\(\int \frac {a+b \log (c (d+e x)^n)}{x^2 (f+g x)} \, dx\) [247]

Optimal result

Integrand size = 25, antiderivative size = 162 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{x^2 (f+g x)} \, dx=\frac {b e n \log (x)}{d f}-\frac {b e n \log (d+e x)}{d f}-\frac {a+b \log \left (c (d+e x)^n\right )}{f x}-\frac {g \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{f^2}+\frac {b g n \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{f^2}-\frac {b g n \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{f^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.87 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{x^2 (f+g x)} \, dx=\frac {\frac {b e f n (\log (x)-\log (d+e x))}{d}-\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{x}-g \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+g \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )+b g n \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )-b g n \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{f^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 162, 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.

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

\(\Big \downarrow \) 2863

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 = 1.62 (sec) , antiderivative size = 338, normalized size of antiderivative = 2.09

method	result	size
risch	\(\frac {b \ln \left (\left (e x +d \right )^{n}\right ) g \ln \left (g x +f \right )}{f^{2}}-\frac {b \ln \left (\left (e x +d \right )^{n}\right )}{f x}-\frac {b \ln \left (\left (e x +d \right )^{n}\right ) g \ln \left (x \right )}{f^{2}}-\frac {b n g \operatorname {dilog}\left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{f^{2}}-\frac {b n g \ln \left (g x +f \right ) \ln \left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{f^{2}}-\frac {b e n \ln \left (e x +d \right )}{d f}+\frac {b e n \ln \left (x \right )}{d f}+\frac {b n g \operatorname {dilog}\left (\frac {e x +d}{d}\right )}{f^{2}}+\frac {b n g \ln \left (x \right ) \ln \left (\frac {e x +d}{d}\right )}{f^{2}}+\left (\frac {i b \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{2}-\frac {i b \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}}{2}+\frac {i b \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+b \ln \left (c \right )+a \right ) \left (\frac {g \ln \left (g x +f \right )}{f^{2}}-\frac {1}{f x}-\frac {g \ln \left (x \right )}{f^{2}}\right )\)	\(338\)

Input:

int((a+b*ln(c*(e*x+d)^n))/x^2/(g*x+f),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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