\(\int \frac {a+b \log (c x^n)}{x (d+e x)^2} \, dx\) [43]

Optimal result

Integrand size = 21, antiderivative size = 80 \[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx=-\frac {e x \left (a+b \log \left (c x^n\right )\right )}{d^2 (d+e x)}-\frac {\log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2}+\frac {b n \log (d+e x)}{d^2}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.20 \[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx=\frac {\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{d+e x}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{b n}-2 b n (\log (x)-\log (d+e x))-2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )-2 b n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{2 d^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.18, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {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.

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

\(\Big \downarrow \) 2789

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

\(\Big \downarrow \) 2751

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 2779

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

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 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]
 

Maple [C] (warning: unable to verify)

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

Time = 0.52 (sec) , antiderivative size = 229, normalized size of antiderivative = 2.86

method	result	size
risch	\(-\frac {b \ln \left (x^{n}\right ) \ln \left (e x +d \right )}{d^{2}}+\frac {b \ln \left (x^{n}\right )}{d \left (e x +d \right )}+\frac {b \ln \left (x^{n}\right ) \ln \left (x \right )}{d^{2}}-\frac {b n \ln \left (x \right )^{2}}{2 d^{2}}+\frac {b n \ln \left (e x +d \right )}{d^{2}}-\frac {b n \ln \left (x \right )}{d^{2}}+\frac {b n \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{d^{2}}+\frac {b n \operatorname {dilog}\left (-\frac {e x}{d}\right )}{d^{2}}+\left (\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+b \ln \left (c \right )+a \right ) \left (-\frac {\ln \left (e x +d \right )}{d^{2}}+\frac {1}{d \left (e x +d \right )}+\frac {\ln \left (x \right )}{d^{2}}\right )\)	\(229\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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