\(\int \frac {a+b \log (c (d+e x)^n)}{(f+\frac {g}{x}) x} \, dx\) [306]

Optimal result

Integrand size = 27, antiderivative size = 63 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (f+\frac {g}{x}\right ) x} \, dx=\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (-\frac {e (g+f x)}{d f-e g}\right )}{f}+\frac {b n \operatorname {PolyLog}\left (2,\frac {f (d+e x)}{d f-e g}\right )}{f} \] Output:

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.98 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (f+\frac {g}{x}\right ) x} \, dx=\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (g+f x)}{-d f+e g}\right )}{f}+\frac {b n \operatorname {PolyLog}\left (2,\frac {f (d+e x)}{d f-e g}\right )}{f} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2005, 2841, 2840, 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.

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

\(\Big \downarrow \) 2005

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

\(\Big \downarrow \) 2841

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

\(\Big \downarrow \) 2840

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2005

Int[(Fx_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[x^(m 
+ n*p)*(b + a/x^n)^p*Fx, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && Neg 
Q[n]
 

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 2840

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_ 
Symbol] :> Simp[1/g   Subst[Int[(a + b*Log[1 + c*e*(x/g)])/x, x], x, f + g* 
x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g + c 
*(e*f - d*g), 0]
 

rule 2841

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

Maple [C] (warning: unable to verify)

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

Time = 0.78 (sec) , antiderivative size = 217, normalized size of antiderivative = 3.44

method	result	size
risch	\(\frac {b \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (f x +g \right )}{f}-\frac {b n \operatorname {dilog}\left (\frac {\left (f x +g \right ) e +d f -e g}{d f -e g}\right )}{f}-\frac {b n \ln \left (f x +g \right ) \ln \left (\frac {\left (f x +g \right ) e +d f -e g}{d f -e g}\right )}{f}+\frac {\left (\frac {i b \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{2}-\frac {i b \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}}{2}+\frac {i b \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+b \ln \left (c \right )+a \right ) \ln \left (f x +g \right )}{f}\)	\(217\)

Input:

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

Output:

b*ln((e*x+d)^n)*ln(f*x+g)/f-b/f*n*dilog(((f*x+g)*e+d*f-e*g)/(d*f-e*g))-b/f 
*n*ln(f*x+g)*ln(((f*x+g)*e+d*f-e*g)/(d*f-e*g))+(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)*ln(f*x+g)/f
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

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

Output:

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