\(\int \frac {(a+b \log (c (d+e x)^n))^3}{(f+g x)^3} \, dx\) [58]

Optimal result

Integrand size = 24, antiderivative size = 342 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x)^3} \, dx=-\frac {3 b e n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 (e f-d g)^2 (f+g x)}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 g (f+g x)^2}+\frac {3 b^2 e^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g (e f-d g)^2}-\frac {3 b e^2 n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (1+\frac {e f-d g}{g (d+e x)}\right )}{2 g (e f-d g)^2}+\frac {3 b^2 e^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e f-d g}{g (d+e x)}\right )}{g (e f-d g)^2}+\frac {3 b^3 e^2 n^3 \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g (e f-d g)^2}+\frac {3 b^3 e^2 n^3 \operatorname {PolyLog}\left (3,-\frac {e f-d g}{g (d+e x)}\right )}{g (e f-d g)^2} \] Output:

-3/2*b*e*n*(e*x+d)*(a+b*ln(c*(e*x+d)^n))^2/(-d*g+e*f)^2/(g*x+f)-1/2*(a+b*l 
n(c*(e*x+d)^n))^3/g/(g*x+f)^2+3*b^2*e^2*n^2*(a+b*ln(c*(e*x+d)^n))*ln(e*(g* 
x+f)/(-d*g+e*f))/g/(-d*g+e*f)^2-3/2*b*e^2*n*(a+b*ln(c*(e*x+d)^n))^2*ln(1+( 
-d*g+e*f)/g/(e*x+d))/g/(-d*g+e*f)^2+3*b^2*e^2*n^2*(a+b*ln(c*(e*x+d)^n))*po 
lylog(2,-(-d*g+e*f)/g/(e*x+d))/g/(-d*g+e*f)^2+3*b^3*e^2*n^3*polylog(2,-g*( 
e*x+d)/(-d*g+e*f))/g/(-d*g+e*f)^2+3*b^3*e^2*n^3*polylog(3,-(-d*g+e*f)/g/(e 
*x+d))/g/(-d*g+e*f)^2
 

Mathematica [A] (verified)

Time = 0.81 (sec) , antiderivative size = 620, normalized size of antiderivative = 1.81 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x)^3} \, dx=-\frac {-3 b e (e f-d g) n (f+g x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+3 b (e f-d g)^2 n \log (d+e x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2-3 b e^2 n (f+g x)^2 \log (d+e x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+(e f-d g)^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^3+3 b e^2 n (f+g x)^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \log (f+g x)+3 b^2 n^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (g (d+e x) (d g-e (2 f+g x)) \log ^2(d+e x)-2 e^2 (f+g x)^2 \log \left (\frac {e (f+g x)}{e f-d g}\right )+2 e (f+g x) \log (d+e x) \left (g (d+e x)+e (f+g x) \log \left (\frac {e (f+g x)}{e f-d g}\right )\right )+2 e^2 (f+g x)^2 \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )\right )+b^3 n^3 \left (g (d+e x) (d g-e (2 f+g x)) \log ^3(d+e x)+3 e (f+g x) \log ^2(d+e x) \left (g (d+e x)+e (f+g x) \log \left (\frac {e (f+g x)}{e f-d g}\right )\right )-6 e^2 (f+g x)^2 \log (d+e x) \left (\log \left (\frac {e (f+g x)}{e f-d g}\right )-\operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )\right )-6 e^2 (f+g x)^2 \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )-6 e^2 (f+g x)^2 \operatorname {PolyLog}\left (3,\frac {g (d+e x)}{-e f+d g}\right )\right )}{2 g (e f-d g)^2 (f+g x)^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 2.37 (sec) , antiderivative size = 325, normalized size of antiderivative = 0.95, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2845, 2858, 27, 2789, 2755, 2754, 2779, 2821, 2838, 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.

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

\(\Big \downarrow \) 2845

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

\(\Big \downarrow \) 2858

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2789

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

\(\Big \downarrow \) 2755

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

\(\Big \downarrow \) 2754

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

\(\Big \downarrow \) 2779

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

\(\Big \downarrow \) 2821

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 2755

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_))^2, x_Sy 
mbol] :> Simp[x*((a + b*Log[c*x^n])^p/(d*(d + e*x))), x] - Simp[b*n*(p/d) 
 Int[(a + b*Log[c*x^n])^(p - 1)/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, 
n, p}, x] && GtQ[p, 0]
 

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 2789

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/ 
(x_), x_Symbol] :> Simp[1/d   Int[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x 
), x], x] - Simp[e/d   Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; Free 
Q[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
 

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 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 2845

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_. 
)*(x_))^(q_.), x_Symbol] :> Simp[(f + g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^ 
n])^p/(g*(q + 1))), x] - Simp[b*e*n*(p/(g*(q + 1)))   Int[(f + g*x)^(q + 1) 
*((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, 
d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && In 
tegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))
 

rule 2858

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ 
.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e   Subst[In 
t[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x, d + 
e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - 
d*g, 0] && (IGtQ[p, 0] || IGtQ[r, 0]) && IntegerQ[2*r]
 

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 [F]

\[\int \frac {{\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{3}}{\left (g x +f \right )^{3}}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 24*int(log((d + e*x)**n*c)/(d*f**3 + 3*d*f**2*g*x + 3*d*f*g**2*x**2 + 
d*g**3*x**3 + e*f**3*x + 3*e*f**2*g*x**2 + 3*e*f*g**2*x**3 + e*g**3*x**4), 
x)*a*b**2*d**4*f**4*g**4*n - 48*int(log((d + e*x)**n*c)/(d*f**3 + 3*d*f**2 
*g*x + 3*d*f*g**2*x**2 + d*g**3*x**3 + e*f**3*x + 3*e*f**2*g*x**2 + 3*e*f* 
g**2*x**3 + e*g**3*x**4),x)*a*b**2*d**4*f**3*g**5*n*x - 24*int(log((d + e* 
x)**n*c)/(d*f**3 + 3*d*f**2*g*x + 3*d*f*g**2*x**2 + d*g**3*x**3 + e*f**3*x 
 + 3*e*f**2*g*x**2 + 3*e*f*g**2*x**3 + e*g**3*x**4),x)*a*b**2*d**4*f**2*g* 
*6*n*x**2 + 72*int(log((d + e*x)**n*c)/(d*f**3 + 3*d*f**2*g*x + 3*d*f*g**2 
*x**2 + d*g**3*x**3 + e*f**3*x + 3*e*f**2*g*x**2 + 3*e*f*g**2*x**3 + e*g** 
3*x**4),x)*a*b**2*d**3*e*f**5*g**3*n + 144*int(log((d + e*x)**n*c)/(d*f**3 
 + 3*d*f**2*g*x + 3*d*f*g**2*x**2 + d*g**3*x**3 + e*f**3*x + 3*e*f**2*g*x* 
*2 + 3*e*f*g**2*x**3 + e*g**3*x**4),x)*a*b**2*d**3*e*f**4*g**4*n*x + 72*in 
t(log((d + e*x)**n*c)/(d*f**3 + 3*d*f**2*g*x + 3*d*f*g**2*x**2 + d*g**3*x* 
*3 + e*f**3*x + 3*e*f**2*g*x**2 + 3*e*f*g**2*x**3 + e*g**3*x**4),x)*a*b**2 
*d**3*e*f**3*g**5*n*x**2 - 72*int(log((d + e*x)**n*c)/(d*f**3 + 3*d*f**2*g 
*x + 3*d*f*g**2*x**2 + d*g**3*x**3 + e*f**3*x + 3*e*f**2*g*x**2 + 3*e*f*g* 
*2*x**3 + e*g**3*x**4),x)*a*b**2*d**2*e**2*f**6*g**2*n - 144*int(log((d + 
e*x)**n*c)/(d*f**3 + 3*d*f**2*g*x + 3*d*f*g**2*x**2 + d*g**3*x**3 + e*f**3 
*x + 3*e*f**2*g*x**2 + 3*e*f*g**2*x**3 + e*g**3*x**4),x)*a*b**2*d**2*e**2* 
f**5*g**3*n*x - 72*int(log((d + e*x)**n*c)/(d*f**3 + 3*d*f**2*g*x + 3*d...