\(\int \frac {a+b \log (c (d+e x)^n)}{x (f+g x)} \, dx\) [246]

Optimal result

Integrand size = 25, antiderivative size = 107 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{x (f+g x)} \, dx=\frac {\log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{f}-\frac {b n \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{f}+\frac {b n \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{f} \] Output:

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.79 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{x (f+g x)} \, dx=\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (\log \left (-\frac {e x}{d}\right )-\log \left (\frac {e (f+g x)}{e f-d g}\right )\right )-b n \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )+b n \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{f} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2863, 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.

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

\(\Big \downarrow \) 2863

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

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

rule 2863

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_)) 
^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a 
 + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a, b, c 
, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]
 

Maple [C] (warning: unable to verify)

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

Time = 1.26 (sec) , antiderivative size = 276, normalized size of antiderivative = 2.58

method	result	size
risch	\(-\frac {b \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (g x +f \right )}{f}+\frac {b \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (x \right )}{f}-\frac {b n \operatorname {dilog}\left (\frac {e x +d}{d}\right )}{f}-\frac {b n \ln \left (x \right ) \ln \left (\frac {e x +d}{d}\right )}{f}+\frac {b n \operatorname {dilog}\left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{f}+\frac {b n \ln \left (g x +f \right ) \ln \left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{f}+\left (\frac {i b \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{2}-\frac {i b \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}}{2}+\frac {i b \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+b \ln \left (c \right )+a \right ) \left (-\frac {\ln \left (g x +f \right )}{f}+\frac {\ln \left (x \right )}{f}\right )\)	\(276\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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