\(\int \frac {(a+b \log (c x^n))^4 \log (d (\frac {1}{d}+f x^m))}{x} \, dx\) [70]

Optimal result

Integrand size = 28, antiderivative size = 137 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d \left (\frac {1}{d}+f x^m\right )\right )}{x} \, dx=-\frac {\left (a+b \log \left (c x^n\right )\right )^4 \operatorname {PolyLog}\left (2,-d f x^m\right )}{m}+\frac {4 b n \left (a+b \log \left (c x^n\right )\right )^3 \operatorname {PolyLog}\left (3,-d f x^m\right )}{m^2}-\frac {12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (4,-d f x^m\right )}{m^3}+\frac {24 b^3 n^3 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (5,-d f x^m\right )}{m^4}-\frac {24 b^4 n^4 \operatorname {PolyLog}\left (6,-d f x^m\right )}{m^5} \] Output:

-(a+b*ln(c*x^n))^4*polylog(2,-d*f*x^m)/m+4*b*n*(a+b*ln(c*x^n))^3*polylog(3 
,-d*f*x^m)/m^2-12*b^2*n^2*(a+b*ln(c*x^n))^2*polylog(4,-d*f*x^m)/m^3+24*b^3 
*n^3*(a+b*ln(c*x^n))*polylog(5,-d*f*x^m)/m^4-24*b^4*n^4*polylog(6,-d*f*x^m 
)/m^5
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1700\) vs. \(2(137)=274\).

Time = 0.78 (sec) , antiderivative size = 1700, normalized size of antiderivative = 12.41 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d \left (\frac {1}{d}+f x^m\right )\right )}{x} \, dx =\text {Too large to display} \] Input:

Integrate[((a + b*Log[c*x^n])^4*Log[d*(d^(-1) + f*x^m)])/x,x]
 

Output:

(-2*a^3*b*m*n*Log[x]^3)/3 + (3*a^2*b^2*m*n^2*Log[x]^4)/2 - (6*a*b^3*m*n^3* 
Log[x]^5)/5 + (b^4*m*n^4*Log[x]^6)/3 - 2*a^2*b^2*m*n*Log[x]^3*Log[c*x^n] + 
 3*a*b^3*m*n^2*Log[x]^4*Log[c*x^n] - (6*b^4*m*n^3*Log[x]^5*Log[c*x^n])/5 - 
 2*a*b^3*m*n*Log[x]^3*Log[c*x^n]^2 + (3*b^4*m*n^2*Log[x]^4*Log[c*x^n]^2)/2 
 - (2*b^4*m*n*Log[x]^3*Log[c*x^n]^3)/3 - 2*a^3*b*n*Log[x]^2*Log[1 + 1/(d*f 
*x^m)] + 4*a^2*b^2*n^2*Log[x]^3*Log[1 + 1/(d*f*x^m)] - 3*a*b^3*n^3*Log[x]^ 
4*Log[1 + 1/(d*f*x^m)] + (4*b^4*n^4*Log[x]^5*Log[1 + 1/(d*f*x^m)])/5 - 6*a 
^2*b^2*n*Log[x]^2*Log[c*x^n]*Log[1 + 1/(d*f*x^m)] + 8*a*b^3*n^2*Log[x]^3*L 
og[c*x^n]*Log[1 + 1/(d*f*x^m)] - 3*b^4*n^3*Log[x]^4*Log[c*x^n]*Log[1 + 1/( 
d*f*x^m)] - 6*a*b^3*n*Log[x]^2*Log[c*x^n]^2*Log[1 + 1/(d*f*x^m)] + 4*b^4*n 
^2*Log[x]^3*Log[c*x^n]^2*Log[1 + 1/(d*f*x^m)] - 2*b^4*n*Log[x]^2*Log[c*x^n 
]^3*Log[1 + 1/(d*f*x^m)] + 2*a^3*b*n*Log[x]^2*Log[1 + d*f*x^m] - 4*a^2*b^2 
*n^2*Log[x]^3*Log[1 + d*f*x^m] + 3*a*b^3*n^3*Log[x]^4*Log[1 + d*f*x^m] - ( 
4*b^4*n^4*Log[x]^5*Log[1 + d*f*x^m])/5 + (a^4*Log[-(d*f*x^m)]*Log[1 + d*f* 
x^m])/m - (4*a^3*b*n*Log[x]*Log[-(d*f*x^m)]*Log[1 + d*f*x^m])/m + (6*a^2*b 
^2*n^2*Log[x]^2*Log[-(d*f*x^m)]*Log[1 + d*f*x^m])/m - (4*a*b^3*n^3*Log[x]^ 
3*Log[-(d*f*x^m)]*Log[1 + d*f*x^m])/m + (b^4*n^4*Log[x]^4*Log[-(d*f*x^m)]* 
Log[1 + d*f*x^m])/m + 6*a^2*b^2*n*Log[x]^2*Log[c*x^n]*Log[1 + d*f*x^m] - 8 
*a*b^3*n^2*Log[x]^3*Log[c*x^n]*Log[1 + d*f*x^m] + 3*b^4*n^3*Log[x]^4*Log[c 
*x^n]*Log[1 + d*f*x^m] + (4*a^3*b*Log[-(d*f*x^m)]*Log[c*x^n]*Log[1 + d*...
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2821, 2830, 2830, 2830, 7143}

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.

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

\(\Big \downarrow \) 2821

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

\(\Big \downarrow \) 2830

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

\(\Big \downarrow \) 2830

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

\(\Big \downarrow \) 2830

\(\displaystyle \frac {4 b n \left (\frac {\operatorname {PolyLog}\left (3,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )^3}{m}-\frac {3 b n \left (\frac {\operatorname {PolyLog}\left (4,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )^2}{m}-\frac {2 b n \left (\frac {\operatorname {PolyLog}\left (5,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )}{m}-\frac {b n \int \frac {\operatorname {PolyLog}\left (5,-d f x^m\right )}{x}dx}{m}\right )}{m}\right )}{m}\right )}{m}-\frac {\operatorname {PolyLog}\left (2,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )^4}{m}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {4 b n \left (\frac {\operatorname {PolyLog}\left (3,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )^3}{m}-\frac {3 b n \left (\frac {\operatorname {PolyLog}\left (4,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )^2}{m}-\frac {2 b n \left (\frac {\operatorname {PolyLog}\left (5,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )}{m}-\frac {b n \operatorname {PolyLog}\left (6,-d f x^m\right )}{m^2}\right )}{m}\right )}{m}\right )}{m}-\frac {\operatorname {PolyLog}\left (2,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )^4}{m}\)

Input:

Int[((a + b*Log[c*x^n])^4*Log[d*(d^(-1) + f*x^m)])/x,x]
 

Output:

-(((a + b*Log[c*x^n])^4*PolyLog[2, -(d*f*x^m)])/m) + (4*b*n*(((a + b*Log[c 
*x^n])^3*PolyLog[3, -(d*f*x^m)])/m - (3*b*n*(((a + b*Log[c*x^n])^2*PolyLog 
[4, -(d*f*x^m)])/m - (2*b*n*(((a + b*Log[c*x^n])*PolyLog[5, -(d*f*x^m)])/m 
 - (b*n*PolyLog[6, -(d*f*x^m)])/m^2))/m))/m))/m
 

Defintions of rubi rules used

rule 2821

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b 
_.))^(p_.))/(x_), x_Symbol] :> Simp[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c 
*x^n])^p/m), x] + Simp[b*n*(p/m)   Int[PolyLog[2, (-d)*f*x^m]*((a + b*Log[c 
*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 
0] && EqQ[d*e, 1]
 

rule 2830

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_ 
.)])/(x_), x_Symbol] :> Simp[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^p/q) 
, x] - Simp[b*n*(p/q)   Int[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^(p - 
1)/x), x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [C] (warning: unable to verify)

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

Time = 0.04 (sec) , antiderivative size = 1968, normalized size of antiderivative = 14.36

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

Input:

int((a+b*ln(c*x^n))^4*ln(d*(1/d+f*x^m))/x,x)
 

Output:

4*b^4*n/m*dilog(d*f*x^m+1)*ln(x)*ln(x^n)^3+b^4*ln(1/d+f*x^m)*ln(x)*ln(x^n) 
^4-1/5*b^4/n*ln(1/d+f*x^m)*ln(x^n)^5+1/5*b^4*ln(d*(1/d+f*x^m))/n*ln(x^n)^5 
-1/5*b^4*n^4*ln(x)^5*ln(d*f*x^m+1)+1/5*b^4*n^4*ln(1/d+f*x^m)*ln(x)^5-b^4*l 
n(x)*ln(d*f*x^m+1)*ln(x^n)^4-b^4/m*dilog(d*f*x^m+1)*ln(x^n)^4-4*b^4*n/m*ln 
(x)*polylog(2,-d*f*x^m)*ln(x^n)^3+6*b^4*n^2/m*ln(x)^2*polylog(2,-d*f*x^m)* 
ln(x^n)^2-4*b^4*n^3/m*ln(x)^3*polylog(2,-d*f*x^m)*ln(x^n)-6*b^4*n^2/m*dilo 
g(d*f*x^m+1)*ln(x)^2*ln(x^n)^2+4*b^4*n^3/m*dilog(d*f*x^m+1)*ln(x)^3*ln(x^n 
)-1/16*(I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n 
)*csgn(I*c)-I*Pi*b*csgn(I*c*x^n)^3+I*Pi*b*csgn(I*c*x^n)^2*csgn(I*c)+2*b*ln 
(c)+2*a)^4/m*dilog(d*f*x^m+1)+1/2*(I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi 
*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-I*Pi*b*csgn(I*c*x^n)^3+I*Pi*b*csgn( 
I*c*x^n)^2*csgn(I*c)+2*b*ln(c)+2*a)^3*b*(1/2*ln(x)^2*n*ln(d*(1/d+f*x^m))+l 
n(d*(1/d+f*x^m))*ln(x)*(ln(x^n)-n*ln(x))-1/2*n*ln(x)^2*ln(d*f*x^m+1)-n/m*l 
n(x)*polylog(2,-d*f*x^m)+n/m^2*polylog(3,-d*f*x^m)-1/m*(ln(x^n)-n*ln(x))*d 
ilog(d*f*x^m+1)-(ln(x^n)-n*ln(x))*ln(x)*ln(d*f*x^m+1))+24*b^4*n^3/m^4*poly 
log(5,-d*f*x^m)*ln(x^n)-b^4*n^3*ln(1/d+f*x^m)*ln(x)^4*ln(x^n)+2*b^4*n^2*ln 
(1/d+f*x^m)*ln(x)^3*ln(x^n)^2-2*b^4*n*ln(1/d+f*x^m)*ln(x)^2*ln(x^n)^3-b^4* 
n^4/m*dilog(d*f*x^m+1)*ln(x)^4+b^4*n^3*ln(x)^4*ln(d*f*x^m+1)*ln(x^n)-2*b^4 
*n^2*ln(x)^3*ln(d*f*x^m+1)*ln(x^n)^2+2*b^4*n*ln(x)^2*ln(d*f*x^m+1)*ln(x^n) 
^3+b^4*n^4/m*ln(x)^4*polylog(2,-d*f*x^m)+4*b^4*n/m^2*polylog(3,-d*f*x^m...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 523 vs. \(2 (136) = 272\).

Time = 0.09 (sec) , antiderivative size = 523, normalized size of antiderivative = 3.82 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d \left (\frac {1}{d}+f x^m\right )\right )}{x} \, dx=-\frac {24 \, b^{4} n^{4} {\rm polylog}\left (6, -d f x^{m}\right ) + {\left (b^{4} m^{4} n^{4} \log \left (x\right )^{4} + b^{4} m^{4} \log \left (c\right )^{4} + 4 \, a b^{3} m^{4} \log \left (c\right )^{3} + 6 \, a^{2} b^{2} m^{4} \log \left (c\right )^{2} + 4 \, a^{3} b m^{4} \log \left (c\right ) + a^{4} m^{4} + 4 \, {\left (b^{4} m^{4} n^{3} \log \left (c\right ) + a b^{3} m^{4} n^{3}\right )} \log \left (x\right )^{3} + 6 \, {\left (b^{4} m^{4} n^{2} \log \left (c\right )^{2} + 2 \, a b^{3} m^{4} n^{2} \log \left (c\right ) + a^{2} b^{2} m^{4} n^{2}\right )} \log \left (x\right )^{2} + 4 \, {\left (b^{4} m^{4} n \log \left (c\right )^{3} + 3 \, a b^{3} m^{4} n \log \left (c\right )^{2} + 3 \, a^{2} b^{2} m^{4} n \log \left (c\right ) + a^{3} b m^{4} n\right )} \log \left (x\right )\right )} {\rm Li}_2\left (-d f x^{m}\right ) - 24 \, {\left (b^{4} m n^{4} \log \left (x\right ) + b^{4} m n^{3} \log \left (c\right ) + a b^{3} m n^{3}\right )} {\rm polylog}\left (5, -d f x^{m}\right ) + 12 \, {\left (b^{4} m^{2} n^{4} \log \left (x\right )^{2} + b^{4} m^{2} n^{2} \log \left (c\right )^{2} + 2 \, a b^{3} m^{2} n^{2} \log \left (c\right ) + a^{2} b^{2} m^{2} n^{2} + 2 \, {\left (b^{4} m^{2} n^{3} \log \left (c\right ) + a b^{3} m^{2} n^{3}\right )} \log \left (x\right )\right )} {\rm polylog}\left (4, -d f x^{m}\right ) - 4 \, {\left (b^{4} m^{3} n^{4} \log \left (x\right )^{3} + b^{4} m^{3} n \log \left (c\right )^{3} + 3 \, a b^{3} m^{3} n \log \left (c\right )^{2} + 3 \, a^{2} b^{2} m^{3} n \log \left (c\right ) + a^{3} b m^{3} n + 3 \, {\left (b^{4} m^{3} n^{3} \log \left (c\right ) + a b^{3} m^{3} n^{3}\right )} \log \left (x\right )^{2} + 3 \, {\left (b^{4} m^{3} n^{2} \log \left (c\right )^{2} + 2 \, a b^{3} m^{3} n^{2} \log \left (c\right ) + a^{2} b^{2} m^{3} n^{2}\right )} \log \left (x\right )\right )} {\rm polylog}\left (3, -d f x^{m}\right )}{m^{5}} \] Input:

integrate((a+b*log(c*x^n))^4*log(d*(1/d+f*x^m))/x,x, algorithm="fricas")
 

Output:

-(24*b^4*n^4*polylog(6, -d*f*x^m) + (b^4*m^4*n^4*log(x)^4 + b^4*m^4*log(c) 
^4 + 4*a*b^3*m^4*log(c)^3 + 6*a^2*b^2*m^4*log(c)^2 + 4*a^3*b*m^4*log(c) + 
a^4*m^4 + 4*(b^4*m^4*n^3*log(c) + a*b^3*m^4*n^3)*log(x)^3 + 6*(b^4*m^4*n^2 
*log(c)^2 + 2*a*b^3*m^4*n^2*log(c) + a^2*b^2*m^4*n^2)*log(x)^2 + 4*(b^4*m^ 
4*n*log(c)^3 + 3*a*b^3*m^4*n*log(c)^2 + 3*a^2*b^2*m^4*n*log(c) + a^3*b*m^4 
*n)*log(x))*dilog(-d*f*x^m) - 24*(b^4*m*n^4*log(x) + b^4*m*n^3*log(c) + a* 
b^3*m*n^3)*polylog(5, -d*f*x^m) + 12*(b^4*m^2*n^4*log(x)^2 + b^4*m^2*n^2*l 
og(c)^2 + 2*a*b^3*m^2*n^2*log(c) + a^2*b^2*m^2*n^2 + 2*(b^4*m^2*n^3*log(c) 
 + a*b^3*m^2*n^3)*log(x))*polylog(4, -d*f*x^m) - 4*(b^4*m^3*n^4*log(x)^3 + 
 b^4*m^3*n*log(c)^3 + 3*a*b^3*m^3*n*log(c)^2 + 3*a^2*b^2*m^3*n*log(c) + a^ 
3*b*m^3*n + 3*(b^4*m^3*n^3*log(c) + a*b^3*m^3*n^3)*log(x)^2 + 3*(b^4*m^3*n 
^2*log(c)^2 + 2*a*b^3*m^3*n^2*log(c) + a^2*b^2*m^3*n^2)*log(x))*polylog(3, 
 -d*f*x^m))/m^5
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d \left (\frac {1}{d}+f x^m\right )\right )}{x} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a+b*ln(c*x**n))**4*ln(d*(1/d+f*x**m))/x,x)
                                                                                    
                                                                                    
 

Output:

Exception raised: TypeError >> Invalid comparison of non-real zoo
 

Maxima [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d \left (\frac {1}{d}+f x^m\right )\right )}{x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{4} \log \left ({\left (f x^{m} + \frac {1}{d}\right )} d\right )}{x} \,d x } \] Input:

integrate((a+b*log(c*x^n))^4*log(d*(1/d+f*x^m))/x,x, algorithm="maxima")
 

Output:

1/5*(b^4*n^4*log(x)^5 + 5*b^4*log(x)*log(x^n)^4 - 5*(b^4*n^3*log(c) + a*b^ 
3*n^3)*log(x)^4 + 10*(b^4*n^2*log(c)^2 + 2*a*b^3*n^2*log(c) + a^2*b^2*n^2) 
*log(x)^3 - 10*(b^4*n*log(x)^2 - 2*(b^4*log(c) + a*b^3)*log(x))*log(x^n)^3 
 + 10*(b^4*n^2*log(x)^3 - 3*(b^4*n*log(c) + a*b^3*n)*log(x)^2 + 3*(b^4*log 
(c)^2 + 2*a*b^3*log(c) + a^2*b^2)*log(x))*log(x^n)^2 - 10*(b^4*n*log(c)^3 
+ 3*a*b^3*n*log(c)^2 + 3*a^2*b^2*n*log(c) + a^3*b*n)*log(x)^2 - 5*(b^4*n^3 
*log(x)^4 - 4*(b^4*n^2*log(c) + a*b^3*n^2)*log(x)^3 + 6*(b^4*n*log(c)^2 + 
2*a*b^3*n*log(c) + a^2*b^2*n)*log(x)^2 - 4*(b^4*log(c)^3 + 3*a*b^3*log(c)^ 
2 + 3*a^2*b^2*log(c) + a^3*b)*log(x))*log(x^n) + 5*(b^4*log(c)^4 + 4*a*b^3 
*log(c)^3 + 6*a^2*b^2*log(c)^2 + 4*a^3*b*log(c) + a^4)*log(x))*log(d*f*x^m 
 + 1) - integrate(1/5*(5*b^4*d*f*m*x^m*log(x)*log(x^n)^4 - 10*(b^4*d*f*m*n 
*log(x)^2 - 2*(b^4*d*f*m*log(c) + a*b^3*d*f*m)*log(x))*x^m*log(x^n)^3 + 10 
*(b^4*d*f*m*n^2*log(x)^3 - 3*(b^4*d*f*m*n*log(c) + a*b^3*d*f*m*n)*log(x)^2 
 + 3*(b^4*d*f*m*log(c)^2 + 2*a*b^3*d*f*m*log(c) + a^2*b^2*d*f*m)*log(x))*x 
^m*log(x^n)^2 - 5*(b^4*d*f*m*n^3*log(x)^4 - 4*(b^4*d*f*m*n^2*log(c) + a*b^ 
3*d*f*m*n^2)*log(x)^3 + 6*(b^4*d*f*m*n*log(c)^2 + 2*a*b^3*d*f*m*n*log(c) + 
 a^2*b^2*d*f*m*n)*log(x)^2 - 4*(b^4*d*f*m*log(c)^3 + 3*a*b^3*d*f*m*log(c)^ 
2 + 3*a^2*b^2*d*f*m*log(c) + a^3*b*d*f*m)*log(x))*x^m*log(x^n) + (b^4*d*f* 
m*n^4*log(x)^5 - 5*(b^4*d*f*m*n^3*log(c) + a*b^3*d*f*m*n^3)*log(x)^4 + 10* 
(b^4*d*f*m*n^2*log(c)^2 + 2*a*b^3*d*f*m*n^2*log(c) + a^2*b^2*d*f*m*n^2)...
 

Giac [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d \left (\frac {1}{d}+f x^m\right )\right )}{x} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{4} \log \left ({\left (f x^{m} + \frac {1}{d}\right )} d\right )}{x} \,d x } \] Input:

integrate((a+b*log(c*x^n))^4*log(d*(1/d+f*x^m))/x,x, algorithm="giac")
 

Output:

integrate((b*log(c*x^n) + a)^4*log((f*x^m + 1/d)*d)/x, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d \left (\frac {1}{d}+f x^m\right )\right )}{x} \, dx=\int \frac {\ln \left (d\,\left (f\,x^m+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^4}{x} \,d x \] Input:

int((log(d*(f*x^m + 1/d))*(a + b*log(c*x^n))^4)/x,x)
 

Output:

int((log(d*(f*x^m + 1/d))*(a + b*log(c*x^n))^4)/x, x)
 

Reduce [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d \left (\frac {1}{d}+f x^m\right )\right )}{x} \, dx=\frac {2 \left (\int \frac {\mathrm {log}\left (x^{m} d f +1\right )}{x^{m} d f x +x}d x \right ) a^{4} m +2 \left (\int \frac {\mathrm {log}\left (x^{m} d f +1\right ) \mathrm {log}\left (x^{n} c \right )^{4}}{x}d x \right ) b^{4} m +8 \left (\int \frac {\mathrm {log}\left (x^{m} d f +1\right ) \mathrm {log}\left (x^{n} c \right )^{3}}{x}d x \right ) a \,b^{3} m +12 \left (\int \frac {\mathrm {log}\left (x^{m} d f +1\right ) \mathrm {log}\left (x^{n} c \right )^{2}}{x}d x \right ) a^{2} b^{2} m +8 \left (\int \frac {\mathrm {log}\left (x^{m} d f +1\right ) \mathrm {log}\left (x^{n} c \right )}{x}d x \right ) a^{3} b m +\mathrm {log}\left (x^{m} d f +1\right )^{2} a^{4}}{2 m} \] Input:

int((a+b*log(c*x^n))^4*log(d*(1/d+f*x^m))/x,x)
 

Output:

(2*int(log(x**m*d*f + 1)/(x**m*d*f*x + x),x)*a**4*m + 2*int((log(x**m*d*f 
+ 1)*log(x**n*c)**4)/x,x)*b**4*m + 8*int((log(x**m*d*f + 1)*log(x**n*c)**3 
)/x,x)*a*b**3*m + 12*int((log(x**m*d*f + 1)*log(x**n*c)**2)/x,x)*a**2*b**2 
*m + 8*int((log(x**m*d*f + 1)*log(x**n*c))/x,x)*a**3*b*m + log(x**m*d*f + 
1)**2*a**4)/(2*m)