\(\int \frac {(a+b \log (c x^n))^3 \log (d (e+f x^2)^m)}{x^4} \, dx\) [120]

Optimal result

Integrand size = 28, antiderivative size = 898 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{x^4} \, dx =\text {Too large to display} \] Output:

-160/27*b^3*f*m*n^3/e/x-4/27*b^3*f^(3/2)*m*n^3*arctan(f^(1/2)*x/e^(1/2))/e 
^(3/2)-52/9*b^2*f*m*n^2*(a+b*ln(c*x^n))/e/x-4/9*b^2*f^(3/2)*m*n^2*arctan(f 
^(1/2)*x/e^(1/2))*(a+b*ln(c*x^n))/e^(3/2)-8/3*b*f*m*n*(a+b*ln(c*x^n))^2/e/ 
x-2/3*b*f^(3/2)*m*n*arctan(f^(1/2)*x/e^(1/2))*(a+b*ln(c*x^n))^2/e^(3/2)-2/ 
3*f*m*(a+b*ln(c*x^n))^3/e/x-2/3*f^(3/2)*m*arctan(f^(1/2)*x/e^(1/2))*(a+b*l 
n(c*x^n))^3/e^(3/2)-2/27*b^3*n^3*ln(d*(f*x^2+e)^m)/x^3-2/9*b^2*n^2*(a+b*ln 
(c*x^n))*ln(d*(f*x^2+e)^m)/x^3-1/3*b*n*(a+b*ln(c*x^n))^2*ln(d*(f*x^2+e)^m) 
/x^3-1/3*(a+b*ln(c*x^n))^3*ln(d*(f*x^2+e)^m)/x^3-2/9*b^3*f^(3/2)*m*n^3*pol 
ylog(2,-f^(1/2)*x/(-e)^(1/2))/(-e)^(3/2)-2/3*b^2*f^(3/2)*m*n^2*(a+b*ln(c*x 
^n))*polylog(2,-f^(1/2)*x/(-e)^(1/2))/(-e)^(3/2)-b*f^(3/2)*m*n*(a+b*ln(c*x 
^n))^2*polylog(2,-f^(1/2)*x/(-e)^(1/2))/(-e)^(3/2)+2/9*b^3*f^(3/2)*m*n^3*p 
olylog(2,f^(1/2)*x/(-e)^(1/2))/(-e)^(3/2)+2/3*b^2*f^(3/2)*m*n^2*(a+b*ln(c* 
x^n))*polylog(2,f^(1/2)*x/(-e)^(1/2))/(-e)^(3/2)+b*f^(3/2)*m*n*(a+b*ln(c*x 
^n))^2*polylog(2,f^(1/2)*x/(-e)^(1/2))/(-e)^(3/2)+2/3*b^3*f^(3/2)*m*n^3*po 
lylog(3,-f^(1/2)*x/(-e)^(1/2))/(-e)^(3/2)+2*b^2*f^(3/2)*m*n^2*(a+b*ln(c*x^ 
n))*polylog(3,-f^(1/2)*x/(-e)^(1/2))/(-e)^(3/2)-2/3*b^3*f^(3/2)*m*n^3*poly 
log(3,f^(1/2)*x/(-e)^(1/2))/(-e)^(3/2)-2*b^2*f^(3/2)*m*n^2*(a+b*ln(c*x^n)) 
*polylog(3,f^(1/2)*x/(-e)^(1/2))/(-e)^(3/2)-2*b^3*f^(3/2)*m*n^3*polylog(4, 
-f^(1/2)*x/(-e)^(1/2))/(-e)^(3/2)+2*b^3*f^(3/2)*m*n^3*polylog(4,f^(1/2)*x/ 
(-e)^(1/2))/(-e)^(3/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.00 (sec) , antiderivative size = 2488, normalized size of antiderivative = 2.77 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{x^4} \, dx=\text {Result too large to show} \] Input:

Integrate[((a + b*Log[c*x^n])^3*Log[d*(e + f*x^2)^m])/x^4,x]
 

Output:

(-18*a^3*Sqrt[e]*f*m*x^2 - 72*a^2*b*Sqrt[e]*f*m*n*x^2 - 156*a*b^2*Sqrt[e]* 
f*m*n^2*x^2 - 160*b^3*Sqrt[e]*f*m*n^3*x^2 - 18*a^3*f^(3/2)*m*x^3*ArcTan[(S 
qrt[f]*x)/Sqrt[e]] - 18*a^2*b*f^(3/2)*m*n*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]] 
- 12*a*b^2*f^(3/2)*m*n^2*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]] - 4*b^3*f^(3/2)*m 
*n^3*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]] + 54*a^2*b*f^(3/2)*m*n*x^3*ArcTan[(Sq 
rt[f]*x)/Sqrt[e]]*Log[x] + 36*a*b^2*f^(3/2)*m*n^2*x^3*ArcTan[(Sqrt[f]*x)/S 
qrt[e]]*Log[x] + 12*b^3*f^(3/2)*m*n^3*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[ 
x] - 54*a*b^2*f^(3/2)*m*n^2*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]^2 - 18* 
b^3*f^(3/2)*m*n^3*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]^2 + 18*b^3*f^(3/2 
)*m*n^3*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]^3 - 54*a^2*b*Sqrt[e]*f*m*x^ 
2*Log[c*x^n] - 144*a*b^2*Sqrt[e]*f*m*n*x^2*Log[c*x^n] - 156*b^3*Sqrt[e]*f* 
m*n^2*x^2*Log[c*x^n] - 54*a^2*b*f^(3/2)*m*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]* 
Log[c*x^n] - 36*a*b^2*f^(3/2)*m*n*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^ 
n] - 12*b^3*f^(3/2)*m*n^2*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n] + 108 
*a*b^2*f^(3/2)*m*n*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]*Log[c*x^n] + 36* 
b^3*f^(3/2)*m*n^2*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]*Log[c*x^n] - 54*b 
^3*f^(3/2)*m*n^2*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]^2*Log[c*x^n] - 54* 
a*b^2*Sqrt[e]*f*m*x^2*Log[c*x^n]^2 - 72*b^3*Sqrt[e]*f*m*n*x^2*Log[c*x^n]^2 
 - 54*a*b^2*f^(3/2)*m*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n]^2 - 18*b^ 
3*f^(3/2)*m*n*x^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n]^2 + 54*b^3*f^(...
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.89 (sec) , antiderivative size = 989, normalized size of antiderivative = 1.10, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, 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.

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

\(\Big \downarrow \) 2825

\(\displaystyle -2 f m \int \left (-\frac {2 b^3 n^3}{27 x^2 \left (f x^2+e\right )}-\frac {2 b^2 \left (a+b \log \left (c x^n\right )\right ) n^2}{9 x^2 \left (f x^2+e\right )}-\frac {b \left (a+b \log \left (c x^n\right )\right )^2 n}{3 x^2 \left (f x^2+e\right )}-\frac {\left (a+b \log \left (c x^n\right )\right )^3}{3 x^2 \left (f x^2+e\right )}\right )dx-\frac {2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{9 x^3}-\frac {b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )}{3 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{3 x^3}-\frac {2 b^3 n^3 \log \left (d \left (e+f x^2\right )^m\right )}{27 x^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 b^3 \log \left (d \left (f x^2+e\right )^m\right ) n^3}{27 x^3}-\frac {2 b^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (f x^2+e\right )^m\right ) n^2}{9 x^3}-\frac {b \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (f x^2+e\right )^m\right ) n}{3 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (f x^2+e\right )^m\right )}{3 x^3}-2 f m \left (\frac {2 b^3 \sqrt {f} \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) n^3}{27 e^{3/2}}-\frac {i b^3 \sqrt {f} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {f} x}{\sqrt {e}}\right ) n^3}{9 e^{3/2}}+\frac {i b^3 \sqrt {f} \operatorname {PolyLog}\left (2,\frac {i \sqrt {f} x}{\sqrt {e}}\right ) n^3}{9 e^{3/2}}-\frac {b^3 \sqrt {f} \operatorname {PolyLog}\left (3,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) n^3}{3 (-e)^{3/2}}+\frac {b^3 \sqrt {f} \operatorname {PolyLog}\left (3,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) n^3}{3 (-e)^{3/2}}+\frac {b^3 \sqrt {f} \operatorname {PolyLog}\left (4,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) n^3}{(-e)^{3/2}}-\frac {b^3 \sqrt {f} \operatorname {PolyLog}\left (4,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) n^3}{(-e)^{3/2}}+\frac {80 b^3 n^3}{27 e x}+\frac {2 b^2 \sqrt {f} \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right ) n^2}{9 e^{3/2}}+\frac {26 b^2 \left (a+b \log \left (c x^n\right )\right ) n^2}{9 e x}+\frac {b^2 \sqrt {f} \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) n^2}{3 (-e)^{3/2}}-\frac {b^2 \sqrt {f} \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) n^2}{3 (-e)^{3/2}}-\frac {b^2 \sqrt {f} \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) n^2}{(-e)^{3/2}}+\frac {b^2 \sqrt {f} \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) n^2}{(-e)^{3/2}}+\frac {4 b \left (a+b \log \left (c x^n\right )\right )^2 n}{3 e x}-\frac {b \sqrt {f} \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) n}{6 (-e)^{3/2}}+\frac {b \sqrt {f} \left (a+b \log \left (c x^n\right )\right )^2 \log \left (\frac {\sqrt {f} x}{\sqrt {-e}}+1\right ) n}{6 (-e)^{3/2}}+\frac {b \sqrt {f} \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) n}{2 (-e)^{3/2}}-\frac {b \sqrt {f} \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) n}{2 (-e)^{3/2}}+\frac {\left (a+b \log \left (c x^n\right )\right )^3}{3 e x}-\frac {\sqrt {f} \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{6 (-e)^{3/2}}+\frac {\sqrt {f} \left (a+b \log \left (c x^n\right )\right )^3 \log \left (\frac {\sqrt {f} x}{\sqrt {-e}}+1\right )}{6 (-e)^{3/2}}\right )\)

Input:

Int[((a + b*Log[c*x^n])^3*Log[d*(e + f*x^2)^m])/x^4,x]
 

Output:

(-2*b^3*n^3*Log[d*(e + f*x^2)^m])/(27*x^3) - (2*b^2*n^2*(a + b*Log[c*x^n]) 
*Log[d*(e + f*x^2)^m])/(9*x^3) - (b*n*(a + b*Log[c*x^n])^2*Log[d*(e + f*x^ 
2)^m])/(3*x^3) - ((a + b*Log[c*x^n])^3*Log[d*(e + f*x^2)^m])/(3*x^3) - 2*f 
*m*((80*b^3*n^3)/(27*e*x) + (2*b^3*Sqrt[f]*n^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]] 
)/(27*e^(3/2)) + (26*b^2*n^2*(a + b*Log[c*x^n]))/(9*e*x) + (2*b^2*Sqrt[f]* 
n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*(a + b*Log[c*x^n]))/(9*e^(3/2)) + (4*b*n*( 
a + b*Log[c*x^n])^2)/(3*e*x) + (a + b*Log[c*x^n])^3/(3*e*x) - (b*Sqrt[f]*n 
*(a + b*Log[c*x^n])^2*Log[1 - (Sqrt[f]*x)/Sqrt[-e]])/(6*(-e)^(3/2)) - (Sqr 
t[f]*(a + b*Log[c*x^n])^3*Log[1 - (Sqrt[f]*x)/Sqrt[-e]])/(6*(-e)^(3/2)) + 
(b*Sqrt[f]*n*(a + b*Log[c*x^n])^2*Log[1 + (Sqrt[f]*x)/Sqrt[-e]])/(6*(-e)^( 
3/2)) + (Sqrt[f]*(a + b*Log[c*x^n])^3*Log[1 + (Sqrt[f]*x)/Sqrt[-e]])/(6*(- 
e)^(3/2)) + (b^2*Sqrt[f]*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((Sqrt[f]*x)/S 
qrt[-e])])/(3*(-e)^(3/2)) + (b*Sqrt[f]*n*(a + b*Log[c*x^n])^2*PolyLog[2, - 
((Sqrt[f]*x)/Sqrt[-e])])/(2*(-e)^(3/2)) - (b^2*Sqrt[f]*n^2*(a + b*Log[c*x^ 
n])*PolyLog[2, (Sqrt[f]*x)/Sqrt[-e]])/(3*(-e)^(3/2)) - (b*Sqrt[f]*n*(a + b 
*Log[c*x^n])^2*PolyLog[2, (Sqrt[f]*x)/Sqrt[-e]])/(2*(-e)^(3/2)) - ((I/9)*b 
^3*Sqrt[f]*n^3*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]])/e^(3/2) + ((I/9)*b^3* 
Sqrt[f]*n^3*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]])/e^(3/2) - (b^3*Sqrt[f]*n^3* 
PolyLog[3, -((Sqrt[f]*x)/Sqrt[-e])])/(3*(-e)^(3/2)) - (b^2*Sqrt[f]*n^2*(a 
+ b*Log[c*x^n])*PolyLog[3, -((Sqrt[f]*x)/Sqrt[-e])])/(-e)^(3/2) + (b^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 [F]

\[\int \frac {{\left (a +b \ln \left (c \,x^{n}\right )\right )}^{3} \ln \left (d \left (f \,x^{2}+e \right )^{m}\right )}{x^{4}}d x\]

Input:

int((a+b*ln(c*x^n))^3*ln(d*(f*x^2+e)^m)/x^4,x)
 

Output:

int((a+b*ln(c*x^n))^3*ln(d*(f*x^2+e)^m)/x^4,x)
 

Fricas [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{x^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{4}} \,d x } \] Input:

integrate((a+b*log(c*x^n))^3*log(d*(f*x^2+e)^m)/x^4,x, algorithm="fricas")
 

Output:

integral((b^3*log(c*x^n)^3 + 3*a*b^2*log(c*x^n)^2 + 3*a^2*b*log(c*x^n) + a 
^3)*log((f*x^2 + e)^m*d)/x^4, x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{x^4} \, dx=\text {Timed out} \] Input:

integrate((a+b*ln(c*x**n))**3*ln(d*(f*x**2+e)**m)/x**4,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{x^4} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((a+b*log(c*x^n))^3*log(d*(f*x^2+e)^m)/x^4,x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{x^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )}{x^{4}} \,d x } \] Input:

integrate((a+b*log(c*x^n))^3*log(d*(f*x^2+e)^m)/x^4,x, algorithm="giac")
 

Output:

integrate((b*log(c*x^n) + a)^3*log((f*x^2 + e)^m*d)/x^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{x^4} \, dx=\int \frac {\ln \left (d\,{\left (f\,x^2+e\right )}^m\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x^4} \,d x \] Input:

int((log(d*(e + f*x^2)^m)*(a + b*log(c*x^n))^3)/x^4,x)
 

Output:

int((log(d*(e + f*x^2)^m)*(a + b*log(c*x^n))^3)/x^4, x)
 

Reduce [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )}{x^4} \, dx =\text {Too large to display} \] Input:

int((a+b*log(c*x^n))^3*log(d*(f*x^2+e)^m)/x^4,x)
 

Output:

( - 18*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**3*f*m*x**3 - 18*sq 
rt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*b*f*m*n*x**3 - 12*sqrt(f) 
*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*b**2*f*m*n**2*x**3 - 4*sqrt(f)*sq 
rt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*b**3*f*m*n**3*x**3 - 18*int(log(x**n*c 
)**3/(e*x**4 + f*x**6),x)*b**3*e**3*m*x**3 - 54*int(log(x**n*c)**2/(e*x**4 
 + f*x**6),x)*a*b**2*e**3*m*x**3 - 18*int(log(x**n*c)**2/(e*x**4 + f*x**6) 
,x)*b**3*e**3*m*n*x**3 - 54*int(log(x**n*c)/(e*x**4 + f*x**6),x)*a**2*b*e* 
*3*m*x**3 - 36*int(log(x**n*c)/(e*x**4 + f*x**6),x)*a*b**2*e**3*m*n*x**3 - 
 12*int(log(x**n*c)/(e*x**4 + f*x**6),x)*b**3*e**3*m*n**2*x**3 - 9*log((e 
+ f*x**2)**m*d)*log(x**n*c)**3*b**3*e**2 - 27*log((e + f*x**2)**m*d)*log(x 
**n*c)**2*a*b**2*e**2 - 9*log((e + f*x**2)**m*d)*log(x**n*c)**2*b**3*e**2* 
n - 27*log((e + f*x**2)**m*d)*log(x**n*c)*a**2*b*e**2 - 18*log((e + f*x**2 
)**m*d)*log(x**n*c)*a*b**2*e**2*n - 6*log((e + f*x**2)**m*d)*log(x**n*c)*b 
**3*e**2*n**2 - 9*log((e + f*x**2)**m*d)*a**3*e**2 - 9*log((e + f*x**2)**m 
*d)*a**2*b*e**2*n - 6*log((e + f*x**2)**m*d)*a*b**2*e**2*n**2 - 2*log((e + 
 f*x**2)**m*d)*b**3*e**2*n**3 - 6*log(x**n*c)**3*b**3*e**2*m - 18*log(x**n 
*c)**2*a*b**2*e**2*m - 12*log(x**n*c)**2*b**3*e**2*m*n - 18*log(x**n*c)*a* 
*2*b*e**2*m - 24*log(x**n*c)*a*b**2*e**2*m*n - 12*log(x**n*c)*b**3*e**2*m* 
n**2 - 18*a**3*e*f*m*x**2 - 6*a**2*b*e**2*m*n - 18*a**2*b*e*f*m*n*x**2 - 8 
*a*b**2*e**2*m*n**2 - 12*a*b**2*e*f*m*n**2*x**2 - 4*b**3*e**2*m*n**3 - ...