\(\int \frac {(a+b \log (c (d+e x)^n)) (f+g \log (c (d+e x)^n))}{x^3} \, dx\) [384]

Optimal result

Integrand size = 32, antiderivative size = 156 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{x^3} \, dx=\frac {b e^2 g n^2 \log (x)}{d^2}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{2 x^2}-\frac {e n (d+e x) \left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right )}{2 d^2 x}-\frac {e^2 n \left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right ) \log \left (1-\frac {d}{d+e x}\right )}{2 d^2}+\frac {b e^2 g n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+e x}\right )}{d^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.61 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{x^3} \, dx=-\frac {a f}{2 x^2}+\frac {1}{2} b e f n \left (-\frac {1}{d x}-\frac {e \log (x)}{d^2}+\frac {e \log (d+e x)}{d^2}\right )+\frac {1}{2} a e g n \left (-\frac {1}{d x}-\frac {e \log (x)}{d^2}+\frac {e \log (d+e x)}{d^2}\right )-\frac {b f \log \left (c (d+e x)^n\right )}{2 x^2}-\frac {a g \log \left (c (d+e x)^n\right )}{2 x^2}-\frac {b g \log ^2\left (c (d+e x)^n\right )}{2 x^2}+b e g n \left (\frac {e n \log (x)}{d^2}-\frac {e n \log (d+e x)}{d^2}-\frac {\log \left (c (d+e x)^n\right )}{d x}-\frac {e \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )}{d^2}+\frac {e \log ^2\left (c (d+e x)^n\right )}{2 d^2 n}-\frac {e n \operatorname {PolyLog}\left (2,\frac {d+e x}{d}\right )}{d^2}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.23 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2883, 2858, 27, 2789, 2751, 16, 2779, 2838}

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.

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

\(\Big \downarrow \) 2883

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

\(\Big \downarrow \) 2858

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2789

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

\(\Big \downarrow \) 2751

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 2779

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, 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 2751

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x 
_Symbol] :> Simp[x*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/d), x] - Simp[b* 
(n/d)   Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q, r}, 
x] && EqQ[r*(q + 1) + 1, 0]
 

rule 2779

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

rule 2789

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/ 
(x_), x_Symbol] :> Simp[1/d   Int[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x 
), x], x] - Simp[e/d   Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; Free 
Q[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 2858

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ 
.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e   Subst[In 
t[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x, d + 
e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - 
d*g, 0] && (IGtQ[p, 0] || IGtQ[r, 0]) && IntegerQ[2*r]
 

rule 2883

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + Log[(c_.) 
*((d_) + (e_.)*(x_))^(n_.)]*(g_.))*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)* 
(a + b*Log[c*(d + e*x)^n])*((f + g*Log[c*(d + e*x)^n])/(m + 1)), x] - Simp[ 
e*(n/(m + 1))   Int[(x^(m + 1)*(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, m}, x] && NeQ[m, -1]
 

Maple [C] (warning: unable to verify)

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

Time = 1.01 (sec) , antiderivative size = 589, normalized size of antiderivative = 3.78

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 (-\frac {\ln \left (\left (e x +d \right )^{n}\right )}{2 x^{2}}+\frac {e n \left (\frac {e \ln \left (e x +d \right )}{d^{2}}-\frac {1}{d x}-\frac {e \ln \left (x \right )}{d^{2}}\right )}{2}\right )-\frac {\ln \left (\left (e x +d \right )^{n}\right )^{2} b g}{2 x^{2}}+\frac {b g \,e^{2} n \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (e x +d \right )}{d^{2}}-\frac {b g e n \ln \left (\left (e x +d \right )^{n}\right )}{d x}-\frac {b g \,e^{2} n \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (x \right )}{d^{2}}-\frac {b g \,e^{2} n^{2} \ln \left (e x +d \right )^{2}}{2 d^{2}}-\frac {b g \,e^{2} n^{2} \ln \left (e x +d \right )}{d^{2}}+\frac {b \,e^{2} g \,n^{2} \ln \left (x \right )}{d^{2}}+\frac {b g \,e^{2} n^{2} \operatorname {dilog}\left (\frac {e x +d}{d}\right )}{d^{2}}+\frac {b g \,e^{2} n^{2} \ln \left (x \right ) \ln \left (\frac {e x +d}{d}\right )}{d^{2}}-\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 )}{8 x^{2}}\)	\(589\)

Input:

int((a+b*ln(c*(e*x+d)^n))*(f+g*ln(c*(e*x+d)^n))/x^3,x,method=_RETURNVERBOS 
E)
 

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)*(-1/2*ln((e*x+d)^n)/x^ 
2+1/2*e*n*(e/d^2*ln(e*x+d)-1/d/x-e/d^2*ln(x)))-1/2/x^2*ln((e*x+d)^n)^2*b*g 
+b*g*e^2*n*ln((e*x+d)^n)/d^2*ln(e*x+d)-b*g*e*n*ln((e*x+d)^n)/d/x-b*g*e^2*n 
*ln((e*x+d)^n)/d^2*ln(x)-1/2*b*g*e^2*n^2/d^2*ln(e*x+d)^2-b*g*e^2*n^2/d^2*l 
n(e*x+d)+b*e^2*g*n^2*ln(x)/d^2+b*g*e^2*n^2/d^2*dilog((e*x+d)/d)+b*g*e^2*n^ 
2/d^2*ln(x)*ln((e*x+d)/d)-1/8*(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*P 
i*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)/x^2
 

Fricas [F]

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

integrate((a+b*log(c*(e*x+d)^n))*(f+g*log(c*(e*x+d)^n))/x^3,x, algorithm=" 
fricas")
 

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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