\(\int x \log (d (\frac {1}{d}+f \sqrt {x})) (a+b \log (c x^n))^2 \, dx\) [60]

Optimal result

Integrand size = 28, antiderivative size = 557 \[ \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {21 b^2 n^2 \sqrt {x}}{4 d^3 f^3}+\frac {a b n x}{2 d^2 f^2}-\frac {7 b^2 n^2 x}{8 d^2 f^2}+\frac {37 b^2 n^2 x^{3/2}}{108 d f}-\frac {3}{16} b^2 n^2 x^2-\frac {b^2 n^2 \log \left (1+d f \sqrt {x}\right )}{4 d^4 f^4}+\frac {1}{4} b^2 n^2 x^2 \log \left (1+d f \sqrt {x}\right )+\frac {b^2 n x \log \left (c x^n\right )}{2 d^2 f^2}-\frac {5 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 f^3}+\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}-\frac {7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{18 d f}+\frac {1}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^4 f^4}-\frac {1}{2} b n x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 d f}-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 d^4 f^4}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {b^2 n^2 \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )}{d^4 f^4}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )}{d^4 f^4}+\frac {4 b^2 n^2 \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right )}{d^4 f^4} \] Output:

21/4*b^2*n^2*x^(1/2)/d^3/f^3+1/2*a*b*n*x/d^2/f^2-7/8*b^2*n^2*x/d^2/f^2+37/ 
108*b^2*n^2*x^(3/2)/d/f-3/16*b^2*n^2*x^2-1/4*b^2*n^2*ln(1+d*f*x^(1/2))/d^4 
/f^4+1/4*b^2*n^2*x^2*ln(1+d*f*x^(1/2))+1/2*b^2*n*x*ln(c*x^n)/d^2/f^2-5/2*b 
*n*x^(1/2)*(a+b*ln(c*x^n))/d^3/f^3+1/4*b*n*x*(a+b*ln(c*x^n))/d^2/f^2-7/18* 
b*n*x^(3/2)*(a+b*ln(c*x^n))/d/f+1/4*b*n*x^2*(a+b*ln(c*x^n))+1/2*b*n*ln(1+d 
*f*x^(1/2))*(a+b*ln(c*x^n))/d^4/f^4-1/2*b*n*x^2*ln(1+d*f*x^(1/2))*(a+b*ln( 
c*x^n))+1/2*x^(1/2)*(a+b*ln(c*x^n))^2/d^3/f^3-1/4*x*(a+b*ln(c*x^n))^2/d^2/ 
f^2+1/6*x^(3/2)*(a+b*ln(c*x^n))^2/d/f-1/8*x^2*(a+b*ln(c*x^n))^2-1/2*ln(1+d 
*f*x^(1/2))*(a+b*ln(c*x^n))^2/d^4/f^4+1/2*x^2*ln(1+d*f*x^(1/2))*(a+b*ln(c* 
x^n))^2+b^2*n^2*polylog(2,-d*f*x^(1/2))/d^4/f^4-2*b*n*(a+b*ln(c*x^n))*poly 
log(2,-d*f*x^(1/2))/d^4/f^4+4*b^2*n^2*polylog(3,-d*f*x^(1/2))/d^4/f^4
 

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 769, normalized size of antiderivative = 1.38 \[ \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {216 a^2 d f \sqrt {x}-1080 a b d f n \sqrt {x}+2268 b^2 d f n^2 \sqrt {x}-108 a^2 d^2 f^2 x+324 a b d^2 f^2 n x-378 b^2 d^2 f^2 n^2 x+72 a^2 d^3 f^3 x^{3/2}-168 a b d^3 f^3 n x^{3/2}+148 b^2 d^3 f^3 n^2 x^{3/2}-54 a^2 d^4 f^4 x^2+108 a b d^4 f^4 n x^2-81 b^2 d^4 f^4 n^2 x^2-216 a^2 \log \left (1+d f \sqrt {x}\right )+216 a b n \log \left (1+d f \sqrt {x}\right )-108 b^2 n^2 \log \left (1+d f \sqrt {x}\right )+216 a^2 d^4 f^4 x^2 \log \left (1+d f \sqrt {x}\right )-216 a b d^4 f^4 n x^2 \log \left (1+d f \sqrt {x}\right )+108 b^2 d^4 f^4 n^2 x^2 \log \left (1+d f \sqrt {x}\right )+432 a b d f \sqrt {x} \log \left (c x^n\right )-1080 b^2 d f n \sqrt {x} \log \left (c x^n\right )-216 a b d^2 f^2 x \log \left (c x^n\right )+324 b^2 d^2 f^2 n x \log \left (c x^n\right )+144 a b d^3 f^3 x^{3/2} \log \left (c x^n\right )-168 b^2 d^3 f^3 n x^{3/2} \log \left (c x^n\right )-108 a b d^4 f^4 x^2 \log \left (c x^n\right )+108 b^2 d^4 f^4 n x^2 \log \left (c x^n\right )-432 a b \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )+216 b^2 n \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )+432 a b d^4 f^4 x^2 \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )-216 b^2 d^4 f^4 n x^2 \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )+216 b^2 d f \sqrt {x} \log ^2\left (c x^n\right )-108 b^2 d^2 f^2 x \log ^2\left (c x^n\right )+72 b^2 d^3 f^3 x^{3/2} \log ^2\left (c x^n\right )-54 b^2 d^4 f^4 x^2 \log ^2\left (c x^n\right )-216 b^2 \log \left (1+d f \sqrt {x}\right ) \log ^2\left (c x^n\right )+216 b^2 d^4 f^4 x^2 \log \left (1+d f \sqrt {x}\right ) \log ^2\left (c x^n\right )+432 b n \left (-2 a+b n-2 b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )+1728 b^2 n^2 \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right )}{432 d^4 f^4} \] Input:

Integrate[x*Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n])^2,x]
 

Output:

(216*a^2*d*f*Sqrt[x] - 1080*a*b*d*f*n*Sqrt[x] + 2268*b^2*d*f*n^2*Sqrt[x] - 
 108*a^2*d^2*f^2*x + 324*a*b*d^2*f^2*n*x - 378*b^2*d^2*f^2*n^2*x + 72*a^2* 
d^3*f^3*x^(3/2) - 168*a*b*d^3*f^3*n*x^(3/2) + 148*b^2*d^3*f^3*n^2*x^(3/2) 
- 54*a^2*d^4*f^4*x^2 + 108*a*b*d^4*f^4*n*x^2 - 81*b^2*d^4*f^4*n^2*x^2 - 21 
6*a^2*Log[1 + d*f*Sqrt[x]] + 216*a*b*n*Log[1 + d*f*Sqrt[x]] - 108*b^2*n^2* 
Log[1 + d*f*Sqrt[x]] + 216*a^2*d^4*f^4*x^2*Log[1 + d*f*Sqrt[x]] - 216*a*b* 
d^4*f^4*n*x^2*Log[1 + d*f*Sqrt[x]] + 108*b^2*d^4*f^4*n^2*x^2*Log[1 + d*f*S 
qrt[x]] + 432*a*b*d*f*Sqrt[x]*Log[c*x^n] - 1080*b^2*d*f*n*Sqrt[x]*Log[c*x^ 
n] - 216*a*b*d^2*f^2*x*Log[c*x^n] + 324*b^2*d^2*f^2*n*x*Log[c*x^n] + 144*a 
*b*d^3*f^3*x^(3/2)*Log[c*x^n] - 168*b^2*d^3*f^3*n*x^(3/2)*Log[c*x^n] - 108 
*a*b*d^4*f^4*x^2*Log[c*x^n] + 108*b^2*d^4*f^4*n*x^2*Log[c*x^n] - 432*a*b*L 
og[1 + d*f*Sqrt[x]]*Log[c*x^n] + 216*b^2*n*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] 
 + 432*a*b*d^4*f^4*x^2*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] - 216*b^2*d^4*f^4*n 
*x^2*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] + 216*b^2*d*f*Sqrt[x]*Log[c*x^n]^2 - 
108*b^2*d^2*f^2*x*Log[c*x^n]^2 + 72*b^2*d^3*f^3*x^(3/2)*Log[c*x^n]^2 - 54* 
b^2*d^4*f^4*x^2*Log[c*x^n]^2 - 216*b^2*Log[1 + d*f*Sqrt[x]]*Log[c*x^n]^2 + 
 216*b^2*d^4*f^4*x^2*Log[1 + d*f*Sqrt[x]]*Log[c*x^n]^2 + 432*b*n*(-2*a + b 
*n - 2*b*Log[c*x^n])*PolyLog[2, -(d*f*Sqrt[x])] + 1728*b^2*n^2*PolyLog[3, 
-(d*f*Sqrt[x])])/(432*d^4*f^4)
 

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 513, normalized size of antiderivative = 0.92, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2824, 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.

\(\displaystyle \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx\)

\(\Big \downarrow \) 2824

\(\displaystyle -2 b n \int \left (-\frac {1}{8} x \left (a+b \log \left (c x^n\right )\right )+\frac {1}{2} x \log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {\log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^4 f^4 x}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{6 d f}+\frac {a+b \log \left (c x^n\right )}{2 d^3 f^3 \sqrt {x}}-\frac {a+b \log \left (c x^n\right )}{4 d^2 f^2}\right )dx-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 d^4 f^4}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 d f}+\frac {1}{2} x^2 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^2\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 d^4 f^4}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{4 d^2 f^2}-2 b n \left (\frac {\operatorname {PolyLog}\left (2,-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4 f^4}-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d^4 f^4}+\frac {5 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{4 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{8 d^2 f^2}+\frac {7 x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{36 d f}+\frac {1}{4} x^2 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {a x}{4 d^2 f^2}-\frac {b x \log \left (c x^n\right )}{4 d^2 f^2}-\frac {b n \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )}{2 d^4 f^4}-\frac {2 b n \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right )}{d^4 f^4}+\frac {b n \log \left (d f \sqrt {x}+1\right )}{8 d^4 f^4}-\frac {21 b n \sqrt {x}}{8 d^3 f^3}+\frac {7 b n x}{16 d^2 f^2}-\frac {37 b n x^{3/2}}{216 d f}-\frac {1}{8} b n x^2 \log \left (d f \sqrt {x}+1\right )+\frac {3}{32} b n x^2\right )+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 d f}+\frac {1}{2} x^2 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^2\)

Input:

Int[x*Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n])^2,x]
 

Output:

(Sqrt[x]*(a + b*Log[c*x^n])^2)/(2*d^3*f^3) - (x*(a + b*Log[c*x^n])^2)/(4*d 
^2*f^2) + (x^(3/2)*(a + b*Log[c*x^n])^2)/(6*d*f) - (x^2*(a + b*Log[c*x^n]) 
^2)/8 - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/(2*d^4*f^4) + (x^2*Log 
[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/2 - 2*b*n*((-21*b*n*Sqrt[x])/(8*d^ 
3*f^3) - (a*x)/(4*d^2*f^2) + (7*b*n*x)/(16*d^2*f^2) - (37*b*n*x^(3/2))/(21 
6*d*f) + (3*b*n*x^2)/32 + (b*n*Log[1 + d*f*Sqrt[x]])/(8*d^4*f^4) - (b*n*x^ 
2*Log[1 + d*f*Sqrt[x]])/8 - (b*x*Log[c*x^n])/(4*d^2*f^2) + (5*Sqrt[x]*(a + 
 b*Log[c*x^n]))/(4*d^3*f^3) - (x*(a + b*Log[c*x^n]))/(8*d^2*f^2) + (7*x^(3 
/2)*(a + b*Log[c*x^n]))/(36*d*f) - (x^2*(a + b*Log[c*x^n]))/8 - (Log[1 + d 
*f*Sqrt[x]]*(a + b*Log[c*x^n]))/(4*d^4*f^4) + (x^2*Log[1 + d*f*Sqrt[x]]*(a 
 + b*Log[c*x^n]))/4 - (b*n*PolyLog[2, -(d*f*Sqrt[x])])/(2*d^4*f^4) + ((a + 
 b*Log[c*x^n])*PolyLog[2, -(d*f*Sqrt[x])])/(d^4*f^4) - (2*b*n*PolyLog[3, - 
(d*f*Sqrt[x])])/(d^4*f^4))
 

Defintions of rubi rules used

rule 2009

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

rule 2824

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

Maple [F]

\[\int x \ln \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}d x\]

Input:

int(x*ln(d*(1/d+f*x^(1/2)))*(a+b*ln(c*x^n))^2,x)
 

Output:

int(x*ln(d*(1/d+f*x^(1/2)))*(a+b*ln(c*x^n))^2,x)
 

Fricas [F]

\[ \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right ) \,d x } \] Input:

integrate(x*log(d*(1/d+f*x^(1/2)))*(a+b*log(c*x^n))^2,x, algorithm="fricas 
")
 

Output:

integral((b^2*x*log(c*x^n)^2 + 2*a*b*x*log(c*x^n) + a^2*x)*log(d*f*sqrt(x) 
 + 1), x)
 

Sympy [F(-1)]

Timed out. \[ \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\text {Timed out} \] Input:

integrate(x*ln(d*(1/d+f*x**(1/2)))*(a+b*ln(c*x**n))**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right ) \,d x } \] Input:

integrate(x*log(d*(1/d+f*x^(1/2)))*(a+b*log(c*x^n))^2,x, algorithm="maxima 
")
 

Output:

integrate((b*log(c*x^n) + a)^2*x*log((f*sqrt(x) + 1/d)*d), x)
 

Giac [F]

\[ \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right ) \,d x } \] Input:

integrate(x*log(d*(1/d+f*x^(1/2)))*(a+b*log(c*x^n))^2,x, algorithm="giac")
 

Output:

integrate((b*log(c*x^n) + a)^2*x*log((f*sqrt(x) + 1/d)*d), x)
 

Mupad [F(-1)]

Timed out. \[ \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\int x\,\ln \left (d\,\left (f\,\sqrt {x}+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \] Input:

int(x*log(d*(f*x^(1/2) + 1/d))*(a + b*log(c*x^n))^2,x)
 

Output:

int(x*log(d*(f*x^(1/2) + 1/d))*(a + b*log(c*x^n))^2, x)
 

Reduce [F]

\[ \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx =\text {Too large to display} \] Input:

int(x*log(d*(1/d+f*x^(1/2)))*(a+b*log(c*x^n))^2,x)
 

Output:

(72*sqrt(x)*log(x**n*c)**2*b**2*d**3*f**3*n*x + 216*sqrt(x)*log(x**n*c)**2 
*b**2*d*f*n + 144*sqrt(x)*log(x**n*c)*a*b*d**3*f**3*n*x + 432*sqrt(x)*log( 
x**n*c)*a*b*d*f*n - 168*sqrt(x)*log(x**n*c)*b**2*d**3*f**3*n**2*x - 1080*s 
qrt(x)*log(x**n*c)*b**2*d*f*n**2 + 72*sqrt(x)*a**2*d**3*f**3*n*x + 216*sqr 
t(x)*a**2*d*f*n - 168*sqrt(x)*a*b*d**3*f**3*n**2*x - 1080*sqrt(x)*a*b*d*f* 
n**2 + 148*sqrt(x)*b**2*d**3*f**3*n**3*x + 2268*sqrt(x)*b**2*d*f*n**3 - 10 
8*int(log(x**n*c)**2/(d**2*f**2*x**2 - x),x)*b**2*n - 216*int(log(x**n*c)/ 
(d**2*f**2*x**2 - x),x)*a*b*n + 108*int(log(x**n*c)/(d**2*f**2*x**2 - x),x 
)*b**2*n**2 + 108*int((sqrt(x)*log(x**n*c)**2)/(d**2*f**2*x**2 - x),x)*b** 
2*d*f*n + 216*int((sqrt(x)*log(x**n*c))/(d**2*f**2*x**2 - x),x)*a*b*d*f*n 
- 108*int((sqrt(x)*log(x**n*c))/(d**2*f**2*x**2 - x),x)*b**2*d*f*n**2 + 21 
6*log(sqrt(x)*d*f + 1)*log(x**n*c)**2*b**2*d**4*f**4*n*x**2 + 432*log(sqrt 
(x)*d*f + 1)*log(x**n*c)*a*b*d**4*f**4*n*x**2 - 216*log(sqrt(x)*d*f + 1)*l 
og(x**n*c)*b**2*d**4*f**4*n**2*x**2 + 216*log(sqrt(x)*d*f + 1)*a**2*d**4*f 
**4*n*x**2 - 216*log(sqrt(x)*d*f + 1)*a**2*n - 216*log(sqrt(x)*d*f + 1)*a* 
b*d**4*f**4*n**2*x**2 + 216*log(sqrt(x)*d*f + 1)*a*b*n**2 + 108*log(sqrt(x 
)*d*f + 1)*b**2*d**4*f**4*n**3*x**2 - 108*log(sqrt(x)*d*f + 1)*b**2*n**3 - 
 36*log(x**n*c)**3*b**2 - 108*log(x**n*c)**2*a*b - 54*log(x**n*c)**2*b**2* 
d**4*f**4*n*x**2 - 108*log(x**n*c)**2*b**2*d**2*f**2*n*x + 54*log(x**n*c)* 
*2*b**2*n - 108*log(x**n*c)*a*b*d**4*f**4*n*x**2 - 216*log(x**n*c)*a*b*...