\(\int \log (d (\frac {1}{d}+f \sqrt {x})) (a+b \log (c x^n))^3 \, dx\) [66]

Optimal result

Integrand size = 27, antiderivative size = 604 \[ \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx=-\frac {90 b^3 n^3 \sqrt {x}}{d f}-6 a b^2 n^2 x+12 b^3 n^3 x-6 b^3 n^3 x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )+\frac {6 b^3 n^3 \log \left (1+d f \sqrt {x}\right )}{d^2 f^2}-6 b^3 n^2 x \log \left (c x^n\right )+\frac {42 b^2 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d f}-3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )+6 b^2 n^2 x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {6 b^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}-\frac {9 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}+3 b n x \left (a+b \log \left (c x^n\right )\right )^2-3 b n x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {3 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2 f^2}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^3+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d^2 f^2}-\frac {12 b^3 n^3 \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )}{d^2 f^2}+\frac {12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )}{d^2 f^2}-\frac {6 b n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )}{d^2 f^2}-\frac {24 b^3 n^3 \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right )}{d^2 f^2}+\frac {24 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right )}{d^2 f^2}-\frac {48 b^3 n^3 \operatorname {PolyLog}\left (4,-d f \sqrt {x}\right )}{d^2 f^2} \] Output:

-90*b^3*n^3*x^(1/2)/d/f-6*a*b^2*n^2*x+12*b^3*n^3*x-6*b^3*n^3*x*ln(d*(1/d+f 
*x^(1/2)))+6*b^3*n^3*ln(1+d*f*x^(1/2))/d^2/f^2-6*b^3*n^2*x*ln(c*x^n)+42*b^ 
2*n^2*x^(1/2)*(a+b*ln(c*x^n))/d/f-3*b^2*n^2*x*(a+b*ln(c*x^n))+6*b^2*n^2*x* 
ln(d*(1/d+f*x^(1/2)))*(a+b*ln(c*x^n))-6*b^2*n^2*ln(1+d*f*x^(1/2))*(a+b*ln( 
c*x^n))/d^2/f^2-9*b*n*x^(1/2)*(a+b*ln(c*x^n))^2/d/f+3*b*n*x*(a+b*ln(c*x^n) 
)^2-3*b*n*x*ln(d*(1/d+f*x^(1/2)))*(a+b*ln(c*x^n))^2+3*b*n*ln(1+d*f*x^(1/2) 
)*(a+b*ln(c*x^n))^2/d^2/f^2+x^(1/2)*(a+b*ln(c*x^n))^3/d/f-1/2*x*(a+b*ln(c* 
x^n))^3+x*ln(d*(1/d+f*x^(1/2)))*(a+b*ln(c*x^n))^3-ln(1+d*f*x^(1/2))*(a+b*l 
n(c*x^n))^3/d^2/f^2-12*b^3*n^3*polylog(2,-d*f*x^(1/2))/d^2/f^2+12*b^2*n^2* 
(a+b*ln(c*x^n))*polylog(2,-d*f*x^(1/2))/d^2/f^2-6*b*n*(a+b*ln(c*x^n))^2*po 
lylog(2,-d*f*x^(1/2))/d^2/f^2-24*b^3*n^3*polylog(3,-d*f*x^(1/2))/d^2/f^2+2 
4*b^2*n^2*(a+b*ln(c*x^n))*polylog(3,-d*f*x^(1/2))/d^2/f^2-48*b^3*n^3*polyl 
og(4,-d*f*x^(1/2))/d^2/f^2
 

Mathematica [A] (verified)

Time = 0.61 (sec) , antiderivative size = 986, normalized size of antiderivative = 1.63 \[ \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx =\text {Too large to display} \] Input:

Integrate[Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n])^3,x]
 

Output:

-1/2*(-2*a^3*d*f*Sqrt[x] + 18*a^2*b*d*f*n*Sqrt[x] - 84*a*b^2*d*f*n^2*Sqrt[ 
x] + 180*b^3*d*f*n^3*Sqrt[x] + a^3*d^2*f^2*x - 6*a^2*b*d^2*f^2*n*x + 18*a* 
b^2*d^2*f^2*n^2*x - 24*b^3*d^2*f^2*n^3*x + 2*a^3*Log[1 + d*f*Sqrt[x]] - 6* 
a^2*b*n*Log[1 + d*f*Sqrt[x]] + 12*a*b^2*n^2*Log[1 + d*f*Sqrt[x]] - 12*b^3* 
n^3*Log[1 + d*f*Sqrt[x]] - 2*a^3*d^2*f^2*x*Log[1 + d*f*Sqrt[x]] + 6*a^2*b* 
d^2*f^2*n*x*Log[1 + d*f*Sqrt[x]] - 12*a*b^2*d^2*f^2*n^2*x*Log[1 + d*f*Sqrt 
[x]] + 12*b^3*d^2*f^2*n^3*x*Log[1 + d*f*Sqrt[x]] - 6*a^2*b*d*f*Sqrt[x]*Log 
[c*x^n] + 36*a*b^2*d*f*n*Sqrt[x]*Log[c*x^n] - 84*b^3*d*f*n^2*Sqrt[x]*Log[c 
*x^n] + 3*a^2*b*d^2*f^2*x*Log[c*x^n] - 12*a*b^2*d^2*f^2*n*x*Log[c*x^n] + 1 
8*b^3*d^2*f^2*n^2*x*Log[c*x^n] + 6*a^2*b*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] - 
 12*a*b^2*n*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] + 12*b^3*n^2*Log[1 + d*f*Sqrt[ 
x]]*Log[c*x^n] - 6*a^2*b*d^2*f^2*x*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] + 12*a* 
b^2*d^2*f^2*n*x*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] - 12*b^3*d^2*f^2*n^2*x*Log 
[1 + d*f*Sqrt[x]]*Log[c*x^n] - 6*a*b^2*d*f*Sqrt[x]*Log[c*x^n]^2 + 18*b^3*d 
*f*n*Sqrt[x]*Log[c*x^n]^2 + 3*a*b^2*d^2*f^2*x*Log[c*x^n]^2 - 6*b^3*d^2*f^2 
*n*x*Log[c*x^n]^2 + 6*a*b^2*Log[1 + d*f*Sqrt[x]]*Log[c*x^n]^2 - 6*b^3*n*Lo 
g[1 + d*f*Sqrt[x]]*Log[c*x^n]^2 - 6*a*b^2*d^2*f^2*x*Log[1 + d*f*Sqrt[x]]*L 
og[c*x^n]^2 + 6*b^3*d^2*f^2*n*x*Log[1 + d*f*Sqrt[x]]*Log[c*x^n]^2 - 2*b^3* 
d*f*Sqrt[x]*Log[c*x^n]^3 + b^3*d^2*f^2*x*Log[c*x^n]^3 + 2*b^3*Log[1 + d*f* 
Sqrt[x]]*Log[c*x^n]^3 - 2*b^3*d^2*f^2*x*Log[1 + d*f*Sqrt[x]]*Log[c*x^n]...
 

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 567, normalized size of antiderivative = 0.94, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2817, 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.

\(\displaystyle \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx\)

\(\Big \downarrow \) 2817

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

\(\Big \downarrow \) 2009

\(\displaystyle -3 b n \left (-\frac {4 b n \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}-\frac {8 b n \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}+\frac {2 \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2 f^2}+\frac {2 b n \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2 f^2}-2 b n x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {14 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d f}+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {3 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}+b n x \left (a+b \log \left (c x^n\right )\right )-x \left (a+b \log \left (c x^n\right )\right )^2+2 a b n x+2 b^2 n x \log \left (c x^n\right )+\frac {4 b^2 n^2 \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )}{d^2 f^2}+\frac {8 b^2 n^2 \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right )}{d^2 f^2}+\frac {16 b^2 n^2 \operatorname {PolyLog}\left (4,-d f \sqrt {x}\right )}{d^2 f^2}-\frac {2 b^2 n^2 \log \left (d f \sqrt {x}+1\right )}{d^2 f^2}+\frac {30 b^2 n^2 \sqrt {x}}{d f}+2 b^2 n^2 x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )-4 b^2 n^2 x\right )-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d^2 f^2}+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^3\)

Input:

Int[Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n])^3,x]
 

Output:

(Sqrt[x]*(a + b*Log[c*x^n])^3)/(d*f) - (x*(a + b*Log[c*x^n])^3)/2 + x*Log[ 
d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n])^3 - (Log[1 + d*f*Sqrt[x]]*(a + 
b*Log[c*x^n])^3)/(d^2*f^2) - 3*b*n*((30*b^2*n^2*Sqrt[x])/(d*f) + 2*a*b*n*x 
 - 4*b^2*n^2*x + 2*b^2*n^2*x*Log[d*(d^(-1) + f*Sqrt[x])] - (2*b^2*n^2*Log[ 
1 + d*f*Sqrt[x]])/(d^2*f^2) + 2*b^2*n*x*Log[c*x^n] - (14*b*n*Sqrt[x]*(a + 
b*Log[c*x^n]))/(d*f) + b*n*x*(a + b*Log[c*x^n]) - 2*b*n*x*Log[d*(d^(-1) + 
f*Sqrt[x])]*(a + b*Log[c*x^n]) + (2*b*n*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c* 
x^n]))/(d^2*f^2) + (3*Sqrt[x]*(a + b*Log[c*x^n])^2)/(d*f) - x*(a + b*Log[c 
*x^n])^2 + x*Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n])^2 - (Log[1 + d 
*f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/(d^2*f^2) + (4*b^2*n^2*PolyLog[2, -(d*f* 
Sqrt[x])])/(d^2*f^2) - (4*b*n*(a + b*Log[c*x^n])*PolyLog[2, -(d*f*Sqrt[x]) 
])/(d^2*f^2) + (2*(a + b*Log[c*x^n])^2*PolyLog[2, -(d*f*Sqrt[x])])/(d^2*f^ 
2) + (8*b^2*n^2*PolyLog[3, -(d*f*Sqrt[x])])/(d^2*f^2) - (8*b*n*(a + b*Log[ 
c*x^n])*PolyLog[3, -(d*f*Sqrt[x])])/(d^2*f^2) + (16*b^2*n^2*PolyLog[4, -(d 
*f*Sqrt[x])])/(d^2*f^2))
 

Defintions of rubi rules used

rule 2009

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

rule 2817

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_. 
)]*(b_.))^(p_.), x_Symbol] :> With[{u = IntHide[Log[d*(e + f*x^m)^r], 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, r, m, n}, x] && IGtQ[p, 0] 
&& RationalQ[m] && (EqQ[p, 1] || (FractionQ[m] && IntegerQ[1/m]) || (EqQ[r, 
 1] && EqQ[m, 1] && EqQ[d*e, 1]))
 

Maple [F]

\[\int \ln \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{3}d x\]

Input:

int(ln(d*(1/d+f*x^(1/2)))*(a+b*ln(c*x^n))^3,x)
 

Output:

int(ln(d*(1/d+f*x^(1/2)))*(a+b*ln(c*x^n))^3,x)
 

Fricas [F]

\[ \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right ) \,d x } \] Input:

integrate(log(d*(1/d+f*x^(1/2)))*(a+b*log(c*x^n))^3,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(d*f*sqrt(x) + 1), x)
 

Sympy [F(-1)]

Timed out. \[ \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx=\text {Timed out} \] Input:

integrate(ln(d*(1/d+f*x**(1/2)))*(a+b*ln(c*x**n))**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right ) \,d x } \] Input:

integrate(log(d*(1/d+f*x^(1/2)))*(a+b*log(c*x^n))^3,x, algorithm="maxima")
 

Output:

(b^3*x*log(x^n)^3 - 3*(b^3*(n - log(c)) - a*b^2)*x*log(x^n)^2 + 3*((2*n^2 
- 2*n*log(c) + log(c)^2)*b^3 - 2*a*b^2*(n - log(c)) + a^2*b)*x*log(x^n) + 
(3*(2*n^2 - 2*n*log(c) + log(c)^2)*a*b^2 - (6*n^3 - 6*n^2*log(c) + 3*n*log 
(c)^2 - log(c)^3)*b^3 - 3*a^2*b*(n - log(c)) + a^3)*x)*log(d*f*sqrt(x) + 1 
) - 1/27*(9*b^3*d*f*x^2*log(x^n)^3 + 9*(3*a*b^2*d*f - (5*d*f*n - 3*d*f*log 
(c))*b^3)*x^2*log(x^n)^2 + 3*(9*a^2*b*d*f - 6*(5*d*f*n - 3*d*f*log(c))*a*b 
^2 + (38*d*f*n^2 - 30*d*f*n*log(c) + 9*d*f*log(c)^2)*b^3)*x^2*log(x^n) + ( 
9*a^3*d*f - 9*(5*d*f*n - 3*d*f*log(c))*a^2*b + 3*(38*d*f*n^2 - 30*d*f*n*lo 
g(c) + 9*d*f*log(c)^2)*a*b^2 - (130*d*f*n^3 - 114*d*f*n^2*log(c) + 45*d*f* 
n*log(c)^2 - 9*d*f*log(c)^3)*b^3)*x^2)/sqrt(x) + integrate(1/2*(b^3*d^2*f^ 
2*x*log(x^n)^3 + 3*(a*b^2*d^2*f^2 - (d^2*f^2*n - d^2*f^2*log(c))*b^3)*x*lo 
g(x^n)^2 + 3*(a^2*b*d^2*f^2 - 2*(d^2*f^2*n - d^2*f^2*log(c))*a*b^2 + (2*d^ 
2*f^2*n^2 - 2*d^2*f^2*n*log(c) + d^2*f^2*log(c)^2)*b^3)*x*log(x^n) + (a^3* 
d^2*f^2 - 3*(d^2*f^2*n - d^2*f^2*log(c))*a^2*b + 3*(2*d^2*f^2*n^2 - 2*d^2* 
f^2*n*log(c) + d^2*f^2*log(c)^2)*a*b^2 - (6*d^2*f^2*n^3 - 6*d^2*f^2*n^2*lo 
g(c) + 3*d^2*f^2*n*log(c)^2 - d^2*f^2*log(c)^3)*b^3)*x)/(d*f*sqrt(x) + 1), 
 x)
 

Giac [F]

\[ \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right ) \,d x } \] Input:

integrate(log(d*(1/d+f*x^(1/2)))*(a+b*log(c*x^n))^3,x, algorithm="giac")
 

Output:

integrate((b*log(c*x^n) + a)^3*log((f*sqrt(x) + 1/d)*d), x)
 

Mupad [F(-1)]

Timed out. \[ \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx=\int \ln \left (d\,\left (f\,\sqrt {x}+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3 \,d x \] Input:

int(log(d*(f*x^(1/2) + 1/d))*(a + b*log(c*x^n))^3,x)
 

Output:

int(log(d*(f*x^(1/2) + 1/d))*(a + b*log(c*x^n))^3, x)
 

Reduce [F]

\[ \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx =\text {Too large to display} \] Input:

int(log(d*(1/d+f*x^(1/2)))*(a+b*log(c*x^n))^3,x)
 

Output:

(8*sqrt(x)*log(x**n*c)**3*b**3*d*f*n + 24*sqrt(x)*log(x**n*c)**2*a*b**2*d* 
f*n - 72*sqrt(x)*log(x**n*c)**2*b**3*d*f*n**2 + 24*sqrt(x)*log(x**n*c)*a** 
2*b*d*f*n - 144*sqrt(x)*log(x**n*c)*a*b**2*d*f*n**2 + 336*sqrt(x)*log(x**n 
*c)*b**3*d*f*n**3 + 8*sqrt(x)*a**3*d*f*n - 72*sqrt(x)*a**2*b*d*f*n**2 + 33 
6*sqrt(x)*a*b**2*d*f*n**3 - 720*sqrt(x)*b**3*d*f*n**4 - 4*int(log(x**n*c)* 
*3/(d**2*f**2*x**2 - x),x)*b**3*n - 12*int(log(x**n*c)**2/(d**2*f**2*x**2 
- x),x)*a*b**2*n + 12*int(log(x**n*c)**2/(d**2*f**2*x**2 - x),x)*b**3*n**2 
 - 12*int(log(x**n*c)/(d**2*f**2*x**2 - x),x)*a**2*b*n + 24*int(log(x**n*c 
)/(d**2*f**2*x**2 - x),x)*a*b**2*n**2 - 24*int(log(x**n*c)/(d**2*f**2*x**2 
 - x),x)*b**3*n**3 + 4*int((sqrt(x)*log(x**n*c)**3)/(d**2*f**2*x**2 - x),x 
)*b**3*d*f*n + 12*int((sqrt(x)*log(x**n*c)**2)/(d**2*f**2*x**2 - x),x)*a*b 
**2*d*f*n - 12*int((sqrt(x)*log(x**n*c)**2)/(d**2*f**2*x**2 - x),x)*b**3*d 
*f*n**2 + 12*int((sqrt(x)*log(x**n*c))/(d**2*f**2*x**2 - x),x)*a**2*b*d*f* 
n - 24*int((sqrt(x)*log(x**n*c))/(d**2*f**2*x**2 - x),x)*a*b**2*d*f*n**2 + 
 24*int((sqrt(x)*log(x**n*c))/(d**2*f**2*x**2 - x),x)*b**3*d*f*n**3 + 8*lo 
g(sqrt(x)*d*f + 1)*log(x**n*c)**3*b**3*d**2*f**2*n*x + 24*log(sqrt(x)*d*f 
+ 1)*log(x**n*c)**2*a*b**2*d**2*f**2*n*x - 24*log(sqrt(x)*d*f + 1)*log(x** 
n*c)**2*b**3*d**2*f**2*n**2*x + 24*log(sqrt(x)*d*f + 1)*log(x**n*c)*a**2*b 
*d**2*f**2*n*x - 48*log(sqrt(x)*d*f + 1)*log(x**n*c)*a*b**2*d**2*f**2*n**2 
*x + 48*log(sqrt(x)*d*f + 1)*log(x**n*c)*b**3*d**2*f**2*n**3*x + 8*log(...