\(\int \frac {\log (f x^m) (a+b \log (c (d+e x)^n))}{x^2} \, dx\) [363]

Optimal result

Integrand size = 24, antiderivative size = 102 \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{x^2} \, dx=\frac {b e m n \log (x)}{d}-\frac {b e n \log \left (1+\frac {d}{e x}\right ) \log \left (f x^m\right )}{d}-\frac {b e m n \log (d+e x)}{d}-\left (\frac {m}{x}+\frac {\log \left (f x^m\right )}{x}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {b e m n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d} \] Output:

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

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.09 \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{x^2} \, dx=-\frac {b e m n x \log ^2(x)+2 \left (m+\log \left (f x^m\right )\right ) \left (a d+b e n x \log (d+e x)+b d \log \left (c (d+e x)^n\right )\right )-2 b e n x \log (x) \left (m+\log \left (f x^m\right )+m \log (d+e x)-m \log \left (1+\frac {e x}{d}\right )\right )+2 b e m n x \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{2 d x} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2873, 47, 14, 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.

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

\(\Big \downarrow \) 2873

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

\(\Big \downarrow \) 47

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

\(\Big \downarrow \) 14

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 2779

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 14

Int[(a_.)/(x_), x_Symbol] :> Simp[a*Log[x], x] /; FreeQ[a, x]
 

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 47

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c 
 - a*d)   Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d)   Int[1/(c + d*x), x 
], x] /; FreeQ[{a, b, c, d}, 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 2873

Int[Log[(f_.)*(x_)^(m_.)]*((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_ 
.))*((g_.)*(x_))^(q_.), x_Symbol] :> Simp[(-(g*(q + 1))^(-1))*(m*((g*x)^(q 
+ 1)/(q + 1)) - (g*x)^(q + 1)*Log[f*x^m])*(a + b*Log[c*(d + e*x)^n]), x] + 
(-Simp[b*e*(n/(g*(q + 1)))   Int[(g*x)^(q + 1)*(Log[f*x^m]/(d + e*x)), x], 
x] + Simp[b*e*m*(n/(g*(q + 1)^2))   Int[(g*x)^(q + 1)/(d + e*x), x], x]) /; 
 FreeQ[{a, b, c, d, e, f, g, m, n, q}, x] && NeQ[q, -1]
 

Maple [C] (warning: unable to verify)

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

Time = 11.25 (sec) , antiderivative size = 709, normalized size of antiderivative = 6.95

method	result	size
risch	\(\left (-\frac {b \ln \left (x^{m}\right )}{x}-\frac {-i \pi b \,\operatorname {csgn}\left (i f \right ) \operatorname {csgn}\left (i x^{m}\right ) \operatorname {csgn}\left (i f \,x^{m}\right )+i \pi b \,\operatorname {csgn}\left (i f \right ) \operatorname {csgn}\left (i f \,x^{m}\right )^{2}+i \pi b \,\operatorname {csgn}\left (i x^{m}\right ) \operatorname {csgn}\left (i f \,x^{m}\right )^{2}-i \pi b \operatorname {csgn}\left (i f \,x^{m}\right )^{3}+2 b \ln \left (f \right )+2 b m}{2 x}\right ) \ln \left (\left (e x +d \right )^{n}\right )+\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}}{4}-\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 )}{4}-\frac {i b \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}}{4}+\frac {i b \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{4}+\frac {b \ln \left (c \right )}{2}+\frac {a}{2}\right ) \left (-\frac {2 \ln \left (x^{m}\right )}{x}-\frac {2 m}{x}-\frac {-i \pi \,\operatorname {csgn}\left (i f \right ) \operatorname {csgn}\left (i x^{m}\right ) \operatorname {csgn}\left (i f \,x^{m}\right )+i \pi \,\operatorname {csgn}\left (i f \right ) \operatorname {csgn}\left (i f \,x^{m}\right )^{2}+i \pi \,\operatorname {csgn}\left (i x^{m}\right ) \operatorname {csgn}\left (i f \,x^{m}\right )^{2}-i \pi \operatorname {csgn}\left (i f \,x^{m}\right )^{3}+2 \ln \left (f \right )}{x}\right )-\frac {i e n b \ln \left (x \right ) \pi \operatorname {csgn}\left (i f \,x^{m}\right )^{3}}{2 d}+\frac {i e n b \ln \left (x \right ) \pi \,\operatorname {csgn}\left (i f \right ) \operatorname {csgn}\left (i f \,x^{m}\right )^{2}}{2 d}-\frac {i e n b \ln \left (e x +d \right ) \pi \,\operatorname {csgn}\left (i x^{m}\right ) \operatorname {csgn}\left (i f \,x^{m}\right )^{2}}{2 d}+\frac {i e n b \ln \left (e x +d \right ) \pi \,\operatorname {csgn}\left (i f \right ) \operatorname {csgn}\left (i x^{m}\right ) \operatorname {csgn}\left (i f \,x^{m}\right )}{2 d}-\frac {e n b \ln \left (e x +d \right ) \ln \left (f \right )}{d}-\frac {b e m n \ln \left (e x +d \right )}{d}-\frac {i e n b \ln \left (e x +d \right ) \pi \,\operatorname {csgn}\left (i f \right ) \operatorname {csgn}\left (i f \,x^{m}\right )^{2}}{2 d}+\frac {i e n b \ln \left (e x +d \right ) \pi \operatorname {csgn}\left (i f \,x^{m}\right )^{3}}{2 d}+\frac {i e n b \ln \left (x \right ) \pi \,\operatorname {csgn}\left (i x^{m}\right ) \operatorname {csgn}\left (i f \,x^{m}\right )^{2}}{2 d}-\frac {i e n b \ln \left (x \right ) \pi \,\operatorname {csgn}\left (i f \right ) \operatorname {csgn}\left (i x^{m}\right ) \operatorname {csgn}\left (i f \,x^{m}\right )}{2 d}+\frac {e n b \ln \left (x \right ) \ln \left (f \right )}{d}+\frac {b e m n \ln \left (x \right )}{d}-\frac {e n b \ln \left (x^{m}\right ) \ln \left (e x +d \right )}{d}+\frac {e n b \ln \left (x^{m}\right ) \ln \left (x \right )}{d}-\frac {e n b m \ln \left (x \right )^{2}}{2 d}+\frac {e n b m \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{d}+\frac {e n b m \operatorname {dilog}\left (-\frac {e x}{d}\right )}{d}\)	\(709\)

Input:

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

Output:

(-b/x*ln(x^m)-1/2*(-I*Pi*b*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)+I*Pi*b*csgn 
(I*f)*csgn(I*f*x^m)^2+I*Pi*b*csgn(I*x^m)*csgn(I*f*x^m)^2-I*Pi*b*csgn(I*f*x 
^m)^3+2*b*ln(f)+2*b*m)/x)*ln((e*x+d)^n)+(1/4*I*b*Pi*csgn(I*(e*x+d)^n)*csgn 
(I*c*(e*x+d)^n)^2-1/4*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*csgn(I* 
c)-1/4*I*b*Pi*csgn(I*c*(e*x+d)^n)^3+1/4*I*b*Pi*csgn(I*c*(e*x+d)^n)^2*csgn( 
I*c)+1/2*b*ln(c)+1/2*a)*(-2*ln(x^m)/x-2*m/x-(-I*Pi*csgn(I*f)*csgn(I*x^m)*c 
sgn(I*f*x^m)+I*Pi*csgn(I*f)*csgn(I*f*x^m)^2+I*Pi*csgn(I*x^m)*csgn(I*f*x^m) 
^2-I*Pi*csgn(I*f*x^m)^3+2*ln(f))/x)-1/2*I*e*n*b/d*ln(x)*Pi*csgn(I*f*x^m)^3 
+1/2*I*e*n*b/d*ln(x)*Pi*csgn(I*f)*csgn(I*f*x^m)^2-1/2*I*e*n*b/d*ln(e*x+d)* 
Pi*csgn(I*x^m)*csgn(I*f*x^m)^2+1/2*I*e*n*b/d*ln(e*x+d)*Pi*csgn(I*f)*csgn(I 
*x^m)*csgn(I*f*x^m)-e*n*b/d*ln(e*x+d)*ln(f)-b*e*m*n*ln(e*x+d)/d-1/2*I*e*n* 
b/d*ln(e*x+d)*Pi*csgn(I*f)*csgn(I*f*x^m)^2+1/2*I*e*n*b/d*ln(e*x+d)*Pi*csgn 
(I*f*x^m)^3+1/2*I*e*n*b/d*ln(x)*Pi*csgn(I*x^m)*csgn(I*f*x^m)^2-1/2*I*e*n*b 
/d*ln(x)*Pi*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)+e*n*b/d*ln(x)*ln(f)+b*e*m* 
n*ln(x)/d-e*n*b*ln(x^m)/d*ln(e*x+d)+e*n*b*ln(x^m)/d*ln(x)-1/2*e*n*b*m/d*ln 
(x)^2+e*n*b*m/d*ln(e*x+d)*ln(-e*x/d)+e*n*b*m/d*dilog(-e*x/d)
 

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.59 \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{x^2} \, dx=-\frac {1}{2} \, {\left (\frac {2 \, {\left (\log \left (\frac {e x}{d} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {e x}{d}\right )\right )} b e n}{d} + \frac {2 \, b e n \log \left (e x + d\right )}{d} - \frac {2 \, b e n x \log \left (e x + d\right ) \log \left (x\right ) - b e n x \log \left (x\right )^{2} + 2 \, b e n x \log \left (x\right ) - 2 \, b d \log \left ({\left (e x + d\right )}^{n}\right ) - 2 \, b d \log \left (c\right ) - 2 \, a d}{d x}\right )} m - {\left (b e n {\left (\frac {\log \left (e x + d\right )}{d} - \frac {\log \left (x\right )}{d}\right )} + \frac {b \log \left ({\left (e x + d\right )}^{n} c\right )}{x} + \frac {a}{x}\right )} \log \left (f x^{m}\right ) \] Input:

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

Output:

-1/2*(2*(log(e*x/d + 1)*log(x) + dilog(-e*x/d))*b*e*n/d + 2*b*e*n*log(e*x 
+ d)/d - (2*b*e*n*x*log(e*x + d)*log(x) - b*e*n*x*log(x)^2 + 2*b*e*n*x*log 
(x) - 2*b*d*log((e*x + d)^n) - 2*b*d*log(c) - 2*a*d)/(d*x))*m - (b*e*n*(lo 
g(e*x + d)/d - log(x)/d) + b*log((e*x + d)^n*c)/x + a/x)*log(f*x^m)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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