\(\int x^2 \log (d (\frac {1}{d}+f \sqrt {x})) (a+b \log (c x^n))^2 \, dx\) [59]

Optimal result

Integrand size = 30, antiderivative size = 708 \[ \int x^2 \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 {86 b^2 n^2 \sqrt {x}}{27 d^5 f^5}+\frac {a b n x}{3 d^4 f^4}-\frac {13 b^2 n^2 x}{27 d^4 f^4}+\frac {14 b^2 n^2 x^{3/2}}{81 d^3 f^3}-\frac {19 b^2 n^2 x^2}{216 d^2 f^2}+\frac {182 b^2 n^2 x^{5/2}}{3375 d f}-\frac {1}{27} b^2 n^2 x^3-\frac {2 b^2 n^2 \log \left (1+d f \sqrt {x}\right )}{27 d^6 f^6}+\frac {2}{27} b^2 n^2 x^3 \log \left (1+d f \sqrt {x}\right )+\frac {b^2 n x \log \left (c x^n\right )}{3 d^4 f^4}-\frac {14 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{9 d^5 f^5}+\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{9 d^4 f^4}-\frac {2 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 d^3 f^3}+\frac {5 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{36 d^2 f^2}-\frac {22 b n x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{225 d f}+\frac {2}{27} b n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {2 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 d^6 f^6}-\frac {2}{9} b n x^3 \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}{3 d^5 f^5}-\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{6 d^4 f^4}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{9 d^3 f^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{12 d^2 f^2}+\frac {x^{5/2} \left (a+b \log \left (c x^n\right )\right )^2}{15 d f}-\frac {1}{18} x^3 \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}{3 d^6 f^6}+\frac {1}{3} x^3 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {4 b^2 n^2 \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )}{9 d^6 f^6}-\frac {4 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )}{3 d^6 f^6}+\frac {8 b^2 n^2 \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right )}{3 d^6 f^6} \] Output:

1/15*x^(5/2)*(a+b*ln(c*x^n))^2/d/f-1/6*x*(a+b*ln(c*x^n))^2/d^4/f^4+1/9*x^( 
3/2)*(a+b*ln(c*x^n))^2/d^3/f^3-1/12*x^2*(a+b*ln(c*x^n))^2/d^2/f^2+1/3*a*b* 
n*x/d^4/f^4+1/3*b^2*n*x*ln(c*x^n)/d^4/f^4+1/9*b*n*x*(a+b*ln(c*x^n))/d^4/f^ 
4-2/9*b*n*x^(3/2)*(a+b*ln(c*x^n))/d^3/f^3+5/36*b*n*x^2*(a+b*ln(c*x^n))/d^2 
/f^2-22/225*b*n*x^(5/2)*(a+b*ln(c*x^n))/d/f-4/3*b*n*(a+b*ln(c*x^n))*polylo 
g(2,-d*f*x^(1/2))/d^6/f^6+2/9*b*n*ln(1+d*f*x^(1/2))*(a+b*ln(c*x^n))/d^6/f^ 
6-14/9*b*n*x^(1/2)*(a+b*ln(c*x^n))/d^5/f^5-19/216*b^2*n^2*x^2/d^2/f^2+182/ 
3375*b^2*n^2*x^(5/2)/d/f-13/27*b^2*n^2*x/d^4/f^4+14/81*b^2*n^2*x^(3/2)/d^3 
/f^3-1/18*x^3*(a+b*ln(c*x^n))^2+1/3*x^3*ln(1+d*f*x^(1/2))*(a+b*ln(c*x^n))^ 
2+8/3*b^2*n^2*polylog(3,-d*f*x^(1/2))/d^6/f^6+4/9*b^2*n^2*polylog(2,-d*f*x 
^(1/2))/d^6/f^6-2/27*b^2*n^2*ln(1+d*f*x^(1/2))/d^6/f^6-2/9*b*n*x^3*ln(1+d* 
f*x^(1/2))*(a+b*ln(c*x^n))+86/27*b^2*n^2*x^(1/2)/d^5/f^5+2/27*b*n*x^3*(a+b 
*ln(c*x^n))+1/3*x^(1/2)*(a+b*ln(c*x^n))^2/d^5/f^5+2/27*b^2*n^2*x^3*ln(1+d* 
f*x^(1/2))-1/3*ln(1+d*f*x^(1/2))*(a+b*ln(c*x^n))^2/d^6/f^6-1/27*b^2*n^2*x^ 
3
 

Mathematica [A] (verified)

Time = 0.66 (sec) , antiderivative size = 995, normalized size of antiderivative = 1.41 \[ \int x^2 \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx =\text {Too large to display} \] Input:

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

Output:

(27000*a^2*d*f*Sqrt[x] - 126000*a*b*d*f*n*Sqrt[x] + 258000*b^2*d*f*n^2*Sqr 
t[x] - 13500*a^2*d^2*f^2*x + 36000*a*b*d^2*f^2*n*x - 39000*b^2*d^2*f^2*n^2 
*x + 9000*a^2*d^3*f^3*x^(3/2) - 18000*a*b*d^3*f^3*n*x^(3/2) + 14000*b^2*d^ 
3*f^3*n^2*x^(3/2) - 6750*a^2*d^4*f^4*x^2 + 11250*a*b*d^4*f^4*n*x^2 - 7125* 
b^2*d^4*f^4*n^2*x^2 + 5400*a^2*d^5*f^5*x^(5/2) - 7920*a*b*d^5*f^5*n*x^(5/2 
) + 4368*b^2*d^5*f^5*n^2*x^(5/2) - 4500*a^2*d^6*f^6*x^3 + 6000*a*b*d^6*f^6 
*n*x^3 - 3000*b^2*d^6*f^6*n^2*x^3 - 27000*a^2*Log[1 + d*f*Sqrt[x]] + 18000 
*a*b*n*Log[1 + d*f*Sqrt[x]] - 6000*b^2*n^2*Log[1 + d*f*Sqrt[x]] + 27000*a^ 
2*d^6*f^6*x^3*Log[1 + d*f*Sqrt[x]] - 18000*a*b*d^6*f^6*n*x^3*Log[1 + d*f*S 
qrt[x]] + 6000*b^2*d^6*f^6*n^2*x^3*Log[1 + d*f*Sqrt[x]] + 54000*a*b*d*f*Sq 
rt[x]*Log[c*x^n] - 126000*b^2*d*f*n*Sqrt[x]*Log[c*x^n] - 27000*a*b*d^2*f^2 
*x*Log[c*x^n] + 36000*b^2*d^2*f^2*n*x*Log[c*x^n] + 18000*a*b*d^3*f^3*x^(3/ 
2)*Log[c*x^n] - 18000*b^2*d^3*f^3*n*x^(3/2)*Log[c*x^n] - 13500*a*b*d^4*f^4 
*x^2*Log[c*x^n] + 11250*b^2*d^4*f^4*n*x^2*Log[c*x^n] + 10800*a*b*d^5*f^5*x 
^(5/2)*Log[c*x^n] - 7920*b^2*d^5*f^5*n*x^(5/2)*Log[c*x^n] - 9000*a*b*d^6*f 
^6*x^3*Log[c*x^n] + 6000*b^2*d^6*f^6*n*x^3*Log[c*x^n] - 54000*a*b*Log[1 + 
d*f*Sqrt[x]]*Log[c*x^n] + 18000*b^2*n*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] + 54 
000*a*b*d^6*f^6*x^3*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] - 18000*b^2*d^6*f^6*n* 
x^3*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] + 27000*b^2*d*f*Sqrt[x]*Log[c*x^n]^2 - 
 13500*b^2*d^2*f^2*x*Log[c*x^n]^2 + 9000*b^2*d^3*f^3*x^(3/2)*Log[c*x^n]...
 

Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 650, normalized size of antiderivative = 0.92, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, 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.

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

\(\Big \downarrow \) 2824

\(\displaystyle -2 b n \int \left (\frac {1}{3} \log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right ) x^2-\frac {1}{18} \left (a+b \log \left (c x^n\right )\right ) x^2+\frac {\left (a+b \log \left (c x^n\right )\right ) x^{3/2}}{15 d f}-\frac {\left (a+b \log \left (c x^n\right )\right ) x}{12 d^2 f^2}+\frac {\left (a+b \log \left (c x^n\right )\right ) \sqrt {x}}{9 d^3 f^3}-\frac {a+b \log \left (c x^n\right )}{6 d^4 f^4}+\frac {a+b \log \left (c x^n\right )}{3 d^5 f^5 \sqrt {x}}-\frac {\log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^6 f^6 x}\right )dx-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 d^6 f^6}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{3 d^5 f^5}-\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{6 d^4 f^4}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{9 d^3 f^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{12 d^2 f^2}+\frac {x^{5/2} \left (a+b \log \left (c x^n\right )\right )^2}{15 d f}+\frac {1}{3} x^3 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{18} x^3 \left (a+b \log \left (c x^n\right )\right )^2\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 d^6 f^6}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{3 d^5 f^5}-\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{6 d^4 f^4}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{9 d^3 f^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{12 d^2 f^2}-2 b n \left (\frac {2 \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^6 f^6}-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{9 d^6 f^6}+\frac {7 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{9 d^5 f^5}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{18 d^4 f^4}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 d^3 f^3}-\frac {5 x^2 \left (a+b \log \left (c x^n\right )\right )}{72 d^2 f^2}+\frac {11 x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{225 d f}+\frac {1}{9} x^3 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{27} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {a x}{6 d^4 f^4}-\frac {b x \log \left (c x^n\right )}{6 d^4 f^4}-\frac {2 b n \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )}{9 d^6 f^6}-\frac {4 b n \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right )}{3 d^6 f^6}+\frac {b n \log \left (d f \sqrt {x}+1\right )}{27 d^6 f^6}-\frac {43 b n \sqrt {x}}{27 d^5 f^5}+\frac {13 b n x}{54 d^4 f^4}-\frac {7 b n x^{3/2}}{81 d^3 f^3}+\frac {19 b n x^2}{432 d^2 f^2}-\frac {91 b n x^{5/2}}{3375 d f}-\frac {1}{27} b n x^3 \log \left (d f \sqrt {x}+1\right )+\frac {1}{54} b n x^3\right )+\frac {x^{5/2} \left (a+b \log \left (c x^n\right )\right )^2}{15 d f}+\frac {1}{3} x^3 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{18} x^3 \left (a+b \log \left (c x^n\right )\right )^2\)

Input:

Int[x^2*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)/(3*d^5*f^5) - (x*(a + b*Log[c*x^n])^2)/(6*d 
^4*f^4) + (x^(3/2)*(a + b*Log[c*x^n])^2)/(9*d^3*f^3) - (x^2*(a + b*Log[c*x 
^n])^2)/(12*d^2*f^2) + (x^(5/2)*(a + b*Log[c*x^n])^2)/(15*d*f) - (x^3*(a + 
 b*Log[c*x^n])^2)/18 - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/(3*d^6* 
f^6) + (x^3*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/3 - 2*b*n*((-43*b*n 
*Sqrt[x])/(27*d^5*f^5) - (a*x)/(6*d^4*f^4) + (13*b*n*x)/(54*d^4*f^4) - (7* 
b*n*x^(3/2))/(81*d^3*f^3) + (19*b*n*x^2)/(432*d^2*f^2) - (91*b*n*x^(5/2))/ 
(3375*d*f) + (b*n*x^3)/54 + (b*n*Log[1 + d*f*Sqrt[x]])/(27*d^6*f^6) - (b*n 
*x^3*Log[1 + d*f*Sqrt[x]])/27 - (b*x*Log[c*x^n])/(6*d^4*f^4) + (7*Sqrt[x]* 
(a + b*Log[c*x^n]))/(9*d^5*f^5) - (x*(a + b*Log[c*x^n]))/(18*d^4*f^4) + (x 
^(3/2)*(a + b*Log[c*x^n]))/(9*d^3*f^3) - (5*x^2*(a + b*Log[c*x^n]))/(72*d^ 
2*f^2) + (11*x^(5/2)*(a + b*Log[c*x^n]))/(225*d*f) - (x^3*(a + b*Log[c*x^n 
]))/27 - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/(9*d^6*f^6) + (x^3*Log[ 
1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/9 - (2*b*n*PolyLog[2, -(d*f*Sqrt[x])] 
)/(9*d^6*f^6) + (2*(a + b*Log[c*x^n])*PolyLog[2, -(d*f*Sqrt[x])])/(3*d^6*f 
^6) - (4*b*n*PolyLog[3, -(d*f*Sqrt[x])])/(3*d^6*f^6))
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

integrate(x^2*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^2*log((f*sqrt(x) + 1/d)*d), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(5400*sqrt(x)*log(x**n*c)**2*b**2*d**5*f**5*n*x**2 + 9000*sqrt(x)*log(x**n 
*c)**2*b**2*d**3*f**3*n*x + 27000*sqrt(x)*log(x**n*c)**2*b**2*d*f*n + 1080 
0*sqrt(x)*log(x**n*c)*a*b*d**5*f**5*n*x**2 + 18000*sqrt(x)*log(x**n*c)*a*b 
*d**3*f**3*n*x + 54000*sqrt(x)*log(x**n*c)*a*b*d*f*n - 7920*sqrt(x)*log(x* 
*n*c)*b**2*d**5*f**5*n**2*x**2 - 18000*sqrt(x)*log(x**n*c)*b**2*d**3*f**3* 
n**2*x - 126000*sqrt(x)*log(x**n*c)*b**2*d*f*n**2 + 5400*sqrt(x)*a**2*d**5 
*f**5*n*x**2 + 9000*sqrt(x)*a**2*d**3*f**3*n*x + 27000*sqrt(x)*a**2*d*f*n 
- 7920*sqrt(x)*a*b*d**5*f**5*n**2*x**2 - 18000*sqrt(x)*a*b*d**3*f**3*n**2* 
x - 126000*sqrt(x)*a*b*d*f*n**2 + 4368*sqrt(x)*b**2*d**5*f**5*n**3*x**2 + 
14000*sqrt(x)*b**2*d**3*f**3*n**3*x + 258000*sqrt(x)*b**2*d*f*n**3 - 13500 
*int(log(x**n*c)**2/(d**2*f**2*x**2 - x),x)*b**2*n - 27000*int(log(x**n*c) 
/(d**2*f**2*x**2 - x),x)*a*b*n + 9000*int(log(x**n*c)/(d**2*f**2*x**2 - x) 
,x)*b**2*n**2 + 13500*int((sqrt(x)*log(x**n*c)**2)/(d**2*f**2*x**2 - x),x) 
*b**2*d*f*n + 27000*int((sqrt(x)*log(x**n*c))/(d**2*f**2*x**2 - x),x)*a*b* 
d*f*n - 9000*int((sqrt(x)*log(x**n*c))/(d**2*f**2*x**2 - x),x)*b**2*d*f*n* 
*2 + 27000*log(sqrt(x)*d*f + 1)*log(x**n*c)**2*b**2*d**6*f**6*n*x**3 + 540 
00*log(sqrt(x)*d*f + 1)*log(x**n*c)*a*b*d**6*f**6*n*x**3 - 18000*log(sqrt( 
x)*d*f + 1)*log(x**n*c)*b**2*d**6*f**6*n**2*x**3 + 27000*log(sqrt(x)*d*f + 
 1)*a**2*d**6*f**6*n*x**3 - 27000*log(sqrt(x)*d*f + 1)*a**2*n - 18000*log( 
sqrt(x)*d*f + 1)*a*b*d**6*f**6*n**2*x**3 + 18000*log(sqrt(x)*d*f + 1)*a...