*b**2*f**2*n*x*log(c) + 3*a*b**2*f**2*x*log(c)**2 + 3*a*b**2*f*g*n
**2*x**2*log(d + e*x)**2 - 3*a*b**2*f*g*n**2*x**2*log(d + e*x) + 3*a*b**2*f*g*n**2*x**2/2 + 6*a*b**2*f*g*n*x**
2*log(c)*log(d + e*x) - 3*a*b**2*f*g*n*x**2*log(c) + 3*a*b**2*f*g*x**2*log(c)**2 + a*b**2*g**2*n**2*x**3*log(d
+ e*x)**2 - 2*a*b**2*g**2*n**2*x**3*log(d + e*x)/3 + 2*a*b**2*g**2*n**2*x**3/9 + 2*a*b**2*g**2*n*x**3*log(c)*
log(d + e*x) - 2*a*b**2*g**2*n*x**3*log(c)/3 + a*b**2*g**2*x**3*log(c)**2 + b**3*d**3*g**2*n**3*log(d + e*x)**
3/(3*e**3) - 11*b**3*d**3*g**2*n**3*log(d + e*x)**2/(6*e**3) + 85*b**3*d**3*g**2*n**3*log(d + e*x)/(18*e**3) +
b**3*d**3*g**2*n**2*log(c)*log(d + e*x)**2/e**3 - 11*b**3*d**3*g**2*n**2*log(c)*log(d + e*x)/(3*e**3) + b**3*
d**3*g**2*n*log(c)**2*log(d + e*x)/e**3 - b**3*d**2*f*g*n**3*log(d + e*x)**3/e**2 + 9*b**3*d**2*f*g*n**3*log(d
+ e*x)**2/(2*e**2) - 21*b**3*d**2*f*g*n**3*log(d + e*x)/(2*e**2) - 3*b**3*d**2*f*g*n**2*log(c)*log(d + e*x)**
2/e**2 + 9*b**3*d**2*f*g*n**2*log(c)*log(d + e*x)/e**2 - 3*b**3*d**2*f*g*n*log(c)**2*log(d + e*x)/e**2 - b**3*
d**2*g**2*n**3*x*log(d + e*x)**2/e**2 + 11*b**3*d**2*g**2*n**3*x*log(d + e*x)/(3*e**2) - 85*b**3*d**2*g**2*n**
3*x/(18*e**2) - 2*b**3*d**2*g**2*n**2*x*log(c)*log(d + e*x)/e**2 + 11*b**3*d**2*g**2*n**2*x*log(c)/(3*e**2) -
b**3*d**2*g**2*n*x*log(c)**2/e**2 + b**3*d*f**2*n**3*log(d + e*x)**3/e - 3*b**3*d*f**2*n**3*log(d + e*x)**2/e
+ 6*b**3*d*f**2*n**3*log(d + e*x)/e + 3*b**3*d*f**2*n**2*log(c)*log(d + e*x)**2/e - 6*b**3*d*f**2*n**2*log(c)*
log(d + e*x)/e + 3*b**3*d*f**2*n*log(c)**2*log(d + e*x)/e + 3*b**3*d*f*g*n**3*x*log(d + e*x)**2/e - 9*b**3*d*f
*g*n**3*x*log(d + e*x)/e + 21*b**3*d*f*g*n**3*x/(2*e) + 6*b**3*d*f*g*n**2*x*log(c)*log(d + e*x)/e - 9*b**3*d*f
*g*n**2*x*log(c)/e + 3*b**3*d*f*g*n*x*log(c)**2/e + b**3*d*g**2*n**3*x**2*log(d + e*x)**2/(2*e) - 5*b**3*d*g**
2*n**3*x**2*log(d + e*x)/(6*e) + 19*b**3*d*g**2*n**3*x**2/(36*e) + b**3*d*g**2*n**2*x**2*log(c)*log(d + e*x)/e
- 5*b**3*d*g**2*n**2*x**2*log(c)/(6*e) + b**3*d*g**2*n*x**2*log(c)**2/(2*e) + b**3*f**2*n**3*x*log(d + e*x)**
3 - 3*b**3*f**2*n**3*x*log(d + e*x)**2 + 6*b**3*f**2*n**3*x*log(d + e*x) - 6*b**3*f**2*n**3*x + 3*b**3*f**2*n*
*2*x*log(c)*log(d + e*x)**2 - 6*b**3*f**2*n**2*x*log(c)*log(d + e*x) + 6*b**3*f**2*n**2*x*log(c) + 3*b**3*f**2
*n*x*log(c)**2*log(d + e*x) - 3*b**3*f**2*n*x*log(c)**2 + b**3*f**2*x*log(c)**3 + b**3*f*g*n**3*x**2*log(d + e
*x)**3 - 3*b**3*f*g*n**3*x**2*log(d + e*x)**2/2 + 3*b**3*f*g*n**3*x**2*log(d + e*x)/2 - 3*b**3*f*g*n**3*x**2/4
+ 3*b**3*f*g*n**2*x**2*log(c)*log(d + e*x)**2 - 3*b**3*f*g*n**2*x**2*log(c)*log(d + e*x) + 3*b**3*f*g*n**2*x*
*2*log(c)/2 + 3*b**3*f*g*n*x**2*log(c)**2*log(d + e*x) - 3*b**3*f*g*n*x**2*log(c)**2/2 + b**3*f*g*x**2*log(c)*
*3 + b**3*g**2*n**3*x**3*log(d + e*x)**3/3 - b**3*g**2*n**3*x**3*log(d + e*x)**2/3 + 2*b**3*g**2*n**3*x**3*log
(d + e*x)/9 - 2*b**3*g**2*n**3*x**3/27 + b**3*g**2*n**2*x**3*log(c)*log(d + e*x)**2 - 2*b**3*g**2*n**2*x**3*lo
g(c)*log(d + e*x)/3 + 2*b**3*g**2*n**2*x**3*log(c)/9 + b**3*g**2*n*x**3*log(c)**2*log(d + e*x) - b**3*g**2*n*x
**3*log(c)**2/3 + b**3*g**2*x**3*log(c)**3/3, Ne(e, 0)), ((a + b*log(c*d**n))**3*(f**2*x + f*g*x**2 + g**2*x**
3/3), True))

________________________________________________________________________________________

Giac [B]  time = 1.35771, size = 4039, normalized size = 9.35 \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)^2*(a+b*log(c*(e*x+d)^n))^3,x, algorithm="giac")

[Out]

1/3*(x*e + d)^3*b^3*g^2*n^3*e^(-3)*log(x*e + d)^3 - (x*e + d)^2*b^3*d*g^2*n^3*e^(-3)*log(x*e + d)^3 + (x*e + d
)*b^3*d^2*g^2*n^3*e^(-3)*log(x*e + d)^3 - 1/3*(x*e + d)^3*b^3*g^2*n^3*e^(-3)*log(x*e + d)^2 + 3/2*(x*e + d)^2*
b^3*d*g^2*n^3*e^(-3)*log(x*e + d)^2 - 3*(x*e + d)*b^3*d^2*g^2*n^3*e^(-3)*log(x*e + d)^2 + (x*e + d)^2*b^3*f*g*
n^3*e^(-2)*log(x*e + d)^3 - 2*(x*e + d)*b^3*d*f*g*n^3*e^(-2)*log(x*e + d)^3 + (x*e + d)^3*b^3*g^2*n^2*e^(-3)*l
og(x*e + d)^2*log(c) - 3*(x*e + d)^2*b^3*d*g^2*n^2*e^(-3)*log(x*e + d)^2*log(c) + 3*(x*e + d)*b^3*d^2*g^2*n^2*
e^(-3)*log(x*e + d)^2*log(c) + 2/9*(x*e + d)^3*b^3*g^2*n^3*e^(-3)*log(x*e + d) - 3/2*(x*e + d)^2*b^3*d*g^2*n^3
*e^(-3)*log(x*e + d) + 6*(x*e + d)*b^3*d^2*g^2*n^3*e^(-3)*log(x*e + d) - 3/2*(x*e + d)^2*b^3*f*g*n^3*e^(-2)*lo
g(x*e + d)^2 + 6*(x*e + d)*b^3*d*f*g*n^3*e^(-2)*log(x*e + d)^2 + (x*e + d)^3*a*b^2*g^2*n^2*e^(-3)*log(x*e + d)
^2 - 3*(x*e + d)^2*a*b^2*d*g^2*n^2*e^(-3)*log(x*e + d)^2 + 3*(x*e + d)*a*b^2*d^2*g^2*n^2*e^(-3)*log(x*e + d)^2
+ (x*e + d)*b^3*f^2*n^3*e^(-1)*log(x*e + d)^3 - 2/3*(x*e + d)^3*b^3*g^2*n^2*e^(-3)*log(x*e + d)*log(c) + 3*(x
*e + d)^2*b^3*d*g^2*n^2*e^(-3)*log(x*e + d)*log(c) - 6*(x*e + d)*b^3*d^2*g^2*n^2*e^(-3)*log(x*e + d)*log(c) +
3*(x*e + d)^2*b^3*f*g*n^2*e^(-2)*log(x*e + d)^2*log(c) - 6*(x*e + d)*b^3*d*f*g*n^2*e^(-2)*log(x*e + d)^2*log(c
) + (x*e + d)^3*b^3*g^2*n*e^(-3)*log(x*e + d)*log(c)^2 - 3*(x*e + d)^2*b^3*d*g^2*n*e^(-3)*log(x*e + d)*log(c)^
2 + 3*(x*e + d)*b^3*d^2*g^2*n*e^(-3)*log(x*e + d)*log(c)^2 - 2/27*(x*e + d)^3*b^3*g^2*n^3*e^(-3) + 3/4*(x*e +
d)^2*b^3*d*g^2*n^3*e^(-3) - 6*(x*e + d)*b^3*d^2*g^2*n^3*e^(-3) + 3/2*(x*e + d)^2*b^3*f*g*n^3*e^(-2)*log(x*e +
d) - 12*(x*e + d)*b^3*d*f*g*n^3*e^(-2)*log(x*e + d) - 2/3*(x*e + d)^3*a*b^2*g^2*n^2*e^(-3)*log(x*e + d) + 3*(x
*e + d)^2*a*b^2*d*g^2*n^2*e^(-3)*log(x*e + d) - 6*(x*e + d)*a*b^2*d^2*g^2*n^2*e^(-3)*log(x*e + d) - 3*(x*e + d
)*b^3*f^2*n^3*e^(-1)*log(x*e + d)^2 + 3*(x*e + d)^2*a*b^2*f*g*n^2*e^(-2)*log(x*e + d)^2 - 6*(x*e + d)*a*b^2*d*
f*g*n^2*e^(-2)*log(x*e + d)^2 + 2/9*(x*e + d)^3*b^3*g^2*n^2*e^(-3)*log(c) - 3/2*(x*e + d)^2*b^3*d*g^2*n^2*e^(-
3)*log(c) + 6*(x*e + d)*b^3*d^2*g^2*n^2*e^(-3)*log(c) - 3*(x*e + d)^2*b^3*f*g*n^2*e^(-2)*log(x*e + d)*log(c) +
12*(x*e + d)*b^3*d*f*g*n^2*e^(-2)*log(x*e + d)*log(c) + 2*(x*e + d)^3*a*b^2*g^2*n*e^(-3)*log(x*e + d)*log(c)
- 6*(x*e + d)^2*a*b^2*d*g^2*n*e^(-3)*log(x*e + d)*log(c) + 6*(x*e + d)*a*b^2*d^2*g^2*n*e^(-3)*log(x*e + d)*log
(c) + 3*(x*e + d)*b^3*f^2*n^2*e^(-1)*log(x*e + d)^2*log(c) - 1/3*(x*e + d)^3*b^3*g^2*n*e^(-3)*log(c)^2 + 3/2*(
x*e + d)^2*b^3*d*g^2*n*e^(-3)*log(c)^2 - 3*(x*e + d)*b^3*d^2*g^2*n*e^(-3)*log(c)^2 + 3*(x*e + d)^2*b^3*f*g*n*e
^(-2)*log(x*e + d)*log(c)^2 - 6*(x*e + d)*b^3*d*f*g*n*e^(-2)*log(x*e + d)*log(c)^2 + 1/3*(x*e + d)^3*b^3*g^2*e
^(-3)*log(c)^3 - (x*e + d)^2*b^3*d*g^2*e^(-3)*log(c)^3 + (x*e + d)*b^3*d^2*g^2*e^(-3)*log(c)^3 - 3/4*(x*e + d)
^2*b^3*f*g*n^3*e^(-2) + 12*(x*e + d)*b^3*d*f*g*n^3*e^(-2) + 2/9*(x*e + d)^3*a*b^2*g^2*n^2*e^(-3) - 3/2*(x*e +
d)^2*a*b^2*d*g^2*n^2*e^(-3) + 6*(x*e + d)*a*b^2*d^2*g^2*n^2*e^(-3) + 6*(x*e + d)*b^3*f^2*n^3*e^(-1)*log(x*e +
d) - 3*(x*e + d)^2*a*b^2*f*g*n^2*e^(-2)*log(x*e + d) + 12*(x*e + d)*a*b^2*d*f*g*n^2*e^(-2)*log(x*e + d) + (x*e
+ d)^3*a^2*b*g^2*n*e^(-3)*log(x*e + d) - 3*(x*e + d)^2*a^2*b*d*g^2*n*e^(-3)*log(x*e + d) + 3*(x*e + d)*a^2*b*
d^2*g^2*n*e^(-3)*log(x*e + d) + 3*(x*e + d)*a*b^2*f^2*n^2*e^(-1)*log(x*e + d)^2 + 3/2*(x*e + d)^2*b^3*f*g*n^2*
e^(-2)*log(c) - 12*(x*e + d)*b^3*d*f*g*n^2*e^(-2)*log(c) - 2/3*(x*e + d)^3*a*b^2*g^2*n*e^(-3)*log(c) + 3*(x*e
+ d)^2*a*b^2*d*g^2*n*e^(-3)*log(c) - 6*(x*e + d)*a*b^2*d^2*g^2*n*e^(-3)*log(c) - 6*(x*e + d)*b^3*f^2*n^2*e^(-1
)*log(x*e + d)*log(c) + 6*(x*e + d)^2*a*b^2*f*g*n*e^(-2)*log(x*e + d)*log(c) - 12*(x*e + d)*a*b^2*d*f*g*n*e^(-
2)*log(x*e + d)*log(c) - 3/2*(x*e + d)^2*b^3*f*g*n*e^(-2)*log(c)^2 + 6*(x*e + d)*b^3*d*f*g*n*e^(-2)*log(c)^2 +
(x*e + d)^3*a*b^2*g^2*e^(-3)*log(c)^2 - 3*(x*e + d)^2*a*b^2*d*g^2*e^(-3)*log(c)^2 + 3*(x*e + d)*a*b^2*d^2*g^2
*e^(-3)*log(c)^2 + 3*(x*e + d)*b^3*f^2*n*e^(-1)*log(x*e + d)*log(c)^2 + (x*e + d)^2*b^3*f*g*e^(-2)*log(c)^3 -
2*(x*e + d)*b^3*d*f*g*e^(-2)*log(c)^3 - 6*(x*e + d)*b^3*f^2*n^3*e^(-1) + 3/2*(x*e + d)^2*a*b^2*f*g*n^2*e^(-2)
- 12*(x*e + d)*a*b^2*d*f*g*n^2*e^(-2) - 1/3*(x*e + d)^3*a^2*b*g^2*n*e^(-3) + 3/2*(x*e + d)^2*a^2*b*d*g^2*n*e^(
-3) - 3*(x*e + d)*a^2*b*d^2*g^2*n*e^(-3) - 6*(x*e + d)*a*b^2*f^2*n^2*e^(-1)*log(x*e + d) + 3*(x*e + d)^2*a^2*b
*f*g*n*e^(-2)*log(x*e + d) - 6*(x*e + d)*a^2*b*d*f*g*n*e^(-2)*log(x*e + d) + 6*(x*e + d)*b^3*f^2*n^2*e^(-1)*lo
g(c) - 3*(x*e + d)^2*a*b^2*f*g*n*e^(-2)*log(c) + 12*(x*e + d)*a*b^2*d*f*g*n*e^(-2)*log(c) + (x*e + d)^3*a^2*b*
g^2*e^(-3)*log(c) - 3*(x*e + d)^2*a^2*b*d*g^2*e^(-3)*log(c) + 3*(x*e + d)*a^2*b*d^2*g^2*e^(-3)*log(c) + 6*(x*e
+ d)*a*b^2*f^2*n*e^(-1)*log(x*e + d)*log(c) - 3*(x*e + d)*b^3*f^2*n*e^(-1)*log(c)^2 + 3*(x*e + d)^2*a*b^2*f*g
*e^(-2)*log(c)^2 - 6*(x*e + d)*a*b^2*d*f*g*e^(-2)*log(c)^2 + (x*e + d)*b^3*f^2*e^(-1)*log(c)^3 + 6*(x*e + d)*a
*b^2*f^2*n^2*e^(-1) - 3/2*(x*e + d)^2*a^2*b*f*g*n*e^(-2) + 6*(x*e + d)*a^2*b*d*f*g*n*e^(-2) + 1/3*(x*e + d)^3*
a^3*g^2*e^(-3) - (x*e + d)^2*a^3*d*g^2*e^(-3) + (x*e + d)*a^3*d^2*g^2*e^(-3) + 3*(x*e + d)*a^2*b*f^2*n*e^(-1)*
log(x*e + d) - 6*(x*e + d)*a*b^2*f^2*n*e^(-1)*log(c) + 3*(x*e + d)^2*a^2*b*f*g*e^(-2)*log(c) - 6*(x*e + d)*a^2
*b*d*f*g*e^(-2)*log(c) + 3*(x*e + d)*a*b^2*f^2*e^(-1)*log(c)^2 - 3*(x*e + d)*a^2*b*f^2*n*e^(-1) + (x*e + d)^2*
a^3*f*g*e^(-2) - 2*(x*e + d)*a^3*d*f*g*e^(-2) + 3*(x*e + d)*a^2*b*f^2*e^(-1)*log(c) + (x*e + d)*a^3*f^2*e^(-1)