\(\int \log (f x^m) (a+b \log (c (d+e x)^n)) \, dx\) [361]

Optimal result

Integrand size = 21, antiderivative size = 99 \[ \int \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=2 b m n x-b n x \log \left (f x^m\right )-\frac {b d m n \log (d+e x)}{e}-x \left (m-\log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {b d n \log \left (f x^m\right ) \log \left (1+\frac {e x}{d}\right )}{e}+\frac {b d m n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e} \] Output:

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.17 \[ \int \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=\frac {\log \left (f x^m\right ) \left (b d n \log (d+e x)+e x \left (a-b n+b \log \left (c (d+e x)^n\right )\right )\right )-m \left (a e x-2 b e n x+b d n (1+\log (x)) \log (d+e x)+b e x \log \left (c (d+e x)^n\right )-b d n \log (x) \log \left (1+\frac {e x}{d}\right )\right )+b d m n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.13, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2869, 49, 2009, 2793, 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.

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

\(\Big \downarrow \) 2869

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

\(\Big \downarrow \) 49

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 2793

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 49

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] 
&& IGtQ[m, 0] && IGtQ[m + n + 2, 0]
 

rule 2009

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

rule 2793

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* 
(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[a + b*Log[c*x^n], 
 (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, 
 f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && Integer 
Q[r]))
 

rule 2869

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

Maple [C] (warning: unable to verify)

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

Time = 11.04 (sec) , antiderivative size = 657, normalized size of antiderivative = 6.64

method	result	size
risch	\(\left (b x \ln \left (x^{m}\right )+\frac {x b \left (-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 )-2 m \right )}{2}\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 (i \pi \,\operatorname {csgn}\left (i f \right ) \operatorname {csgn}\left (i f \,x^{m}\right )^{2} x +i \pi \,\operatorname {csgn}\left (i x^{m}\right ) \operatorname {csgn}\left (i f \,x^{m}\right )^{2} x +2 x \ln \left (f \right )-i \pi \operatorname {csgn}\left (i f \,x^{m}\right )^{3} x -i \pi \,\operatorname {csgn}\left (i f \right ) \operatorname {csgn}\left (i x^{m}\right ) \operatorname {csgn}\left (i f \,x^{m}\right ) x +2 x \ln \left (x^{m}\right )-2 x m \right )-\frac {i n b d \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 e}-\frac {i n b d \ln \left (e x +d \right ) \pi \operatorname {csgn}\left (i f \,x^{m}\right )^{3}}{2 e}+\frac {i n b d \ln \left (e x +d \right ) \pi \,\operatorname {csgn}\left (i f \right ) \operatorname {csgn}\left (i f \,x^{m}\right )^{2}}{2 e}-\frac {i n b x \pi \,\operatorname {csgn}\left (i x^{m}\right ) \operatorname {csgn}\left (i f \,x^{m}\right )^{2}}{2}-n b x \ln \left (f \right )+2 b m n x -\frac {i n b x \pi \,\operatorname {csgn}\left (i f \right ) \operatorname {csgn}\left (i f \,x^{m}\right )^{2}}{2}+\frac {i n b x \pi \operatorname {csgn}\left (i f \,x^{m}\right )^{3}}{2}+\frac {i n b d \ln \left (e x +d \right ) \pi \,\operatorname {csgn}\left (i x^{m}\right ) \operatorname {csgn}\left (i f \,x^{m}\right )^{2}}{2 e}+\frac {i n b x \pi \,\operatorname {csgn}\left (i f \right ) \operatorname {csgn}\left (i x^{m}\right ) \operatorname {csgn}\left (i f \,x^{m}\right )}{2}+\frac {n b d \ln \left (e x +d \right ) \ln \left (f \right )}{e}-\frac {b d m n \ln \left (e x +d \right )}{e}-n b \ln \left (x^{m}\right ) x +\frac {n b \ln \left (x^{m}\right ) d \ln \left (e x +d \right )}{e}+\frac {n b m d}{e}-\frac {n b m d \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e}-\frac {n b m d \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e}\)	\(657\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.40 \[ \int \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=-{\left (\frac {{\left (\log \left (e x + d\right ) \log \left (-\frac {e x + d}{d} + 1\right ) + {\rm Li}_2\left (\frac {e x + d}{d}\right )\right )} b d n}{e} + \frac {b d n \log \left (e x + d\right ) + b e x \log \left ({\left (e x + d\right )}^{n}\right ) - {\left ({\left (2 \, e n - e \log \left (c\right )\right )} b - a e\right )} x}{e}\right )} m - {\left (b e n {\left (\frac {x}{e} - \frac {d \log \left (e x + d\right )}{e^{2}}\right )} - b x \log \left ({\left (e x + d\right )}^{n} c\right ) - a x\right )} \log \left (f x^{m}\right ) \] Input:

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=\frac {-2 \left (\int \frac {\mathrm {log}\left (x^{m} f \right )}{e \,x^{2}+d x}d x \right ) b \,d^{2} m n +2 \,\mathrm {log}\left (\left (e x +d \right )^{n} c \right ) \mathrm {log}\left (x^{m} f \right ) b e m x -2 \,\mathrm {log}\left (\left (e x +d \right )^{n} c \right ) b d \,m^{2}-2 \,\mathrm {log}\left (\left (e x +d \right )^{n} c \right ) b e \,m^{2} x +\mathrm {log}\left (x^{m} f \right )^{2} b d n +2 \,\mathrm {log}\left (x^{m} f \right ) a e m x -2 \,\mathrm {log}\left (x^{m} f \right ) b e m n x -2 a e \,m^{2} x +4 b e \,m^{2} n x}{2 e m} \] Input:

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

Output:

( - 2*int(log(x**m*f)/(d*x + e*x**2),x)*b*d**2*m*n + 2*log((d + e*x)**n*c) 
*log(x**m*f)*b*e*m*x - 2*log((d + e*x)**n*c)*b*d*m**2 - 2*log((d + e*x)**n 
*c)*b*e*m**2*x + log(x**m*f)**2*b*d*n + 2*log(x**m*f)*a*e*m*x - 2*log(x**m 
*f)*b*e*m*n*x - 2*a*e*m**2*x + 4*b*e*m**2*n*x)/(2*e*m)