\(\int (a+b \log (c x^n)) \log (d (e+f x)^m) \, dx\) [79]

Optimal result

Integrand size = 21, antiderivative size = 117 \[ \int \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right ) \, dx=2 b m n x-m x \left (a+b \log \left (c x^n\right )\right )-\frac {b n (e+f x) \log \left (d (e+f x)^m\right )}{f}-\frac {b e n \log \left (-\frac {f x}{e}\right ) \log \left (d (e+f x)^m\right )}{f}+\frac {(e+f x) \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{f}-\frac {b e m n \operatorname {PolyLog}\left (2,1+\frac {f x}{e}\right )}{f} \] Output:

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.30 \[ \int \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right ) \, dx=\frac {-a f m x+2 b f m n x-b e m n \log (e+f x)-b e m n \log (x) \log (e+f x)+a e \log \left (d (e+f x)^m\right )+a f x \log \left (d (e+f x)^m\right )-b f n x \log \left (d (e+f x)^m\right )+b \log \left (c x^n\right ) \left (e m \log (e+f x)+f x \left (-m+\log \left (d (e+f x)^m\right )\right )\right )+b e m n \log (x) \log \left (1+\frac {f x}{e}\right )+b e m n \operatorname {PolyLog}\left (2,-\frac {f x}{e}\right )}{f} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.95, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2817, 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.

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

\(\Big \downarrow \) 2817

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

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

rule 2817

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_. 
)]*(b_.))^(p_.), x_Symbol] :> With[{u = IntHide[Log[d*(e + f*x^m)^r], x]}, 
Simp[(a + b*Log[c*x^n])^p   u, x] - Simp[b*n*p   Int[(a + b*Log[c*x^n])^(p 
- 1)/x   u, x], x]] /; FreeQ[{a, b, c, d, e, f, r, m, n}, x] && IGtQ[p, 0] 
&& RationalQ[m] && (EqQ[p, 1] || (FractionQ[m] && IntegerQ[1/m]) || (EqQ[r, 
 1] && EqQ[m, 1] && EqQ[d*e, 1]))
 

Maple [C] (warning: unable to verify)

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

Time = 10.82 (sec) , antiderivative size = 686, normalized size of antiderivative = 5.86

method	result	size
risch	\(\left (b x \ln \left (x^{n}\right )+\frac {x \left (i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+2 b \ln \left (c \right )-2 n b +2 a \right )}{2}\right ) \ln \left (\left (f x +e \right )^{m}\right )+\left (-\frac {i \pi \,\operatorname {csgn}\left (i d \right ) \operatorname {csgn}\left (i \left (f x +e \right )^{m}\right ) \operatorname {csgn}\left (i d \left (f x +e \right )^{m}\right )}{4}+\frac {i \pi \,\operatorname {csgn}\left (i d \right ) \operatorname {csgn}\left (i d \left (f x +e \right )^{m}\right )^{2}}{4}+\frac {i \pi \,\operatorname {csgn}\left (i \left (f x +e \right )^{m}\right ) \operatorname {csgn}\left (i d \left (f x +e \right )^{m}\right )^{2}}{4}-\frac {i \pi \operatorname {csgn}\left (i d \left (f x +e \right )^{m}\right )^{3}}{4}+\frac {\ln \left (d \right )}{2}\right ) \left (i \pi b x \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b x \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )+2 a x +2 \ln \left (c \right ) b x +2 b x \ln \left (x^{n}\right )-2 b n x -i \pi b x \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-i \pi b x \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )\right )+\frac {i m x \pi b \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i m x \pi b \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+\frac {i m e \ln \left (f x +e \right ) \pi b \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{2 f}-\frac {i m e \ln \left (f x +e \right ) \pi b \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2 f}-x \ln \left (c \right ) b m +2 b m n x -a m x -\frac {i m x \pi b \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i m e \ln \left (f x +e \right ) \pi b \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{2 f}+\frac {i m e \ln \left (f x +e \right ) \pi b \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2 f}+\frac {i m x \pi b \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+\frac {m e \ln \left (f x +e \right ) b \ln \left (c \right )}{f}-\frac {m b n e \ln \left (f x +e \right )}{f}+\frac {a m e \ln \left (f x +e \right )}{f}-m b \ln \left (x^{n}\right ) x +\frac {m b \ln \left (x^{n}\right ) e \ln \left (f x +e \right )}{f}+\frac {m b n e}{f}-\frac {m b n e \ln \left (f x +e \right ) \ln \left (-\frac {f x}{e}\right )}{f}-\frac {m b n e \operatorname {dilog}\left (-\frac {f x}{e}\right )}{f}\)	\(686\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.61 \[ \int \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right ) \, dx=\frac {{\left (\log \left (\frac {f x}{e} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {f x}{e}\right )\right )} b e m n}{f} + \frac {{\left (a e m - {\left (e m n - e m \log \left (c\right )\right )} b\right )} \log \left (f x + e\right )}{f} - \frac {b e m n \log \left (f x + e\right ) \log \left (x\right ) + {\left ({\left (f m - f \log \left (d\right )\right )} a - {\left (2 \, f m n - f n \log \left (d\right ) - {\left (f m - f \log \left (d\right )\right )} \log \left (c\right )\right )} b\right )} x - {\left (b f x \log \left (x^{n}\right ) - {\left ({\left (f n - f \log \left (c\right )\right )} b - a f\right )} x\right )} \log \left ({\left (f x + e\right )}^{m}\right ) - {\left (b e m \log \left (f x + e\right ) - {\left (f m - f \log \left (d\right )\right )} b x\right )} \log \left (x^{n}\right )}{f} \] Input:

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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