\(\int \frac {(a+b \log (c (d+e x)^n)) (f+g \log (c (d+e x)^n))}{x^2} \, dx\) [383]

Optimal result

Integrand size = 32, antiderivative size = 96 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{x^2} \, dx=-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{x}+\frac {e n \left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right ) \log \left (1-\frac {d}{d+e x}\right )}{d}-\frac {2 b e g n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+e x}\right )}{d} \] Output:

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.88 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{x^2} \, dx=-\frac {a f}{x}+\frac {b e f n \log (x)}{d}+\frac {a e g n \log (x)}{d}-\frac {b e f n \log (d+e x)}{d}-\frac {a e g n \log (d+e x)}{d}-\frac {b f \log \left (c (d+e x)^n\right )}{x}-\frac {a g \log \left (c (d+e x)^n\right )}{x}+\frac {2 b e g n \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )}{d}-\frac {b e g \log ^2\left (c (d+e x)^n\right )}{d}-\frac {b g \log ^2\left (c (d+e x)^n\right )}{x}+\frac {2 b e g n^2 \operatorname {PolyLog}\left (2,\frac {d+e x}{d}\right )}{d} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2883, 2858, 25, 27, 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.

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

\(\Big \downarrow \) 2883

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

\(\Big \downarrow \) 2858

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2779

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

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 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 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 = 0.99 (sec) , antiderivative size = 517, normalized size of antiderivative = 5.39

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 )}{x}+e n \left (-\frac {\ln \left (e x +d \right )}{d}+\frac {\ln \left (x \right )}{d}\right )\right )-\frac {\ln \left (\left (e x +d \right )^{n}\right )^{2} b g}{x}-\frac {2 b g e n \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (e x +d \right )}{d}+\frac {2 b g e n \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (x \right )}{d}-\frac {2 b g e \,n^{2} \operatorname {dilog}\left (\frac {e x +d}{d}\right )}{d}-\frac {2 b g e \,n^{2} \ln \left (x \right ) \ln \left (\frac {e x +d}{d}\right )}{d}+\frac {b g e \,n^{2} \ln \left (e x +d \right )^{2}}{d}-\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 )}{4 x}\)	\(517\)

Input:

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

Fricas [F]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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