\(\int \frac {(a+b \log (c x^n))^3}{x (d+e x)} \, dx\) [121]

Optimal result

Integrand size = 23, antiderivative size = 113 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)} \, dx=-\frac {\log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d}+\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {d}{e x}\right )}{d}+\frac {6 b^3 n^3 \operatorname {PolyLog}\left (4,-\frac {d}{e x}\right )}{d} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(243\) vs. \(2(113)=226\).

Time = 0.24 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.15 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)} \, dx=\frac {4 \log (x) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^3-4 \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^3 \log (d+e x)+6 b n \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^2 \left (\log ^2(x)-2 \left (\log (x) \log \left (1+\frac {e x}{d}\right )+\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )\right )\right )-4 b^2 n^2 \left (-a+b n \log (x)-b \log \left (c x^n\right )\right ) \left (\log ^2(x) \left (\log (x)-3 \log \left (1+\frac {e x}{d}\right )\right )-6 \log (x) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+6 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )\right )+b^3 n^3 \left (\log ^4(x)-4 \log ^3(x) \log \left (1+\frac {e x}{d}\right )-12 \log ^2(x) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+24 \log (x) \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )-24 \operatorname {PolyLog}\left (4,-\frac {e x}{d}\right )\right )}{4 d} \] Input:

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

Output:

(4*Log[x]*(a - b*n*Log[x] + b*Log[c*x^n])^3 - 4*(a - b*n*Log[x] + b*Log[c* 
x^n])^3*Log[d + e*x] + 6*b*n*(a - b*n*Log[x] + b*Log[c*x^n])^2*(Log[x]^2 - 
 2*(Log[x]*Log[1 + (e*x)/d] + PolyLog[2, -((e*x)/d)])) - 4*b^2*n^2*(-a + b 
*n*Log[x] - b*Log[c*x^n])*(Log[x]^2*(Log[x] - 3*Log[1 + (e*x)/d]) - 6*Log[ 
x]*PolyLog[2, -((e*x)/d)] + 6*PolyLog[3, -((e*x)/d)]) + b^3*n^3*(Log[x]^4 
- 4*Log[x]^3*Log[1 + (e*x)/d] - 12*Log[x]^2*PolyLog[2, -((e*x)/d)] + 24*Lo 
g[x]*PolyLog[3, -((e*x)/d)] - 24*PolyLog[4, -((e*x)/d)]))/(4*d)
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.92, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2779, 2821, 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.

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

\(\Big \downarrow \) 2779

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

\(\Big \downarrow \) 2821

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

\(\Big \downarrow \) 2830

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

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

Defintions of rubi rules used

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 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 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 = 0.98 (sec) , antiderivative size = 967, normalized size of antiderivative = 8.56

method	result	size
risch	\(\text {Expression too large to display}\)	\(967\)

Input:

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

Output:

-b^3*ln(x^n)^3/d*ln(e*x+d)+b^3*ln(x^n)^3/d*ln(x)-3/2*b^3*n/d*ln(x^n)^2*ln( 
x)^2+b^3/d*n^2*ln(x^n)*ln(x)^3-1/4*b^3/d*ln(x)^4*n^3+3*b^3/d*ln(x)^2*ln(e* 
x+d)*ln(-e*x/d)*n^3+3*b^3/d*ln(x)^2*dilog(-e*x/d)*n^3-6*b^3/d*ln(x)*ln(x^n 
)*ln(e*x+d)*ln(-e*x/d)*n^2-6*b^3/d*ln(x)*ln(x^n)*dilog(-e*x/d)*n^2+3*b^3*n 
/d*ln(x^n)^2*ln(e*x+d)*ln(-e*x/d)+3*b^3*n/d*ln(x^n)^2*dilog(-e*x/d)-2*b^3/ 
d*n^3*ln(e*x+d)*ln(x)^3+2*b^3/d*n^3*ln(x)^3*ln(1+e*x/d)+3*b^3/d*n^3*ln(x)^ 
2*polylog(2,-e*x/d)-6*b^3/d*n^3*polylog(4,-e*x/d)+3*b^3/d*n^2*ln(x)^2*ln(x 
^n)*ln(e*x+d)-3*b^3/d*n^2*ln(x)^2*ln(x^n)*ln(1+e*x/d)-6*b^3/d*n^2*ln(x)*ln 
(x^n)*polylog(2,-e*x/d)+6*b^3/d*n^2*ln(x^n)*polylog(3,-e*x/d)+1/8*(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*P 
i*b*csgn(I*c*x^n)^3+I*Pi*b*csgn(I*c*x^n)^2*csgn(I*c)+2*b*ln(c)+2*a)^3*(-1/ 
d*ln(e*x+d)+1/d*ln(x))+3/2*(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*a)*b^2*(-ln(x^n)^2/d*ln(e*x+d)+ln(x^n)^2/d*ln(x) 
-2*n*(1/2/d*ln(x^n)*ln(x)^2-1/6/d*ln(x)^3*n-1/d*((ln(x^n)-n*ln(x))*(dilog( 
-e*x/d)+ln(e*x+d)*ln(-e*x/d))+n*(1/2*ln(e*x+d)*ln(x)^2-1/2*ln(x)^2*ln(1+e* 
x/d)-ln(x)*polylog(2,-e*x/d)+polylog(3,-e*x/d)))))+3/4*(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*a)^2*b*(-ln(x^n)/d*l 
n(e*x+d)+ln(x^n)/d*ln(x)-n*(1/2/d*ln(x)^2-1/d*ln(e*x+d)*ln(-e*x/d)-1/d*...
 

Fricas [F]

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

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

Sympy [F]

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

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

Output:

Integral((a + b*log(c*x**n))**3/(x*(d + e*x)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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