\(\int \frac {(a+b \log (c x^n))^3 \log (d (e+f x)^m)}{x} \, dx\) [93]

Optimal result

Integrand size = 26, antiderivative size = 161 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d (e+f x)^m\right )}{4 b n}-\frac {m \left (a+b \log \left (c x^n\right )\right )^4 \log \left (1+\frac {f x}{e}\right )}{4 b n}-m \left (a+b \log \left (c x^n\right )\right )^3 \operatorname {PolyLog}\left (2,-\frac {f x}{e}\right )+3 b m n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (3,-\frac {f x}{e}\right )-6 b^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (4,-\frac {f x}{e}\right )+6 b^3 m n^3 \operatorname {PolyLog}\left (5,-\frac {f x}{e}\right ) \] Output:

1/4*(a+b*ln(c*x^n))^4*ln(d*(f*x+e)^m)/b/n-1/4*m*(a+b*ln(c*x^n))^4*ln(1+f*x 
/e)/b/n-m*(a+b*ln(c*x^n))^3*polylog(2,-f*x/e)+3*b*m*n*(a+b*ln(c*x^n))^2*po 
lylog(3,-f*x/e)-6*b^2*m*n^2*(a+b*ln(c*x^n))*polylog(4,-f*x/e)+6*b^3*m*n^3* 
polylog(5,-f*x/e)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(602\) vs. \(2(161)=322\).

Time = 0.31 (sec) , antiderivative size = 602, normalized size of antiderivative = 3.74 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x} \, dx=a^3 \log (x) \log \left (d (e+f x)^m\right )-\frac {3}{2} a^2 b n \log ^2(x) \log \left (d (e+f x)^m\right )+a b^2 n^2 \log ^3(x) \log \left (d (e+f x)^m\right )-\frac {1}{4} b^3 n^3 \log ^4(x) \log \left (d (e+f x)^m\right )+3 a^2 b \log (x) \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-3 a b^2 n \log ^2(x) \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+b^3 n^2 \log ^3(x) \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+3 a b^2 \log (x) \log ^2\left (c x^n\right ) \log \left (d (e+f x)^m\right )-\frac {3}{2} b^3 n \log ^2(x) \log ^2\left (c x^n\right ) \log \left (d (e+f x)^m\right )+b^3 \log (x) \log ^3\left (c x^n\right ) \log \left (d (e+f x)^m\right )-a^3 m \log (x) \log \left (1+\frac {f x}{e}\right )+\frac {3}{2} a^2 b m n \log ^2(x) \log \left (1+\frac {f x}{e}\right )-a b^2 m n^2 \log ^3(x) \log \left (1+\frac {f x}{e}\right )+\frac {1}{4} b^3 m n^3 \log ^4(x) \log \left (1+\frac {f x}{e}\right )-3 a^2 b m \log (x) \log \left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )+3 a b^2 m n \log ^2(x) \log \left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )-b^3 m n^2 \log ^3(x) \log \left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )-3 a b^2 m \log (x) \log ^2\left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )+\frac {3}{2} b^3 m n \log ^2(x) \log ^2\left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )-b^3 m \log (x) \log ^3\left (c x^n\right ) \log \left (1+\frac {f x}{e}\right )-m \left (a+b \log \left (c x^n\right )\right )^3 \operatorname {PolyLog}\left (2,-\frac {f x}{e}\right )+3 b m n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (3,-\frac {f x}{e}\right )-6 a b^2 m n^2 \operatorname {PolyLog}\left (4,-\frac {f x}{e}\right )-6 b^3 m n^2 \log \left (c x^n\right ) \operatorname {PolyLog}\left (4,-\frac {f x}{e}\right )+6 b^3 m n^3 \operatorname {PolyLog}\left (5,-\frac {f x}{e}\right ) \] Input:

Integrate[((a + b*Log[c*x^n])^3*Log[d*(e + f*x)^m])/x,x]
 

Output:

a^3*Log[x]*Log[d*(e + f*x)^m] - (3*a^2*b*n*Log[x]^2*Log[d*(e + f*x)^m])/2 
+ a*b^2*n^2*Log[x]^3*Log[d*(e + f*x)^m] - (b^3*n^3*Log[x]^4*Log[d*(e + f*x 
)^m])/4 + 3*a^2*b*Log[x]*Log[c*x^n]*Log[d*(e + f*x)^m] - 3*a*b^2*n*Log[x]^ 
2*Log[c*x^n]*Log[d*(e + f*x)^m] + b^3*n^2*Log[x]^3*Log[c*x^n]*Log[d*(e + f 
*x)^m] + 3*a*b^2*Log[x]*Log[c*x^n]^2*Log[d*(e + f*x)^m] - (3*b^3*n*Log[x]^ 
2*Log[c*x^n]^2*Log[d*(e + f*x)^m])/2 + b^3*Log[x]*Log[c*x^n]^3*Log[d*(e + 
f*x)^m] - a^3*m*Log[x]*Log[1 + (f*x)/e] + (3*a^2*b*m*n*Log[x]^2*Log[1 + (f 
*x)/e])/2 - a*b^2*m*n^2*Log[x]^3*Log[1 + (f*x)/e] + (b^3*m*n^3*Log[x]^4*Lo 
g[1 + (f*x)/e])/4 - 3*a^2*b*m*Log[x]*Log[c*x^n]*Log[1 + (f*x)/e] + 3*a*b^2 
*m*n*Log[x]^2*Log[c*x^n]*Log[1 + (f*x)/e] - b^3*m*n^2*Log[x]^3*Log[c*x^n]* 
Log[1 + (f*x)/e] - 3*a*b^2*m*Log[x]*Log[c*x^n]^2*Log[1 + (f*x)/e] + (3*b^3 
*m*n*Log[x]^2*Log[c*x^n]^2*Log[1 + (f*x)/e])/2 - b^3*m*Log[x]*Log[c*x^n]^3 
*Log[1 + (f*x)/e] - m*(a + b*Log[c*x^n])^3*PolyLog[2, -((f*x)/e)] + 3*b*m* 
n*(a + b*Log[c*x^n])^2*PolyLog[3, -((f*x)/e)] - 6*a*b^2*m*n^2*PolyLog[4, - 
((f*x)/e)] - 6*b^3*m*n^2*Log[c*x^n]*PolyLog[4, -((f*x)/e)] + 6*b^3*m*n^3*P 
olyLog[5, -((f*x)/e)]
 

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2822, 2754, 2821, 2830, 2830, 7143}

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.

\(\displaystyle \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x} \, dx\)

\(\Big \downarrow \) 2822

\(\displaystyle \frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d (e+f x)^m\right )}{4 b n}-\frac {f m \int \frac {\left (a+b \log \left (c x^n\right )\right )^4}{e+f x}dx}{4 b n}\)

\(\Big \downarrow \) 2754

\(\displaystyle \frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d (e+f x)^m\right )}{4 b n}-\frac {f m \left (\frac {\log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^4}{f}-\frac {4 b n \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (\frac {f x}{e}+1\right )}{x}dx}{f}\right )}{4 b n}\)

\(\Big \downarrow \) 2821

\(\displaystyle \frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d (e+f x)^m\right )}{4 b n}-\frac {f m \left (\frac {\log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^4}{f}-\frac {4 b n \left (3 b n \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {f x}{e}\right )}{x}dx-\operatorname {PolyLog}\left (2,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3\right )}{f}\right )}{4 b n}\)

\(\Big \downarrow \) 2830

\(\displaystyle \frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d (e+f x)^m\right )}{4 b n}-\frac {f m \left (\frac {\log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^4}{f}-\frac {4 b n \left (3 b n \left (\operatorname {PolyLog}\left (3,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2-2 b n \int \frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {f x}{e}\right )}{x}dx\right )-\operatorname {PolyLog}\left (2,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3\right )}{f}\right )}{4 b n}\)

\(\Big \downarrow \) 2830

\(\displaystyle \frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d (e+f x)^m\right )}{4 b n}-\frac {f m \left (\frac {\log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^4}{f}-\frac {4 b n \left (3 b n \left (\operatorname {PolyLog}\left (3,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2-2 b n \left (\operatorname {PolyLog}\left (4,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \int \frac {\operatorname {PolyLog}\left (4,-\frac {f x}{e}\right )}{x}dx\right )\right )-\operatorname {PolyLog}\left (2,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3\right )}{f}\right )}{4 b n}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d (e+f x)^m\right )}{4 b n}-\frac {f m \left (\frac {\log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^4}{f}-\frac {4 b n \left (3 b n \left (\operatorname {PolyLog}\left (3,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2-2 b n \left (\operatorname {PolyLog}\left (4,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \operatorname {PolyLog}\left (5,-\frac {f x}{e}\right )\right )\right )-\operatorname {PolyLog}\left (2,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3\right )}{f}\right )}{4 b n}\)

Input:

Int[((a + b*Log[c*x^n])^3*Log[d*(e + f*x)^m])/x,x]
 

Output:

((a + b*Log[c*x^n])^4*Log[d*(e + f*x)^m])/(4*b*n) - (f*m*(((a + b*Log[c*x^ 
n])^4*Log[1 + (f*x)/e])/f - (4*b*n*(-((a + b*Log[c*x^n])^3*PolyLog[2, -((f 
*x)/e)]) + 3*b*n*((a + b*Log[c*x^n])^2*PolyLog[3, -((f*x)/e)] - 2*b*n*((a 
+ b*Log[c*x^n])*PolyLog[4, -((f*x)/e)] - b*n*PolyLog[5, -((f*x)/e)]))))/f) 
)/(4*b*n)
 

Defintions of rubi rules used

rule 2754

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symb 
ol] :> Simp[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^p/e), x] - Simp[b*n*(p/e) 
  Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, 
b, c, d, e, n}, x] && IGtQ[p, 0]
 

rule 2821

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b 
_.))^(p_.))/(x_), x_Symbol] :> Simp[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c 
*x^n])^p/m), x] + Simp[b*n*(p/m)   Int[PolyLog[2, (-d)*f*x^m]*((a + b*Log[c 
*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 
0] && EqQ[d*e, 1]
 

rule 2822

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

rule 2830

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_ 
.)])/(x_), x_Symbol] :> Simp[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^p/q) 
, x] - Simp[b*n*(p/q)   Int[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^(p - 
1)/x), x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [C] (warning: unable to verify)

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

Time = 281.68 (sec) , antiderivative size = 15171, normalized size of antiderivative = 94.23

method	result	size
risch	\(\text {Expression too large to display}\)	\(15171\)

Input:

int((a+b*ln(c*x^n))^3*ln(d*(f*x+e)^m)/x,x,method=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [F]

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

integrate((a+b*log(c*x^n))^3*log(d*(f*x+e)^m)/x,x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

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

integrate((a+b*ln(c*x**n))**3*ln(d*(f*x+e)**m)/x,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((a+b*log(c*x^n))^3*log(d*(f*x+e)^m)/x,x, algorithm="maxima")
 

Output:

-1/4*(b^3*n^3*log(x)^4 - 4*b^3*log(x)*log(x^n)^3 - 4*(b^3*n^2*log(c) + a*b 
^2*n^2)*log(x)^3 + 6*(b^3*n*log(c)^2 + 2*a*b^2*n*log(c) + a^2*b*n)*log(x)^ 
2 + 6*(b^3*n*log(x)^2 - 2*(b^3*log(c) + a*b^2)*log(x))*log(x^n)^2 - 4*(b^3 
*n^2*log(x)^3 - 3*(b^3*n*log(c) + a*b^2*n)*log(x)^2 + 3*(b^3*log(c)^2 + 2* 
a*b^2*log(c) + a^2*b)*log(x))*log(x^n) - 4*(b^3*log(c)^3 + 3*a*b^2*log(c)^ 
2 + 3*a^2*b*log(c) + a^3)*log(x))*log((f*x + e)^m) - integrate(-1/4*(b^3*f 
*m*n^3*x*log(x)^4 + 4*b^3*e*log(c)^3*log(d) + 12*a*b^2*e*log(c)^2*log(d) + 
 12*a^2*b*e*log(c)*log(d) + 4*a^3*e*log(d) - 4*(b^3*f*m*n^2*log(c) + a*b^2 
*f*m*n^2)*x*log(x)^3 + 6*(b^3*f*m*n*log(c)^2 + 2*a*b^2*f*m*n*log(c) + a^2* 
b*f*m*n)*x*log(x)^2 - 4*(b^3*f*m*x*log(x) - b^3*f*x*log(d) - b^3*e*log(d)) 
*log(x^n)^3 - 4*(b^3*f*m*log(c)^3 + 3*a*b^2*f*m*log(c)^2 + 3*a^2*b*f*m*log 
(c) + a^3*f*m)*x*log(x) + 6*(b^3*f*m*n*x*log(x)^2 + 2*b^3*e*log(c)*log(d) 
+ 2*a*b^2*e*log(d) - 2*(b^3*f*m*log(c) + a*b^2*f*m)*x*log(x) + 2*(b^3*f*lo 
g(c)*log(d) + a*b^2*f*log(d))*x)*log(x^n)^2 + 4*(b^3*f*log(c)^3*log(d) + 3 
*a*b^2*f*log(c)^2*log(d) + 3*a^2*b*f*log(c)*log(d) + a^3*f*log(d))*x - 4*( 
b^3*f*m*n^2*x*log(x)^3 - 3*b^3*e*log(c)^2*log(d) - 6*a*b^2*e*log(c)*log(d) 
 - 3*a^2*b*e*log(d) - 3*(b^3*f*m*n*log(c) + a*b^2*f*m*n)*x*log(x)^2 + 3*(b 
^3*f*m*log(c)^2 + 2*a*b^2*f*m*log(c) + a^2*b*f*m)*x*log(x) - 3*(b^3*f*log( 
c)^2*log(d) + 2*a*b^2*f*log(c)*log(d) + a^2*b*f*log(d))*x)*log(x^n))/(f*x^ 
2 + e*x), x)
 

Giac [F]

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

integrate((a+b*log(c*x^n))^3*log(d*(f*x+e)^m)/x,x, algorithm="giac")
 

Output:

integrate((b*log(c*x^n) + a)^3*log((f*x + e)^m*d)/x, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x} \, dx=\frac {2 \left (\int \frac {\mathrm {log}\left (\left (f x +e \right )^{m} d \right )}{f \,x^{2}+e x}d x \right ) a^{3} e m +2 \left (\int \frac {\mathrm {log}\left (\left (f x +e \right )^{m} d \right ) \mathrm {log}\left (x^{n} c \right )^{3}}{x}d x \right ) b^{3} m +6 \left (\int \frac {\mathrm {log}\left (\left (f x +e \right )^{m} d \right ) \mathrm {log}\left (x^{n} c \right )^{2}}{x}d x \right ) a \,b^{2} m +6 \left (\int \frac {\mathrm {log}\left (\left (f x +e \right )^{m} d \right ) \mathrm {log}\left (x^{n} c \right )}{x}d x \right ) a^{2} b m +\mathrm {log}\left (\left (f x +e \right )^{m} d \right )^{2} a^{3}}{2 m} \] Input:

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

Output:

(2*int(log((e + f*x)**m*d)/(e*x + f*x**2),x)*a**3*e*m + 2*int((log((e + f* 
x)**m*d)*log(x**n*c)**3)/x,x)*b**3*m + 6*int((log((e + f*x)**m*d)*log(x**n 
*c)**2)/x,x)*a*b**2*m + 6*int((log((e + f*x)**m*d)*log(x**n*c))/x,x)*a**2* 
b*m + log((e + f*x)**m*d)**2*a**3)/(2*m)