\(\int \frac {(a+b \log (c (d+e x)^n))^3}{f+g x} \, dx\) [56]

Optimal result

Integrand size = 24, antiderivative size = 158 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{f+g x} \, dx=\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g}+\frac {3 b n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g}-\frac {6 b^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right )}{g}+\frac {6 b^3 n^3 \operatorname {PolyLog}\left (4,-\frac {g (d+e x)}{e f-d g}\right )}{g} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(335\) vs. \(2(158)=316\).

Time = 0.31 (sec) , antiderivative size = 335, normalized size of antiderivative = 2.12 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{f+g x} \, dx=\frac {\left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^3 \log (f+g x)+3 b n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \left (\log (d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )+\operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )\right )+6 b^2 n^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (\frac {1}{2} \log ^2(d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )+\log (d+e x) \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )-\operatorname {PolyLog}\left (3,\frac {g (d+e x)}{-e f+d g}\right )\right )+b^3 n^3 \left (\log ^3(d+e x) \log \left (\frac {e (f+g x)}{e f-d g}\right )+3 \log ^2(d+e x) \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )-6 \log (d+e x) \operatorname {PolyLog}\left (3,\frac {g (d+e x)}{-e f+d g}\right )+6 \operatorname {PolyLog}\left (4,\frac {g (d+e x)}{-e f+d g}\right )\right )}{g} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.94, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {2843, 2881, 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.

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

\(\Big \downarrow \) 2843

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

\(\Big \downarrow \) 2881

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

\(\Big \downarrow \) 2821

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

\(\Big \downarrow \) 2830

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

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

Defintions of rubi rules used

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 2843

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

rule 2881

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log 
[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Sym 
bol] :> Simp[1/e   Subst[Int[(k*(x/d))^r*(a + b*Log[c*x^n])^p*(f + g*Log[h* 
((e*i - d*j)/e + j*(x/e))^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, 
 f, g, h, i, j, k, l, n, p, r}, x] && EqQ[e*k - d*l, 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 = 4.47 (sec) , antiderivative size = 1396, normalized size of antiderivative = 8.84

method	result	size
risch	\(\text {Expression too large to display}\)	\(1396\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{f+g x} \, dx=\frac {4 \left (\int \frac {\mathrm {log}\left (\left (e x +d \right )^{n} c \right )^{3}}{e g \,x^{2}+d g x +e f x +d f}d x \right ) b^{3} d g n -4 \left (\int \frac {\mathrm {log}\left (\left (e x +d \right )^{n} c \right )^{3}}{e g \,x^{2}+d g x +e f x +d f}d x \right ) b^{3} e f n +12 \left (\int \frac {\mathrm {log}\left (\left (e x +d \right )^{n} c \right )^{2}}{e g \,x^{2}+d g x +e f x +d f}d x \right ) a \,b^{2} d g n -12 \left (\int \frac {\mathrm {log}\left (\left (e x +d \right )^{n} c \right )^{2}}{e g \,x^{2}+d g x +e f x +d f}d x \right ) a \,b^{2} e f n +12 \left (\int \frac {\mathrm {log}\left (\left (e x +d \right )^{n} c \right )}{e g \,x^{2}+d g x +e f x +d f}d x \right ) a^{2} b d g n -12 \left (\int \frac {\mathrm {log}\left (\left (e x +d \right )^{n} c \right )}{e g \,x^{2}+d g x +e f x +d f}d x \right ) a^{2} b e f n +4 \,\mathrm {log}\left (g x +f \right ) a^{3} n +\mathrm {log}\left (\left (e x +d \right )^{n} c \right )^{4} b^{3}+4 \mathrm {log}\left (\left (e x +d \right )^{n} c \right )^{3} a \,b^{2}+6 \mathrm {log}\left (\left (e x +d \right )^{n} c \right )^{2} a^{2} b}{4 g n} \] Input:

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

Output:

(4*int(log((d + e*x)**n*c)**3/(d*f + d*g*x + e*f*x + e*g*x**2),x)*b**3*d*g 
*n - 4*int(log((d + e*x)**n*c)**3/(d*f + d*g*x + e*f*x + e*g*x**2),x)*b**3 
*e*f*n + 12*int(log((d + e*x)**n*c)**2/(d*f + d*g*x + e*f*x + e*g*x**2),x) 
*a*b**2*d*g*n - 12*int(log((d + e*x)**n*c)**2/(d*f + d*g*x + e*f*x + e*g*x 
**2),x)*a*b**2*e*f*n + 12*int(log((d + e*x)**n*c)/(d*f + d*g*x + e*f*x + e 
*g*x**2),x)*a**2*b*d*g*n - 12*int(log((d + e*x)**n*c)/(d*f + d*g*x + e*f*x 
 + e*g*x**2),x)*a**2*b*e*f*n + 4*log(f + g*x)*a**3*n + log((d + e*x)**n*c) 
**4*b**3 + 4*log((d + e*x)**n*c)**3*a*b**2 + 6*log((d + e*x)**n*c)**2*a**2 
*b)/(4*g*n)