3.45 \(\int (f+g x)^2 (a+b \log (c (d+e x)^n))^2 \, dx\)
Optimal. Leaf size=287 \[ -\frac{2 b n (e f-d g)^3 \log (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e^3 g}-\frac{2 b n (d+e x) (e f-d g)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^3}-\frac{b g n (d+e x)^2 (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^3}-\frac{2 b g^2 n (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 e^3}+\frac{(f+g x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g}+\frac{2 b^2 n^2 x (e f-d g)^2}{e^2}+\frac{b^2 g n^2 (d+e x)^2 (e f-d g)}{2 e^3}+\frac{b^2 n^2 (e f-d g)^3 \log ^2(d+e x)}{3 e^3 g}+\frac{2 b^2 g^2 n^2 (d+e x)^3}{27 e^3} \]
[Out]
(2*b^2*(e*f - d*g)^2*n^2*x)/e^2 + (b^2*g*(e*f - d*g)*n^2*(d + e*x)^2)/(2*e^3) + (2*b^2*g^2*n^2*(d + e*x)^3)/(2
7*e^3) + (b^2*(e*f - d*g)^3*n^2*Log[d + e*x]^2)/(3*e^3*g) - (2*b*(e*f - d*g)^2*n*(d + e*x)*(a + b*Log[c*(d + e
*x)^n]))/e^3 - (b*g*(e*f - d*g)*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/e^3 - (2*b*g^2*n*(d + e*x)^3*(a + b*
Log[c*(d + e*x)^n]))/(9*e^3) - (2*b*(e*f - d*g)^3*n*Log[d + e*x]*(a + b*Log[c*(d + e*x)^n]))/(3*e^3*g) + ((f +
g*x)^3*(a + b*Log[c*(d + e*x)^n])^2)/(3*g)
________________________________________________________________________________________
Rubi [A] time = 0.412581, antiderivative size = 243, normalized size of antiderivative =
0.85, number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules
used = {2398, 2411, 43, 2334, 12, 14, 2301} \[ -\frac{b n \left (\frac{9 g^2 (d+e x)^2 (e f-d g)}{e^3}+\frac{18 g (d+e x) (e f-d g)^2}{e^3}+\frac{6 (e f-d g)^3 \log (d+e x)}{e^3}+\frac{2 g^3 (d+e x)^3}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 g}+\frac{(f+g x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g}+\frac{2 b^2 n^2 x (e f-d g)^2}{e^2}+\frac{b^2 g n^2 (d+e x)^2 (e f-d g)}{2 e^3}+\frac{b^2 n^2 (e f-d g)^3 \log ^2(d+e x)}{3 e^3 g}+\frac{2 b^2 g^2 n^2 (d+e x)^3}{27 e^3} \]
Antiderivative was successfully verified.
[In]
Int[(f + g*x)^2*(a + b*Log[c*(d + e*x)^n])^2,x]
[Out]
(2*b^2*(e*f - d*g)^2*n^2*x)/e^2 + (b^2*g*(e*f - d*g)*n^2*(d + e*x)^2)/(2*e^3) + (2*b^2*g^2*n^2*(d + e*x)^3)/(2
7*e^3) + (b^2*(e*f - d*g)^3*n^2*Log[d + e*x]^2)/(3*e^3*g) - (b*n*((18*g*(e*f - d*g)^2*(d + e*x))/e^3 + (9*g^2*
(e*f - d*g)*(d + e*x)^2)/e^3 + (2*g^3*(d + e*x)^3)/e^3 + (6*(e*f - d*g)^3*Log[d + e*x])/e^3)*(a + b*Log[c*(d +
e*x)^n]))/(9*g) + ((f + g*x)^3*(a + b*Log[c*(d + e*x)^n])^2)/(3*g)
Rule 2398
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((
f + g*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n])^p)/(g*(q + 1)), x] - Dist[(b*e*n*p)/(g*(q + 1)), Int[((f + g*x)^(q
+ 1)*(a + b*Log[c*(d + e*x)^n])^(p - 1))/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*
f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && IntegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))
Rule 2411
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]
Rule 43
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Rule 2334
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]
] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] && !(EqQ[q, 1] && EqQ[m, -1])
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 14
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&& !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Rule 2301
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]
Rubi steps
\begin{align*} \int (f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx &=\frac{(f+g x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g}-\frac{(2 b e n) \int \frac{(f+g x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx}{3 g}\\ &=\frac{(f+g x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g}-\frac{(2 b n) \operatorname{Subst}\left (\int \frac{\left (\frac{e f-d g}{e}+\frac{g x}{e}\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+e x\right )}{3 g}\\ &=-\frac{b n \left (\frac{18 g (e f-d g)^2 (d+e x)}{e^3}+\frac{9 g^2 (e f-d g) (d+e x)^2}{e^3}+\frac{2 g^3 (d+e x)^3}{e^3}+\frac{6 (e f-d g)^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 g}+\frac{(f+g x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g}+\frac{\left (2 b^2 n^2\right ) \operatorname{Subst}\left (\int \frac{g x \left (18 e^2 f^2+9 e f g (-4 d+x)+g^2 \left (18 d^2-9 d x+2 x^2\right )\right )+6 (e f-d g)^3 \log (x)}{6 e^3 x} \, dx,x,d+e x\right )}{3 g}\\ &=-\frac{b n \left (\frac{18 g (e f-d g)^2 (d+e x)}{e^3}+\frac{9 g^2 (e f-d g) (d+e x)^2}{e^3}+\frac{2 g^3 (d+e x)^3}{e^3}+\frac{6 (e f-d g)^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 g}+\frac{(f+g x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g}+\frac{\left (b^2 n^2\right ) \operatorname{Subst}\left (\int \frac{g x \left (18 e^2 f^2+9 e f g (-4 d+x)+g^2 \left (18 d^2-9 d x+2 x^2\right )\right )+6 (e f-d g)^3 \log (x)}{x} \, dx,x,d+e x\right )}{9 e^3 g}\\ &=-\frac{b n \left (\frac{18 g (e f-d g)^2 (d+e x)}{e^3}+\frac{9 g^2 (e f-d g) (d+e x)^2}{e^3}+\frac{2 g^3 (d+e x)^3}{e^3}+\frac{6 (e f-d g)^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 g}+\frac{(f+g x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g}+\frac{\left (b^2 n^2\right ) \operatorname{Subst}\left (\int \left (g \left (18 (e f-d g)^2+9 g (e f-d g) x+2 g^2 x^2\right )+\frac{6 (e f-d g)^3 \log (x)}{x}\right ) \, dx,x,d+e x\right )}{9 e^3 g}\\ &=-\frac{b n \left (\frac{18 g (e f-d g)^2 (d+e x)}{e^3}+\frac{9 g^2 (e f-d g) (d+e x)^2}{e^3}+\frac{2 g^3 (d+e x)^3}{e^3}+\frac{6 (e f-d g)^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 g}+\frac{(f+g x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g}+\frac{\left (b^2 n^2\right ) \operatorname{Subst}\left (\int \left (18 (e f-d g)^2+9 g (e f-d g) x+2 g^2 x^2\right ) \, dx,x,d+e x\right )}{9 e^3}+\frac{\left (2 b^2 (e f-d g)^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,d+e x\right )}{3 e^3 g}\\ &=\frac{2 b^2 (e f-d g)^2 n^2 x}{e^2}+\frac{b^2 g (e f-d g) n^2 (d+e x)^2}{2 e^3}+\frac{2 b^2 g^2 n^2 (d+e x)^3}{27 e^3}+\frac{b^2 (e f-d g)^3 n^2 \log ^2(d+e x)}{3 e^3 g}-\frac{b n \left (\frac{18 g (e f-d g)^2 (d+e x)}{e^3}+\frac{9 g^2 (e f-d g) (d+e x)^2}{e^3}+\frac{2 g^3 (d+e x)^3}{e^3}+\frac{6 (e f-d g)^3 \log (d+e x)}{e^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 g}+\frac{(f+g x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g}\\ \end{align*}
Mathematica [A] time = 0.149358, size = 247, normalized size = 0.86 \[ \frac{4 b g^2 n \left (b e n x \left (3 d^2+3 d e x+e^2 x^2\right )-3 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )\right )+54 g (d+e x)^2 (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^2+54 (d+e x) (e f-d g)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2-108 b n (e f-d g)^2 \left (e x (a-b n)+b (d+e x) \log \left (c (d+e x)^n\right )\right )+27 b g n (e f-d g) \left (b e n x (2 d+e x)-2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )\right )+18 g^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{54 e^3} \]
Antiderivative was successfully verified.
[In]
Integrate[(f + g*x)^2*(a + b*Log[c*(d + e*x)^n])^2,x]
[Out]
(54*(e*f - d*g)^2*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2 + 54*g*(e*f - d*g)*(d + e*x)^2*(a + b*Log[c*(d + e*x)
^n])^2 + 18*g^2*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n])^2 - 108*b*(e*f - d*g)^2*n*(e*(a - b*n)*x + b*(d + e*x)*
Log[c*(d + e*x)^n]) + 27*b*g*(e*f - d*g)*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*(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])))/(54*e^3)
________________________________________________________________________________________
Maple [C] time = 0.769, size = 4597, normalized size = 16. \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))^2,x)
[Out]
1/3*(g*x+f)^3*b^2/g*ln((e*x+d)^n)^2+2/e*g*a*b*d*f*n*x+1/3*ln(c)^2*b^2*g^2*x^3+ln(c)^2*b^2*f^2*x+2/27*b^2*g^2*n
^2*x^3+2*b^2*f^2*n^2*x+1/3*a^2*g^2*x^3+a^2*f^2*x+a^2*f*g*x^2+1/9*b*(-9*I*Pi*b*e^3*f^2*g*x*csgn(I*c)*csgn(I*(e*
x+d)^n)*csgn(I*c*(e*x+d)^n)+6*a*e^3*g^3*x^3+6*ln(e*x+d)*b*d^3*g^3*n-6*ln(e*x+d)*b*e^3*f^3*n+6*ln(c)*b*e^3*g^3*
x^3-2*b*e^3*g^3*n*x^3+18*a*e^3*f*g^2*x^2+18*a*e^3*f^2*g*x+3*b*d*e^2*g^3*n*x^2-9*b*e^3*f*g^2*n*x^2-6*b*d^2*e*g^
3*n*x-18*b*e^3*f^2*g*n*x+18*ln(c)*b*e^3*f^2*g*x+18*ln(c)*b*e^3*f*g^2*x^2+18*b*d*e^2*f*g^2*n*x-18*ln(e*x+d)*b*d
^2*e*f*g^2*n+18*ln(e*x+d)*b*d*e^2*f^2*g*n+3*I*Pi*b*e^3*g^3*x^3*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+3*I*Pi*b*e^3*g^
3*x^3*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-9*I*Pi*b*e^3*f*g^2*x^2*csgn(I*c*(e*x+d)^n)^3-3*I*Pi*b*e^3*g^3*x^
3*csgn(I*c*(e*x+d)^n)^3-3*I*Pi*b*e^3*g^3*x^3*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+9*I*Pi*b*e^3*f*g^
2*x^2*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+9*I*Pi*b*e^3*f*g^2*x^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2+9*I*Pi*b*
e^3*f^2*g*x*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+9*I*Pi*b*e^3*f^2*g*x*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-9*I*P
i*b*e^3*f*g^2*x^2*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-9*I*Pi*b*e^3*f^2*g*x*csgn(I*c*(e*x+d)^n)^3)/
e^3/g*ln((e*x+d)^n)-1/12*g^2*Pi^2*b^2*x^3*csgn(I*c*(e*x+d)^n)^6+1/3/g*b^2*f^3*n^2*ln(e*x+d)^2-2/9*ln(c)*b^2*g^
2*n*x^3+ln(c)^2*b^2*f*g*x^2+2/3*ln(c)*a*b*g^2*x^3-2*b^2*n*ln(c)*f^2*x+2*ln(c)*a*b*f^2*x-1/4*Pi^2*b^2*f^2*x*csg
n(I*c*(e*x+d)^n)^6+1/6*g^2*Pi^2*b^2*x^3*csgn(I*c)*csgn(I*(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)^3-1/4*g*Pi^2*b^2*f*x
^2*csgn(I*(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)^4+1/2*g*Pi^2*b^2*f*x^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^5-1/12
*g^2*Pi^2*b^2*x^3*csgn(I*c)^2*csgn(I*(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)^2+1/2*g*Pi^2*b^2*f*x^2*csgn(I*c)*csgn(I*
c*(e*x+d)^n)^5-1/4*g*Pi^2*b^2*f*x^2*csgn(I*c)^2*csgn(I*c*(e*x+d)^n)^4-1/3*g^2*Pi^2*b^2*x^3*csgn(I*c)*csgn(I*(e
*x+d)^n)*csgn(I*c*(e*x+d)^n)^4+1/2*Pi^2*b^2*f^2*x*csgn(I*c)^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^3+1/2*Pi^2
*b^2*f^2*x*csgn(I*c)*csgn(I*(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)^3-Pi^2*b^2*f^2*x*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn
(I*c*(e*x+d)^n)^4+I*b^2*n*Pi*f^2*x*csgn(I*c*(e*x+d)^n)^3+2/3/e^3*g^2*ln(c)*ln(e*x+d)*b^2*d^3*n+2/3/e^3*g^2*ln(
e*x+d)*a*b*d^3*n+1/e^2*g*b^2*d^2*f*n^2*ln(e*x+d)^2+1/3/e*g^2*ln(c)*b^2*d*n*x^2-2/3/e^2*g^2*ln(c)*b^2*d^2*n*x+3
/e^2*g*ln(e*x+d)*b^2*d^2*f*n^2-1/4*Pi^2*b^2*f^2*x*csgn(I*c)^2*csgn(I*(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)^2+1/9*I*
g^2*b^2*n*Pi*x^3*csgn(I*c*(e*x+d)^n)^3-I*ln(c)*Pi*b^2*f^2*x*csgn(I*c*(e*x+d)^n)^3-I*Pi*a*b*f^2*x*csgn(I*c*(e*x
+d)^n)^3-1/3*I*g^2*ln(c)*Pi*b^2*x^3*csgn(I*c*(e*x+d)^n)^3-1/3*I*g^2*Pi*a*b*x^3*csgn(I*c*(e*x+d)^n)^3+I/e^2*g*P
i*ln(e*x+d)*b^2*d^2*f*n*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-I/e*g*Pi*b^2*d*f*n*x*csgn(I*c)*csgn(I*
(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-5/18/e*g^2*b^2*d*n^2*x^2+11/9/e^2*g^2*b^2*d^2*n^2*x-11/9/e^3*g^2*ln(e*x+d)*b^2*
d^3*n^2-1/3/e^3*g^2*b^2*d^3*n^2*ln(e*x+d)^2+1/2*Pi^2*b^2*f^2*x*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^5-1/4*Pi^
2*b^2*f^2*x*csgn(I*c)^2*csgn(I*c*(e*x+d)^n)^4+1/2*Pi^2*b^2*f^2*x*csgn(I*c)*csgn(I*c*(e*x+d)^n)^5-2/e^2*g*ln(c)
*ln(e*x+d)*b^2*d^2*f*n-2/e^2*g*ln(e*x+d)*a*b*d^2*f*n+2/e*g*ln(c)*b^2*d*f*n*x-g*Pi^2*b^2*f*x^2*csgn(I*c)*csgn(I
*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^4-1/4*g*Pi^2*b^2*f*x^2*csgn(I*c)^2*csgn(I*(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)^2+1
/2*g*Pi^2*b^2*f*x^2*csgn(I*c)^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^3+1/2*g*Pi^2*b^2*f*x^2*csgn(I*c)*csgn(I*
(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)^3-2/9*a*b*g^2*n*x^3+1/2*b^2*f*g*n^2*x^2-2*a*b*f^2*n*x-a*b*f*g*n*x^2-1/4*Pi^2*
b^2*f^2*x*csgn(I*(e*x+d)^n)^2*csgn(I*c*(e*x+d)^n)^4-ln(c)*b^2*f*g*n*x^2+2*ln(c)*a*b*f*g*x^2-1/4*g*Pi^2*b^2*f*x
^2*csgn(I*c*(e*x+d)^n)^6+1/6*g^2*Pi^2*b^2*x^3*csgn(I*c)*csgn(I*c*(e*x+d)^n)^5-1/12*g^2*Pi^2*b^2*x^3*csgn(I*c)^
2*csgn(I*c*(e*x+d)^n)^4-1/e*b^2*d*f^2*n^2*ln(e*x+d)^2-1/12*g^2*Pi^2*b^2*x^3*csgn(I*(e*x+d)^n)^2*csgn(I*c*(e*x+
d)^n)^4+1/6*g^2*Pi^2*b^2*x^3*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^5-2/e*ln(e*x+d)*b^2*d*f^2*n^2+I*g*ln(c)*Pi*
b^2*f*x^2*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+I*g*ln(c)*Pi*b^2*f*x^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2+I*g*P
i*a*b*f*x^2*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+I*g*Pi*a*b*f*x^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2+I*b^2*n*P
i*f^2*x*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-1/6*I/e*g^2*Pi*b^2*d*n*x^2*csgn(I*c*(e*x+d)^n)^3+I*Pi*
a*b*f^2*x*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+I*Pi*a*b*f^2*x*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2+I*ln(c)*Pi*b^
2*f^2*x*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+I*ln(c)*Pi*b^2*f^2*x*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2+1/3*I*g^2
*Pi*a*b*x^3*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+1/3*I*g^2*ln(c)*Pi*b^2*x^3*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2-I*g*Pi*
a*b*f*x^2*csgn(I*c*(e*x+d)^n)^3+1/3*I*g^2*Pi*a*b*x^3*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*g*ln(c)*Pi*b^2*
f*x^2*csgn(I*c*(e*x+d)^n)^3+1/3*I*g^2*ln(c)*Pi*b^2*x^3*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-1/9*I*g^2*b^2*n
*Pi*x^3*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2-1/9*I*g^2*b^2*n*Pi*x^3*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2+1/2*I*g
*b^2*n*Pi*f*x^2*csgn(I*c*(e*x+d)^n)^3-I*b^2*n*Pi*f^2*x*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2-I*b^2*n*Pi*f^2*x*csgn(I
*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-2/3/e^2*g^2*a*b*d^2*n*x+1/3/e*g^2*a*b*d*n*x^2-3/e*g*b^2*d*f*n^2*x+2/e*ln(c)*
ln(e*x+d)*b^2*d*f^2*n+2/e*ln(e*x+d)*a*b*d*f^2*n+1/6*g^2*Pi^2*b^2*x^3*csgn(I*c)^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e
*x+d)^n)^3-I/e*Pi*ln(e*x+d)*b^2*d*f^2*n*csgn(I*c*(e*x+d)^n)^3+1/9*I*g^2*b^2*n*Pi*x^3*csgn(I*c)*csgn(I*(e*x+d)^
n)*csgn(I*c*(e*x+d)^n)-1/2*I*g*b^2*n*Pi*f*x^2*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2-1/2*I*g*b^2*n*Pi*f*x^2*csgn(I*(e
*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*Pi*a*b*f^2*x*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-I*ln(c)*Pi*b^2*f
^2*x*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-1/3*I/e^3*g^2*Pi*ln(e*x+d)*b^2*d^3*n*csgn(I*c*(e*x+d)^n)^
3+1/3*I/e^2*g^2*Pi*b^2*d^2*n*x*csgn(I*c*(e*x+d)^n)^3-1/3*I*g^2*ln(c)*Pi*b^2*x^3*csgn(I*c)*csgn(I*(e*x+d)^n)*cs
gn(I*c*(e*x+d)^n)-1/3*I*g^2*Pi*a*b*x^3*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+I/e^2*g*Pi*ln(e*x+d)*b^
2*d^2*f*n*csgn(I*c*(e*x+d)^n)^3+I/e*Pi*ln(e*x+d)*b^2*d*f^2*n*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+I/e*Pi*ln(e*x+d)*
b^2*d*f^2*n*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2+1/6*I/e*g^2*Pi*b^2*d*n*x^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x
+d)^n)^2+1/3*I/e^3*g^2*Pi*ln(e*x+d)*b^2*d^3*n*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+1/3*I/e^3*g^2*Pi*ln(e*x+d)*b^2*d
^3*n*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-1/3*I/e^2*g^2*Pi*b^2*d^2*n*x*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2-1/3*
I/e^2*g^2*Pi*b^2*d^2*n*x*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2+1/6*I/e*g^2*Pi*b^2*d*n*x^2*csgn(I*c)*csgn(I*c
*(e*x+d)^n)^2-I/e*g*Pi*b^2*d*f*n*x*csgn(I*c*(e*x+d)^n)^3+1/2*I*g*b^2*n*Pi*f*x^2*csgn(I*c)*csgn(I*(e*x+d)^n)*cs
gn(I*c*(e*x+d)^n)-I*g*ln(c)*Pi*b^2*f*x^2*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-I*g*Pi*a*b*f*x^2*csgn
(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+I/e*g*Pi*b^2*d*f*n*x*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I/e*P
i*ln(e*x+d)*b^2*d*f^2*n*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-1/3*I/e^3*g^2*Pi*ln(e*x+d)*b^2*d^3*n*c
sgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+1/3*I/e^2*g^2*Pi*b^2*d^2*n*x*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I
*c*(e*x+d)^n)-I/e^2*g*Pi*ln(e*x+d)*b^2*d^2*f*n*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-1/6*I/e*g^2*Pi*b^2*d*n*
x^2*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-I/e^2*g*Pi*ln(e*x+d)*b^2*d^2*f*n*csgn(I*c)*csgn(I*c*(e*x+d
)^n)^2+I/e*g*Pi*b^2*d*f*n*x*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2
________________________________________________________________________________________
Maxima [B] time = 1.20479, size = 748, normalized size = 2.61 \begin{align*} \frac{1}{3} \, b^{2} g^{2} x^{3} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + \frac{2}{3} \, a b g^{2} x^{3} \log \left ({\left (e x + d\right )}^{n} c\right ) + b^{2} f g x^{2} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + \frac{1}{3} \, a^{2} g^{2} x^{3} - 2 \, a b e f^{2} n{\left (\frac{x}{e} - \frac{d \log \left (e x + d\right )}{e^{2}}\right )} + \frac{1}{9} \, a b e g^{2} n{\left (\frac{6 \, d^{3} \log \left (e x + d\right )}{e^{4}} - \frac{2 \, e^{2} x^{3} - 3 \, d e x^{2} + 6 \, d^{2} x}{e^{3}}\right )} - a b e f g n{\left (\frac{2 \, d^{2} \log \left (e x + d\right )}{e^{3}} + \frac{e x^{2} - 2 \, d x}{e^{2}}\right )} + 2 \, a b f g x^{2} \log \left ({\left (e x + d\right )}^{n} c\right ) + b^{2} f^{2} x \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + a^{2} f g x^{2} + 2 \, a b f^{2} x \log \left ({\left (e x + d\right )}^{n} c\right ) -{\left (2 \, e n{\left (\frac{x}{e} - \frac{d \log \left (e x + d\right )}{e^{2}}\right )} \log \left ({\left (e x + d\right )}^{n} c\right ) + \frac{{\left (d \log \left (e x + d\right )^{2} - 2 \, e x + 2 \, d \log \left (e x + d\right )\right )} n^{2}}{e}\right )} b^{2} f^{2} - \frac{1}{2} \,{\left (2 \, e n{\left (\frac{2 \, d^{2} \log \left (e x + d\right )}{e^{3}} + \frac{e x^{2} - 2 \, d x}{e^{2}}\right )} \log \left ({\left (e x + d\right )}^{n} c\right ) - \frac{{\left (e^{2} x^{2} + 2 \, d^{2} \log \left (e x + d\right )^{2} - 6 \, d e x + 6 \, d^{2} \log \left (e x + d\right )\right )} n^{2}}{e^{2}}\right )} b^{2} f g + \frac{1}{54} \,{\left (6 \, e n{\left (\frac{6 \, d^{3} \log \left (e x + d\right )}{e^{4}} - \frac{2 \, e^{2} x^{3} - 3 \, d e x^{2} + 6 \, d^{2} x}{e^{3}}\right )} \log \left ({\left (e x + d\right )}^{n} c\right ) + \frac{{\left (4 \, e^{3} x^{3} - 15 \, d e^{2} x^{2} - 18 \, d^{3} \log \left (e x + d\right )^{2} + 66 \, d^{2} e x - 66 \, d^{3} \log \left (e x + d\right )\right )} n^{2}}{e^{3}}\right )} b^{2} g^{2} + a^{2} f^{2} x \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))^2,x, algorithm="maxima")
[Out]
1/3*b^2*g^2*x^3*log((e*x + d)^n*c)^2 + 2/3*a*b*g^2*x^3*log((e*x + d)^n*c) + b^2*f*g*x^2*log((e*x + d)^n*c)^2 +
1/3*a^2*g^2*x^3 - 2*a*b*e*f^2*n*(x/e - d*log(e*x + d)/e^2) + 1/9*a*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) - a*b*e*f*g*n*(2*d^2*log(e*x + d)/e^3 + (e*x^2 - 2*d*x)/e^2) + 2*a*b*f*g*x^2*
log((e*x + d)^n*c) + b^2*f^2*x*log((e*x + d)^n*c)^2 + a^2*f*g*x^2 + 2*a*b*f^2*x*log((e*x + d)^n*c) - (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)*b^2*f^2 - 1
/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)*b^2*f*g + 1/54*(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)*b^2*g^2 + a^2*f^2*x
________________________________________________________________________________________
Fricas [B] time = 2.28045, size = 1571, normalized size = 5.47 \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))^2,x, algorithm="fricas")
[Out]
1/54*(2*(2*b^2*e^3*g^2*n^2 - 6*a*b*e^3*g^2*n + 9*a^2*e^3*g^2)*x^3 + 3*(18*a^2*e^3*f*g + (9*b^2*e^3*f*g - 5*b^2
*d*e^2*g^2)*n^2 - 6*(3*a*b*e^3*f*g - a*b*d*e^2*g^2)*n)*x^2 + 18*(b^2*e^3*g^2*n^2*x^3 + 3*b^2*e^3*f*g*n^2*x^2 +
3*b^2*e^3*f^2*n^2*x + (3*b^2*d*e^2*f^2 - 3*b^2*d^2*e*f*g + b^2*d^3*g^2)*n^2)*log(e*x + d)^2 + 18*(b^2*e^3*g^2
*x^3 + 3*b^2*e^3*f*g*x^2 + 3*b^2*e^3*f^2*x)*log(c)^2 + 6*(9*a^2*e^3*f^2 + (18*b^2*e^3*f^2 - 27*b^2*d*e^2*f*g +
11*b^2*d^2*e*g^2)*n^2 - 6*(3*a*b*e^3*f^2 - 3*a*b*d*e^2*f*g + a*b*d^2*e*g^2)*n)*x - 6*(2*(b^2*e^3*g^2*n^2 - 3*
a*b*e^3*g^2*n)*x^3 + (18*b^2*d*e^2*f^2 - 27*b^2*d^2*e*f*g + 11*b^2*d^3*g^2)*n^2 - 3*(6*a*b*e^3*f*g*n - (3*b^2*
e^3*f*g - b^2*d*e^2*g^2)*n^2)*x^2 - 6*(3*a*b*d*e^2*f^2 - 3*a*b*d^2*e*f*g + a*b*d^3*g^2)*n - 6*(3*a*b*e^3*f^2*n
- (3*b^2*e^3*f^2 - 3*b^2*d*e^2*f*g + b^2*d^2*e*g^2)*n^2)*x - 6*(b^2*e^3*g^2*n*x^3 + 3*b^2*e^3*f*g*n*x^2 + 3*b
^2*e^3*f^2*n*x + (3*b^2*d*e^2*f^2 - 3*b^2*d^2*e*f*g + b^2*d^3*g^2)*n)*log(c))*log(e*x + d) - 6*(2*(b^2*e^3*g^2
*n - 3*a*b*e^3*g^2)*x^3 - 3*(6*a*b*e^3*f*g - (3*b^2*e^3*f*g - b^2*d*e^2*g^2)*n)*x^2 - 6*(3*a*b*e^3*f^2 - (3*b^
2*e^3*f^2 - 3*b^2*d*e^2*f*g + b^2*d^2*e*g^2)*n)*x)*log(c))/e^3
________________________________________________________________________________________
Sympy [A] time = 9.54133, size = 1103, normalized size = 3.84 \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))**2,x)
[Out]
Piecewise((a**2*f**2*x + a**2*f*g*x**2 + a**2*g**2*x**3/3 + 2*a*b*d**3*g**2*n*log(d + e*x)/(3*e**3) - 2*a*b*d*
*2*f*g*n*log(d + e*x)/e**2 - 2*a*b*d**2*g**2*n*x/(3*e**2) + 2*a*b*d*f**2*n*log(d + e*x)/e + 2*a*b*d*f*g*n*x/e
+ a*b*d*g**2*n*x**2/(3*e) + 2*a*b*f**2*n*x*log(d + e*x) - 2*a*b*f**2*n*x + 2*a*b*f**2*x*log(c) + 2*a*b*f*g*n*x
**2*log(d + e*x) - a*b*f*g*n*x**2 + 2*a*b*f*g*x**2*log(c) + 2*a*b*g**2*n*x**3*log(d + e*x)/3 - 2*a*b*g**2*n*x*
*3/9 + 2*a*b*g**2*x**3*log(c)/3 + b**2*d**3*g**2*n**2*log(d + e*x)**2/(3*e**3) - 11*b**2*d**3*g**2*n**2*log(d
+ e*x)/(9*e**3) + 2*b**2*d**3*g**2*n*log(c)*log(d + e*x)/(3*e**3) - b**2*d**2*f*g*n**2*log(d + e*x)**2/e**2 +
3*b**2*d**2*f*g*n**2*log(d + e*x)/e**2 - 2*b**2*d**2*f*g*n*log(c)*log(d + e*x)/e**2 - 2*b**2*d**2*g**2*n**2*x*
log(d + e*x)/(3*e**2) + 11*b**2*d**2*g**2*n**2*x/(9*e**2) - 2*b**2*d**2*g**2*n*x*log(c)/(3*e**2) + b**2*d*f**2
*n**2*log(d + e*x)**2/e - 2*b**2*d*f**2*n**2*log(d + e*x)/e + 2*b**2*d*f**2*n*log(c)*log(d + e*x)/e + 2*b**2*d
*f*g*n**2*x*log(d + e*x)/e - 3*b**2*d*f*g*n**2*x/e + 2*b**2*d*f*g*n*x*log(c)/e + b**2*d*g**2*n**2*x**2*log(d +
e*x)/(3*e) - 5*b**2*d*g**2*n**2*x**2/(18*e) + b**2*d*g**2*n*x**2*log(c)/(3*e) + b**2*f**2*n**2*x*log(d + e*x)
**2 - 2*b**2*f**2*n**2*x*log(d + e*x) + 2*b**2*f**2*n**2*x + 2*b**2*f**2*n*x*log(c)*log(d + e*x) - 2*b**2*f**2
*n*x*log(c) + b**2*f**2*x*log(c)**2 + b**2*f*g*n**2*x**2*log(d + e*x)**2 - b**2*f*g*n**2*x**2*log(d + e*x) + b
**2*f*g*n**2*x**2/2 + 2*b**2*f*g*n*x**2*log(c)*log(d + e*x) - b**2*f*g*n*x**2*log(c) + b**2*f*g*x**2*log(c)**2
+ b**2*g**2*n**2*x**3*log(d + e*x)**2/3 - 2*b**2*g**2*n**2*x**3*log(d + e*x)/9 + 2*b**2*g**2*n**2*x**3/27 + 2
*b**2*g**2*n*x**3*log(c)*log(d + e*x)/3 - 2*b**2*g**2*n*x**3*log(c)/9 + b**2*g**2*x**3*log(c)**2/3, Ne(e, 0)),
((a + b*log(c*d**n))**2*(f**2*x + f*g*x**2 + g**2*x**3/3), True))
________________________________________________________________________________________
Giac [B] time = 1.37615, size = 1808, normalized size = 6.3 \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))^2,x, algorithm="giac")
[Out]
1/3*(x*e + d)^3*b^2*g^2*n^2*e^(-3)*log(x*e + d)^2 - (x*e + d)^2*b^2*d*g^2*n^2*e^(-3)*log(x*e + d)^2 + (x*e + d
)*b^2*d^2*g^2*n^2*e^(-3)*log(x*e + d)^2 - 2/9*(x*e + d)^3*b^2*g^2*n^2*e^(-3)*log(x*e + d) + (x*e + d)^2*b^2*d*
g^2*n^2*e^(-3)*log(x*e + d) - 2*(x*e + d)*b^2*d^2*g^2*n^2*e^(-3)*log(x*e + d) + (x*e + d)^2*b^2*f*g*n^2*e^(-2)
*log(x*e + d)^2 - 2*(x*e + d)*b^2*d*f*g*n^2*e^(-2)*log(x*e + d)^2 + 2/3*(x*e + d)^3*b^2*g^2*n*e^(-3)*log(x*e +
d)*log(c) - 2*(x*e + d)^2*b^2*d*g^2*n*e^(-3)*log(x*e + d)*log(c) + 2*(x*e + d)*b^2*d^2*g^2*n*e^(-3)*log(x*e +
d)*log(c) + 2/27*(x*e + d)^3*b^2*g^2*n^2*e^(-3) - 1/2*(x*e + d)^2*b^2*d*g^2*n^2*e^(-3) + 2*(x*e + d)*b^2*d^2*
g^2*n^2*e^(-3) - (x*e + d)^2*b^2*f*g*n^2*e^(-2)*log(x*e + d) + 4*(x*e + d)*b^2*d*f*g*n^2*e^(-2)*log(x*e + d) +
2/3*(x*e + d)^3*a*b*g^2*n*e^(-3)*log(x*e + d) - 2*(x*e + d)^2*a*b*d*g^2*n*e^(-3)*log(x*e + d) + 2*(x*e + d)*a
*b*d^2*g^2*n*e^(-3)*log(x*e + d) + (x*e + d)*b^2*f^2*n^2*e^(-1)*log(x*e + d)^2 - 2/9*(x*e + d)^3*b^2*g^2*n*e^(
-3)*log(c) + (x*e + d)^2*b^2*d*g^2*n*e^(-3)*log(c) - 2*(x*e + d)*b^2*d^2*g^2*n*e^(-3)*log(c) + 2*(x*e + d)^2*b
^2*f*g*n*e^(-2)*log(x*e + d)*log(c) - 4*(x*e + d)*b^2*d*f*g*n*e^(-2)*log(x*e + d)*log(c) + 1/3*(x*e + d)^3*b^2
*g^2*e^(-3)*log(c)^2 - (x*e + d)^2*b^2*d*g^2*e^(-3)*log(c)^2 + (x*e + d)*b^2*d^2*g^2*e^(-3)*log(c)^2 + 1/2*(x*
e + d)^2*b^2*f*g*n^2*e^(-2) - 4*(x*e + d)*b^2*d*f*g*n^2*e^(-2) - 2/9*(x*e + d)^3*a*b*g^2*n*e^(-3) + (x*e + d)^
2*a*b*d*g^2*n*e^(-3) - 2*(x*e + d)*a*b*d^2*g^2*n*e^(-3) - 2*(x*e + d)*b^2*f^2*n^2*e^(-1)*log(x*e + d) + 2*(x*e
+ d)^2*a*b*f*g*n*e^(-2)*log(x*e + d) - 4*(x*e + d)*a*b*d*f*g*n*e^(-2)*log(x*e + d) - (x*e + d)^2*b^2*f*g*n*e^
(-2)*log(c) + 4*(x*e + d)*b^2*d*f*g*n*e^(-2)*log(c) + 2/3*(x*e + d)^3*a*b*g^2*e^(-3)*log(c) - 2*(x*e + d)^2*a*
b*d*g^2*e^(-3)*log(c) + 2*(x*e + d)*a*b*d^2*g^2*e^(-3)*log(c) + 2*(x*e + d)*b^2*f^2*n*e^(-1)*log(x*e + d)*log(
c) + (x*e + d)^2*b^2*f*g*e^(-2)*log(c)^2 - 2*(x*e + d)*b^2*d*f*g*e^(-2)*log(c)^2 + 2*(x*e + d)*b^2*f^2*n^2*e^(
-1) - (x*e + d)^2*a*b*f*g*n*e^(-2) + 4*(x*e + d)*a*b*d*f*g*n*e^(-2) + 1/3*(x*e + d)^3*a^2*g^2*e^(-3) - (x*e +
d)^2*a^2*d*g^2*e^(-3) + (x*e + d)*a^2*d^2*g^2*e^(-3) + 2*(x*e + d)*a*b*f^2*n*e^(-1)*log(x*e + d) - 2*(x*e + d)
*b^2*f^2*n*e^(-1)*log(c) + 2*(x*e + d)^2*a*b*f*g*e^(-2)*log(c) - 4*(x*e + d)*a*b*d*f*g*e^(-2)*log(c) + (x*e +
d)*b^2*f^2*e^(-1)*log(c)^2 - 2*(x*e + d)*a*b*f^2*n*e^(-1) + (x*e + d)^2*a^2*f*g*e^(-2) - 2*(x*e + d)*a^2*d*f*g
*e^(-2) + 2*(x*e + d)*a*b*f^2*e^(-1)*log(c) + (x*e + d)*a^2*f^2*e^(-1)