\(\int \frac {(a+b \log (c (d+e x)^n)) (f+g \log (c (d+e x)^n))}{x} \, dx\) [382]

Optimal result

Integrand size = 32, antiderivative size = 158 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{x} \, dx=\log (x) \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )-\frac {\log (x) \left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right )^2}{4 b g}+\frac {\log \left (-\frac {e x}{d}\right ) \left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right )^2}{4 b g}+n \left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )-2 b g n^2 \operatorname {PolyLog}\left (3,1+\frac {e x}{d}\right ) \] Output:

ln(x)*(a+b*ln(c*(e*x+d)^n))*(f+g*ln(c*(e*x+d)^n))-1/4*ln(x)*(b*f+a*g+2*b*g 
*ln(c*(e*x+d)^n))^2/b/g+1/4*ln(-e*x/d)*(b*f+a*g+2*b*g*ln(c*(e*x+d)^n))^2/b 
/g+n*(b*f+a*g+2*b*g*ln(c*(e*x+d)^n))*polylog(2,1+e*x/d)-2*b*g*n^2*polylog( 
3,1+e*x/d)
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.44 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{x} \, dx=a f \log (x)+b f \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )+a g \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )+b g \log (x) \left (-n \log (d+e x)+\log \left (c (d+e x)^n\right )\right )^2+2 b g n \left (-n \log (d+e x)+\log \left (c (d+e x)^n\right )\right ) \left (\log (x) \left (\log (d+e x)-\log \left (1+\frac {e x}{d}\right )\right )-\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )\right )+b f n \operatorname {PolyLog}\left (2,\frac {d+e x}{d}\right )+a g n \operatorname {PolyLog}\left (2,\frac {d+e x}{d}\right )+2 b g n^2 \left (\frac {1}{2} \log ^2(d+e x) \log \left (1-\frac {d+e x}{d}\right )+\log (d+e x) \operatorname {PolyLog}\left (2,\frac {d+e x}{d}\right )-\operatorname {PolyLog}\left (3,\frac {d+e x}{d}\right )\right ) \] Input:

Integrate[((a + b*Log[c*(d + e*x)^n])*(f + g*Log[c*(d + e*x)^n]))/x,x]
 

Output:

a*f*Log[x] + b*f*Log[-((e*x)/d)]*Log[c*(d + e*x)^n] + a*g*Log[-((e*x)/d)]* 
Log[c*(d + e*x)^n] + b*g*Log[x]*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n])^2 
 + 2*b*g*n*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n])*(Log[x]*(Log[d + e*x] 
- Log[1 + (e*x)/d]) - PolyLog[2, -((e*x)/d)]) + b*f*n*PolyLog[2, (d + e*x) 
/d] + a*g*n*PolyLog[2, (d + e*x)/d] + 2*b*g*n^2*((Log[d + e*x]^2*Log[1 - ( 
d + e*x)/d])/2 + Log[d + e*x]*PolyLog[2, (d + e*x)/d] - PolyLog[3, (d + e* 
x)/d])
 

Rubi [A] (verified)

Time = 1.23 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.25, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2882, 2881, 2822, 25, 27, 2754, 2821, 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.

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

\(\Big \downarrow \) 2882

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

\(\Big \downarrow \) 2881

\(\displaystyle \log (x) \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )-n \int \frac {\left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {d+e x}{e}-\frac {d}{e}\right )}{d+e x}d(d+e x)\)

\(\Big \downarrow \) 2822

\(\displaystyle \log (x) \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )-n \left (\frac {\log \left (\frac {d+e x}{e}-\frac {d}{e}\right ) \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )^2}{4 b g n}-\frac {\int \frac {\left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right )^2}{x}d(d+e x)}{4 b e g n}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \log (x) \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )-n \left (\frac {\int -\frac {\left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right )^2}{x}d(d+e x)}{4 b e g n}+\frac {\log \left (\frac {d+e x}{e}-\frac {d}{e}\right ) \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )^2}{4 b g n}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \log (x) \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )-n \left (\frac {\int -\frac {\left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right )^2}{e x}d(d+e x)}{4 b g n}+\frac {\log \left (\frac {d+e x}{e}-\frac {d}{e}\right ) \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )^2}{4 b g n}\right )\)

\(\Big \downarrow \) 2754

\(\displaystyle \log (x) \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )-n \left (\frac {4 b g n \int \frac {\left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right ) \log \left (1-\frac {d+e x}{d}\right )}{d+e x}d(d+e x)-\log \left (1-\frac {d+e x}{d}\right ) \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )^2}{4 b g n}+\frac {\log \left (\frac {d+e x}{e}-\frac {d}{e}\right ) \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )^2}{4 b g n}\right )\)

\(\Big \downarrow \) 2821

\(\displaystyle \log (x) \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )-n \left (\frac {4 b g n \left (2 b g n \int \frac {\operatorname {PolyLog}\left (2,\frac {d+e x}{d}\right )}{d+e x}d(d+e x)-\operatorname {PolyLog}\left (2,\frac {d+e x}{d}\right ) \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )\right )-\log \left (1-\frac {d+e x}{d}\right ) \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )^2}{4 b g n}+\frac {\log \left (\frac {d+e x}{e}-\frac {d}{e}\right ) \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )^2}{4 b g n}\right )\)

\(\Big \downarrow \) 7143

\(\displaystyle \log (x) \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )-n \left (\frac {4 b g n \left (2 b g n \operatorname {PolyLog}\left (3,\frac {d+e x}{d}\right )-\operatorname {PolyLog}\left (2,\frac {d+e x}{d}\right ) \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )\right )-\log \left (1-\frac {d+e x}{d}\right ) \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )^2}{4 b g n}+\frac {\log \left (\frac {d+e x}{e}-\frac {d}{e}\right ) \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )^2}{4 b g n}\right )\)

Input:

Int[((a + b*Log[c*(d + e*x)^n])*(f + g*Log[c*(d + e*x)^n]))/x,x]
 

Output:

Log[x]*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[c*(d + e*x)^n]) - n*(((b*f + 
a*g + 2*b*g*Log[c*(d + e*x)^n])^2*Log[-(d/e) + (d + e*x)/e])/(4*b*g*n) + ( 
-((b*f + a*g + 2*b*g*Log[c*(d + e*x)^n])^2*Log[1 - (d + e*x)/d]) + 4*b*g*n 
*(-((b*f + a*g + 2*b*g*Log[c*(d + e*x)^n])*PolyLog[2, (d + e*x)/d]) + 2*b* 
g*n*PolyLog[3, (d + e*x)/d]))/(4*b*g*n))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2754

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symb 
ol] :> Simp[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^p/e), x] - Simp[b*n*(p/e) 
  Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, 
b, c, d, e, n}, x] && IGtQ[p, 0]
 

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 2822

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

rule 2881

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log 
[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Sym 
bol] :> Simp[1/e   Subst[Int[(k*(x/d))^r*(a + b*Log[c*x^n])^p*(f + g*Log[h* 
((e*i - d*j)/e + j*(x/e))^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, 
 f, g, h, i, j, k, l, n, p, r}, x] && EqQ[e*k - d*l, 0]
 

rule 2882

Int[(((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + Log[(c_. 
)*((d_) + (e_.)*(x_))^(n_.)]*(g_.)))/(x_), x_Symbol] :> Simp[Log[x]*(a + b* 
Log[c*(d + e*x)^n])*(f + g*Log[c*(d + e*x)^n]), x] - Simp[e*n   Int[(Log[x] 
*(b*f + a*g + 2*b*g*Log[c*(d + e*x)^n]))/(d + e*x), x], x] /; FreeQ[{a, b, 
c, d, e, f, g, n}, x]
 

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 = 1.83 (sec) , antiderivative size = 626, normalized size of antiderivative = 3.96

method	result	size
risch	\(\left (i \pi b g \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-i \pi b g \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right )-i \pi b g \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}+i \pi b g \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+2 b \ln \left (c \right ) g +a g +b f \right ) \left (\ln \left (\left (e x +d \right )^{n}\right ) \ln \left (x \right )-e n \left (\frac {\operatorname {dilog}\left (\frac {e x +d}{d}\right )}{e}+\frac {\ln \left (x \right ) \ln \left (\frac {e x +d}{d}\right )}{e}\right )\right )+\ln \left (e x +d \right )^{2} \ln \left (e x \right ) b g \,n^{2}+\ln \left (e x +d \right )^{2} \ln \left (1-\frac {e x +d}{d}\right ) b g \,n^{2}-2 \ln \left (e x +d \right )^{2} \ln \left (-\frac {e x}{d}\right ) b g \,n^{2}-2 \ln \left (e x +d \right ) \ln \left (e x \right ) \ln \left (\left (e x +d \right )^{n}\right ) b g n +2 \ln \left (e x +d \right ) \operatorname {polylog}\left (2, \frac {e x +d}{d}\right ) b g \,n^{2}-2 \ln \left (e x +d \right ) \operatorname {dilog}\left (-\frac {e x}{d}\right ) b g \,n^{2}+2 \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right ) \ln \left (\left (e x +d \right )^{n}\right ) b g n +\ln \left (e x \right ) \ln \left (\left (e x +d \right )^{n}\right )^{2} b g -2 \operatorname {polylog}\left (3, \frac {e x +d}{d}\right ) b g \,n^{2}+2 \operatorname {dilog}\left (-\frac {e x}{d}\right ) \ln \left (\left (e x +d \right )^{n}\right ) b g n +\frac {\left (i b \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-i b \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right )-i b \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}+i b \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+2 b \ln \left (c \right )+2 a \right ) \left (i g \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-i g \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right )-i g \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}+i g \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+2 g \ln \left (c \right )+2 f \right ) \ln \left (x \right )}{4}\)	\(626\)

Input:

int((a+b*ln(c*(e*x+d)^n))*(f+g*ln(c*(e*x+d)^n))/x,x,method=_RETURNVERBOSE)
 

Output:

(I*Pi*b*g*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*Pi*b*g*csgn(I*(e*x+d)^ 
n)*csgn(I*c*(e*x+d)^n)*csgn(I*c)-I*Pi*b*g*csgn(I*c*(e*x+d)^n)^3+I*Pi*b*g*c 
sgn(I*c*(e*x+d)^n)^2*csgn(I*c)+2*b*ln(c)*g+a*g+b*f)*(ln((e*x+d)^n)*ln(x)-e 
*n*(dilog((e*x+d)/d)/e+ln(x)*ln((e*x+d)/d)/e))+ln(e*x+d)^2*ln(e*x)*b*g*n^2 
+ln(e*x+d)^2*ln(1-(e*x+d)/d)*b*g*n^2-2*ln(e*x+d)^2*ln(-e*x/d)*b*g*n^2-2*ln 
(e*x+d)*ln(e*x)*ln((e*x+d)^n)*b*g*n+2*ln(e*x+d)*polylog(2,(e*x+d)/d)*b*g*n 
^2-2*ln(e*x+d)*dilog(-e*x/d)*b*g*n^2+2*ln(e*x+d)*ln(-e*x/d)*ln((e*x+d)^n)* 
b*g*n+ln(e*x)*ln((e*x+d)^n)^2*b*g-2*polylog(3,(e*x+d)/d)*b*g*n^2+2*dilog(- 
e*x/d)*ln((e*x+d)^n)*b*g*n+1/4*(I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^ 
n)^2-I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*csgn(I*c)-I*b*Pi*csgn(I* 
c*(e*x+d)^n)^3+I*b*Pi*csgn(I*c*(e*x+d)^n)^2*csgn(I*c)+2*b*ln(c)+2*a)*(I*g* 
Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*g*Pi*csgn(I*(e*x+d)^n)*csgn(I 
*c*(e*x+d)^n)*csgn(I*c)-I*g*Pi*csgn(I*c*(e*x+d)^n)^3+I*g*Pi*csgn(I*c*(e*x+ 
d)^n)^2*csgn(I*c)+2*g*ln(c)+2*f)*ln(x)
 

Fricas [F]

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

integrate((a+b*log(c*(e*x+d)^n))*(f+g*log(c*(e*x+d)^n))/x,x, algorithm="fr 
icas")
 

Output:

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

Sympy [F]

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

integrate((a+b*ln(c*(e*x+d)**n))*(f+g*ln(c*(e*x+d)**n))/x,x)
 

Output:

Integral((a + b*log(c*(d + e*x)**n))*(f + g*log(c*(d + e*x)**n))/x, x)
 

Maxima [F]

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

integrate((a+b*log(c*(e*x+d)^n))*(f+g*log(c*(e*x+d)^n))/x,x, algorithm="ma 
xima")
 

Output:

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

Giac [F]

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

integrate((a+b*log(c*(e*x+d)^n))*(f+g*log(c*(e*x+d)^n))/x,x, algorithm="gi 
ac")
 

Output:

integrate((b*log((e*x + d)^n*c) + a)*(g*log((e*x + d)^n*c) + f)/x, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{x} \, dx=\frac {6 \left (\int \frac {\mathrm {log}\left (\left (e x +d \right )^{n} c \right )^{2}}{e \,x^{2}+d x}d x \right ) b d g n +6 \left (\int \frac {\mathrm {log}\left (\left (e x +d \right )^{n} c \right )}{e \,x^{2}+d x}d x \right ) a d g n +6 \left (\int \frac {\mathrm {log}\left (\left (e x +d \right )^{n} c \right )}{e \,x^{2}+d x}d x \right ) b d f n +2 \mathrm {log}\left (\left (e x +d \right )^{n} c \right )^{3} b g +3 \mathrm {log}\left (\left (e x +d \right )^{n} c \right )^{2} a g +3 \mathrm {log}\left (\left (e x +d \right )^{n} c \right )^{2} b f +6 \,\mathrm {log}\left (x \right ) a f n}{6 n} \] Input:

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

Output:

(6*int(log((d + e*x)**n*c)**2/(d*x + e*x**2),x)*b*d*g*n + 6*int(log((d + e 
*x)**n*c)/(d*x + e*x**2),x)*a*d*g*n + 6*int(log((d + e*x)**n*c)/(d*x + e*x 
**2),x)*b*d*f*n + 2*log((d + e*x)**n*c)**3*b*g + 3*log((d + e*x)**n*c)**2* 
a*g + 3*log((d + e*x)**n*c)**2*b*f + 6*log(x)*a*f*n)/(6*n)