\(\int x (a+b \log (c x^n))^3 \log (d (e+f x)^m) \, dx\) [91]

Optimal result

Integrand size = 24, antiderivative size = 603 \[ \int x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right ) \, dx=\frac {21 a b^2 e m n^2 x}{4 f}-\frac {45 b^3 e m n^3 x}{8 f}+\frac {3}{4} b^3 m n^3 x^2+\frac {21 b^3 e m n^2 x \log \left (c x^n\right )}{4 f}-\frac {9}{8} b^2 m n^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {9 b e m n x \left (a+b \log \left (c x^n\right )\right )^2}{4 f}+\frac {3}{4} b m n x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac {e m x \left (a+b \log \left (c x^n\right )\right )^3}{2 f}-\frac {1}{4} m x^2 \left (a+b \log \left (c x^n\right )\right )^3+\frac {3 b^3 e^2 m n^3 \log (e+f x)}{8 f^2}-\frac {3}{8} b^3 n^3 x^2 \log \left (d (e+f x)^m\right )+\frac {3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-\frac {3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )-\frac {3 b^2 e^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x}{e}\right )}{4 f^2}+\frac {3 b e^2 m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x}{e}\right )}{4 f^2}-\frac {e^2 m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {f x}{e}\right )}{2 f^2}-\frac {3 b^3 e^2 m n^3 \operatorname {PolyLog}\left (2,-\frac {f x}{e}\right )}{4 f^2}+\frac {3 b^2 e^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {f x}{e}\right )}{2 f^2}-\frac {3 b e^2 m n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {f x}{e}\right )}{2 f^2}-\frac {3 b^3 e^2 m n^3 \operatorname {PolyLog}\left (3,-\frac {f x}{e}\right )}{2 f^2}+\frac {3 b^2 e^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {f x}{e}\right )}{f^2}-\frac {3 b^3 e^2 m n^3 \operatorname {PolyLog}\left (4,-\frac {f x}{e}\right )}{f^2} \] Output:

21/4*a*b^2*e*m*n^2*x/f-45/8*b^3*e*m*n^3*x/f+3/4*b^3*m*n^3*x^2+21/4*b^3*e*m 
*n^2*x*ln(c*x^n)/f-9/8*b^2*m*n^2*x^2*(a+b*ln(c*x^n))-9/4*b*e*m*n*x*(a+b*ln 
(c*x^n))^2/f+3/4*b*m*n*x^2*(a+b*ln(c*x^n))^2+1/2*e*m*x*(a+b*ln(c*x^n))^3/f 
-1/4*m*x^2*(a+b*ln(c*x^n))^3+3/8*b^3*e^2*m*n^3*ln(f*x+e)/f^2-3/8*b^3*n^3*x 
^2*ln(d*(f*x+e)^m)+3/4*b^2*n^2*x^2*(a+b*ln(c*x^n))*ln(d*(f*x+e)^m)-3/4*b*n 
*x^2*(a+b*ln(c*x^n))^2*ln(d*(f*x+e)^m)+1/2*x^2*(a+b*ln(c*x^n))^3*ln(d*(f*x 
+e)^m)-3/4*b^2*e^2*m*n^2*(a+b*ln(c*x^n))*ln(1+f*x/e)/f^2+3/4*b*e^2*m*n*(a+ 
b*ln(c*x^n))^2*ln(1+f*x/e)/f^2-1/2*e^2*m*(a+b*ln(c*x^n))^3*ln(1+f*x/e)/f^2 
-3/4*b^3*e^2*m*n^3*polylog(2,-f*x/e)/f^2+3/2*b^2*e^2*m*n^2*(a+b*ln(c*x^n)) 
*polylog(2,-f*x/e)/f^2-3/2*b*e^2*m*n*(a+b*ln(c*x^n))^2*polylog(2,-f*x/e)/f 
^2-3/2*b^3*e^2*m*n^3*polylog(3,-f*x/e)/f^2+3*b^2*e^2*m*n^2*(a+b*ln(c*x^n)) 
*polylog(3,-f*x/e)/f^2-3*b^3*e^2*m*n^3*polylog(4,-f*x/e)/f^2
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1431\) vs. \(2(603)=1206\).

Time = 0.66 (sec) , antiderivative size = 1431, normalized size of antiderivative = 2.37 \[ \int x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

(4*a^3*e*f*m*x - 18*a^2*b*e*f*m*n*x + 42*a*b^2*e*f*m*n^2*x - 45*b^3*e*f*m* 
n^3*x - 2*a^3*f^2*m*x^2 + 6*a^2*b*f^2*m*n*x^2 - 9*a*b^2*f^2*m*n^2*x^2 + 6* 
b^3*f^2*m*n^3*x^2 + 12*a^2*b*e*f*m*x*Log[c*x^n] - 36*a*b^2*e*f*m*n*x*Log[c 
*x^n] + 42*b^3*e*f*m*n^2*x*Log[c*x^n] - 6*a^2*b*f^2*m*x^2*Log[c*x^n] + 12* 
a*b^2*f^2*m*n*x^2*Log[c*x^n] - 9*b^3*f^2*m*n^2*x^2*Log[c*x^n] + 12*a*b^2*e 
*f*m*x*Log[c*x^n]^2 - 18*b^3*e*f*m*n*x*Log[c*x^n]^2 - 6*a*b^2*f^2*m*x^2*Lo 
g[c*x^n]^2 + 6*b^3*f^2*m*n*x^2*Log[c*x^n]^2 + 4*b^3*e*f*m*x*Log[c*x^n]^3 - 
 2*b^3*f^2*m*x^2*Log[c*x^n]^3 - 4*a^3*e^2*m*Log[e + f*x] + 6*a^2*b*e^2*m*n 
*Log[e + f*x] - 6*a*b^2*e^2*m*n^2*Log[e + f*x] + 3*b^3*e^2*m*n^3*Log[e + f 
*x] + 12*a^2*b*e^2*m*n*Log[x]*Log[e + f*x] - 12*a*b^2*e^2*m*n^2*Log[x]*Log 
[e + f*x] + 6*b^3*e^2*m*n^3*Log[x]*Log[e + f*x] - 12*a*b^2*e^2*m*n^2*Log[x 
]^2*Log[e + f*x] + 6*b^3*e^2*m*n^3*Log[x]^2*Log[e + f*x] + 4*b^3*e^2*m*n^3 
*Log[x]^3*Log[e + f*x] - 12*a^2*b*e^2*m*Log[c*x^n]*Log[e + f*x] + 12*a*b^2 
*e^2*m*n*Log[c*x^n]*Log[e + f*x] - 6*b^3*e^2*m*n^2*Log[c*x^n]*Log[e + f*x] 
 + 24*a*b^2*e^2*m*n*Log[x]*Log[c*x^n]*Log[e + f*x] - 12*b^3*e^2*m*n^2*Log[ 
x]*Log[c*x^n]*Log[e + f*x] - 12*b^3*e^2*m*n^2*Log[x]^2*Log[c*x^n]*Log[e + 
f*x] - 12*a*b^2*e^2*m*Log[c*x^n]^2*Log[e + f*x] + 6*b^3*e^2*m*n*Log[c*x^n] 
^2*Log[e + f*x] + 12*b^3*e^2*m*n*Log[x]*Log[c*x^n]^2*Log[e + f*x] - 4*b^3* 
e^2*m*Log[c*x^n]^3*Log[e + f*x] + 4*a^3*f^2*x^2*Log[d*(e + f*x)^m] - 6*a^2 
*b*f^2*n*x^2*Log[d*(e + f*x)^m] + 6*a*b^2*f^2*n^2*x^2*Log[d*(e + f*x)^m...
 

Rubi [A] (verified)

Time = 1.30 (sec) , antiderivative size = 601, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2825, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right ) \, dx\)

\(\Big \downarrow \) 2825

\(\displaystyle -f m \int \left (-\frac {3 b^3 x^2 n^3}{8 (e+f x)}+\frac {3 b^2 x^2 \left (a+b \log \left (c x^n\right )\right ) n^2}{4 (e+f x)}-\frac {3 b x^2 \left (a+b \log \left (c x^n\right )\right )^2 n}{4 (e+f x)}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{2 (e+f x)}\right )dx+\frac {3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-\frac {3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )-\frac {3}{8} b^3 n^3 x^2 \log \left (d (e+f x)^m\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3}{4} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-f m \left (-\frac {3 b^2 e^2 n^2 \operatorname {PolyLog}\left (2,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^3}-\frac {3 b^2 e^2 n^2 \operatorname {PolyLog}\left (3,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^3}+\frac {3 b^2 e^2 n^2 \log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{4 f^3}+\frac {9 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{8 f}-\frac {21 a b^2 e n^2 x}{4 f^2}+\frac {3 b e^2 n \operatorname {PolyLog}\left (2,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 f^3}-\frac {3 b e^2 n \log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 f^3}+\frac {e^2 \log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 f^3}+\frac {9 b e n x \left (a+b \log \left (c x^n\right )\right )^2}{4 f^2}-\frac {e x \left (a+b \log \left (c x^n\right )\right )^3}{2 f^2}-\frac {3 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 f}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 f}-\frac {21 b^3 e n^2 x \log \left (c x^n\right )}{4 f^2}+\frac {3 b^3 e^2 n^3 \operatorname {PolyLog}\left (2,-\frac {f x}{e}\right )}{4 f^3}+\frac {3 b^3 e^2 n^3 \operatorname {PolyLog}\left (3,-\frac {f x}{e}\right )}{2 f^3}+\frac {3 b^3 e^2 n^3 \operatorname {PolyLog}\left (4,-\frac {f x}{e}\right )}{f^3}-\frac {3 b^3 e^2 n^3 \log (e+f x)}{8 f^3}+\frac {45 b^3 e n^3 x}{8 f^2}-\frac {3 b^3 n^3 x^2}{4 f}\right )-\frac {3}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d (e+f x)^m\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )-\frac {3}{8} b^3 n^3 x^2 \log \left (d (e+f x)^m\right )\)

Input:

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

Output:

(-3*b^3*n^3*x^2*Log[d*(e + f*x)^m])/8 + (3*b^2*n^2*x^2*(a + b*Log[c*x^n])* 
Log[d*(e + f*x)^m])/4 - (3*b*n*x^2*(a + b*Log[c*x^n])^2*Log[d*(e + f*x)^m] 
)/4 + (x^2*(a + b*Log[c*x^n])^3*Log[d*(e + f*x)^m])/2 - f*m*((-21*a*b^2*e* 
n^2*x)/(4*f^2) + (45*b^3*e*n^3*x)/(8*f^2) - (3*b^3*n^3*x^2)/(4*f) - (21*b^ 
3*e*n^2*x*Log[c*x^n])/(4*f^2) + (9*b^2*n^2*x^2*(a + b*Log[c*x^n]))/(8*f) + 
 (9*b*e*n*x*(a + b*Log[c*x^n])^2)/(4*f^2) - (3*b*n*x^2*(a + b*Log[c*x^n])^ 
2)/(4*f) - (e*x*(a + b*Log[c*x^n])^3)/(2*f^2) + (x^2*(a + b*Log[c*x^n])^3) 
/(4*f) - (3*b^3*e^2*n^3*Log[e + f*x])/(8*f^3) + (3*b^2*e^2*n^2*(a + b*Log[ 
c*x^n])*Log[1 + (f*x)/e])/(4*f^3) - (3*b*e^2*n*(a + b*Log[c*x^n])^2*Log[1 
+ (f*x)/e])/(4*f^3) + (e^2*(a + b*Log[c*x^n])^3*Log[1 + (f*x)/e])/(2*f^3) 
+ (3*b^3*e^2*n^3*PolyLog[2, -((f*x)/e)])/(4*f^3) - (3*b^2*e^2*n^2*(a + b*L 
og[c*x^n])*PolyLog[2, -((f*x)/e)])/(2*f^3) + (3*b*e^2*n*(a + b*Log[c*x^n]) 
^2*PolyLog[2, -((f*x)/e)])/(2*f^3) + (3*b^3*e^2*n^3*PolyLog[3, -((f*x)/e)] 
)/(2*f^3) - (3*b^2*e^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, -((f*x)/e)])/f^3 
+ (3*b^3*e^2*n^3*PolyLog[4, -((f*x)/e)])/f^3)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2825

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_. 
)]*(b_.))^(p_.)*((g_.)*(x_))^(q_.), x_Symbol] :> With[{u = IntHide[(g*x)^q* 
(a + b*Log[c*x^n])^p, x]}, Simp[Log[d*(e + f*x^m)^r]   u, x] - Simp[f*m*r 
 Int[x^(m - 1)/(e + f*x^m)   u, x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m 
, n, q}, x] && IGtQ[p, 0] && RationalQ[m] && RationalQ[q]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.20 (sec) , antiderivative size = 19601, normalized size of antiderivative = 32.51

\[\text {output too large to display}\]

Input:

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

Output:

result too large to display
 

Fricas [F]

\[ \int x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x \log \left ({\left (f x + e\right )}^{m} d\right ) \,d x } \] Input:

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

Output:

integral((b^3*x*log(c*x^n)^3 + 3*a*b^2*x*log(c*x^n)^2 + 3*a^2*b*x*log(c*x^ 
n) + a^3*x)*log((f*x + e)^m*d), x)
 

Sympy [F(-1)]

Timed out. \[ \int x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right ) \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

\[ \int x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x \log \left ({\left (f x + e\right )}^{m} d\right ) \,d x } \] Input:

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

Output:

1/8*(2*(2*b^3*e*f*m*x - 2*b^3*e^2*m*log(f*x + e) - (f^2*m - 2*f^2*log(d))* 
b^3*x^2)*log(x^n)^3 + (4*b^3*f^2*x^2*log(x^n)^3 + 6*(2*a*b^2*f^2 - (f^2*n 
- 2*f^2*log(c))*b^3)*x^2*log(x^n)^2 + 6*(2*a^2*b*f^2 - 2*(f^2*n - 2*f^2*lo 
g(c))*a*b^2 + (f^2*n^2 - 2*f^2*n*log(c) + 2*f^2*log(c)^2)*b^3)*x^2*log(x^n 
) + (4*a^3*f^2 - 6*(f^2*n - 2*f^2*log(c))*a^2*b + 6*(f^2*n^2 - 2*f^2*n*log 
(c) + 2*f^2*log(c)^2)*a*b^2 - (3*f^2*n^3 - 6*f^2*n^2*log(c) + 6*f^2*n*log( 
c)^2 - 4*f^2*log(c)^3)*b^3)*x^2)*log((f*x + e)^m))/f^2 + integrate(-1/8*(( 
4*(f^3*m - 2*f^3*log(d))*a^3 - 6*(f^3*m*n - 2*(f^3*m - 2*f^3*log(d))*log(c 
))*a^2*b + 6*(f^3*m*n^2 - 2*f^3*m*n*log(c) + 2*(f^3*m - 2*f^3*log(d))*log( 
c)^2)*a*b^2 - (3*f^3*m*n^3 - 6*f^3*m*n^2*log(c) + 6*f^3*m*n*log(c)^2 - 4*( 
f^3*m - 2*f^3*log(d))*log(c)^3)*b^3)*x^3 - 8*(b^3*e*f^2*log(c)^3*log(d) + 
3*a*b^2*e*f^2*log(c)^2*log(d) + 3*a^2*b*e*f^2*log(c)*log(d) + a^3*e*f^2*lo 
g(d))*x^2 + 6*(2*b^3*e^2*f*m*n*x + 2*((f^3*m - 2*f^3*log(d))*a*b^2 - (f^3* 
m*n - f^3*n*log(d) - (f^3*m - 2*f^3*log(d))*log(c))*b^3)*x^3 - (4*a*b^2*e* 
f^2*log(d) - (e*f^2*m*n + 2*e*f^2*n*log(d) - 4*e*f^2*log(c)*log(d))*b^3)*x 
^2 - 2*(b^3*e^2*f*m*n*x + b^3*e^3*m*n)*log(f*x + e))*log(x^n)^2 + 6*((2*(f 
^3*m - 2*f^3*log(d))*a^2*b - 2*(f^3*m*n - 2*(f^3*m - 2*f^3*log(d))*log(c)) 
*a*b^2 + (f^3*m*n^2 - 2*f^3*m*n*log(c) + 2*(f^3*m - 2*f^3*log(d))*log(c)^2 
)*b^3)*x^3 - 4*(b^3*e*f^2*log(c)^2*log(d) + 2*a*b^2*e*f^2*log(c)*log(d) + 
a^2*b*e*f^2*log(d))*x^2)*log(x^n))/(f^3*x^2 + e*f^2*x), x)
 

Giac [F]

\[ \int x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x \log \left ({\left (f x + e\right )}^{m} d\right ) \,d x } \] Input:

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

Output:

integrate((b*log(c*x^n) + a)^3*x*log((f*x + e)^m*d), x)
 

Mupad [F(-1)]

Timed out. \[ \int x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right ) \, dx=\int x\,\ln \left (d\,{\left (e+f\,x\right )}^m\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3 \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

(4*int(log(x**n*c)**3/(e*x + f*x**2),x)*b**3*e**3*m*n + 12*int(log(x**n*c) 
**2/(e*x + f*x**2),x)*a*b**2*e**3*m*n - 6*int(log(x**n*c)**2/(e*x + f*x**2 
),x)*b**3*e**3*m*n**2 + 12*int(log(x**n*c)/(e*x + f*x**2),x)*a**2*b*e**3*m 
*n - 12*int(log(x**n*c)/(e*x + f*x**2),x)*a*b**2*e**3*m*n**2 + 6*int(log(x 
**n*c)/(e*x + f*x**2),x)*b**3*e**3*m*n**3 + 4*log((e + f*x)**m*d)*log(x**n 
*c)**3*b**3*f**2*n*x**2 + 12*log((e + f*x)**m*d)*log(x**n*c)**2*a*b**2*f** 
2*n*x**2 - 6*log((e + f*x)**m*d)*log(x**n*c)**2*b**3*f**2*n**2*x**2 + 12*l 
og((e + f*x)**m*d)*log(x**n*c)*a**2*b*f**2*n*x**2 - 12*log((e + f*x)**m*d) 
*log(x**n*c)*a*b**2*f**2*n**2*x**2 + 6*log((e + f*x)**m*d)*log(x**n*c)*b** 
3*f**2*n**3*x**2 - 4*log((e + f*x)**m*d)*a**3*e**2*n + 4*log((e + f*x)**m* 
d)*a**3*f**2*n*x**2 + 6*log((e + f*x)**m*d)*a**2*b*e**2*n**2 - 6*log((e + 
f*x)**m*d)*a**2*b*f**2*n**2*x**2 - 6*log((e + f*x)**m*d)*a*b**2*e**2*n**3 
+ 6*log((e + f*x)**m*d)*a*b**2*f**2*n**3*x**2 + 3*log((e + f*x)**m*d)*b**3 
*e**2*n**4 - 3*log((e + f*x)**m*d)*b**3*f**2*n**4*x**2 - log(x**n*c)**4*b* 
*3*e**2*m - 4*log(x**n*c)**3*a*b**2*e**2*m + 2*log(x**n*c)**3*b**3*e**2*m* 
n + 4*log(x**n*c)**3*b**3*e*f*m*n*x - 2*log(x**n*c)**3*b**3*f**2*m*n*x**2 
- 6*log(x**n*c)**2*a**2*b*e**2*m + 6*log(x**n*c)**2*a*b**2*e**2*m*n + 12*l 
og(x**n*c)**2*a*b**2*e*f*m*n*x - 6*log(x**n*c)**2*a*b**2*f**2*m*n*x**2 - 3 
*log(x**n*c)**2*b**3*e**2*m*n**2 - 18*log(x**n*c)**2*b**3*e*f*m*n**2*x + 6 
*log(x**n*c)**2*b**3*f**2*m*n**2*x**2 + 12*log(x**n*c)*a**2*b*e*f*m*n*x...