\(\int \frac {(A+B \log (e (a+b x)^n (c+d x)^{-n}))^3}{(a+b x)^5} \, dx\) [171]

Optimal result

Integrand size = 33, antiderivative size = 830 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^5} \, dx=\frac {6 B^3 d^3 n^3 (c+d x)}{(b c-a d)^4 (a+b x)}-\frac {9 b B^3 d^2 n^3 (c+d x)^2}{8 (b c-a d)^4 (a+b x)^2}+\frac {2 b^2 B^3 d n^3 (c+d x)^3}{9 (b c-a d)^4 (a+b x)^3}-\frac {3 b^3 B^3 n^3 (c+d x)^4}{128 (b c-a d)^4 (a+b x)^4}+\frac {6 B^2 d^3 n^2 (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{(b c-a d)^4 (a+b x)}-\frac {9 b B^2 d^2 n^2 (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{4 (b c-a d)^4 (a+b x)^2}+\frac {2 b^2 B^2 d n^2 (c+d x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{3 (b c-a d)^4 (a+b x)^3}-\frac {3 b^3 B^2 n^2 (c+d x)^4 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{32 (b c-a d)^4 (a+b x)^4}+\frac {3 B d^3 n (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(b c-a d)^4 (a+b x)}-\frac {9 b B d^2 n (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{4 (b c-a d)^4 (a+b x)^2}+\frac {b^2 B d n (c+d x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(b c-a d)^4 (a+b x)^3}-\frac {3 b^3 B n (c+d x)^4 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{16 (b c-a d)^4 (a+b x)^4}+\frac {d^3 (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(b c-a d)^4 (a+b x)}-\frac {3 b d^2 (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{2 (b c-a d)^4 (a+b x)^2}+\frac {b^2 d (c+d x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(b c-a d)^4 (a+b x)^3}-\frac {b^3 (c+d x)^4 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{4 (b c-a d)^4 (a+b x)^4} \] Output:

6*B^3*d^3*n^3*(d*x+c)/(-a*d+b*c)^4/(b*x+a)-9/8*b*B^3*d^2*n^3*(d*x+c)^2/(-a 
*d+b*c)^4/(b*x+a)^2+2/9*b^2*B^3*d*n^3*(d*x+c)^3/(-a*d+b*c)^4/(b*x+a)^3-3/1 
28*b^3*B^3*n^3*(d*x+c)^4/(-a*d+b*c)^4/(b*x+a)^4+6*B^2*d^3*n^2*(d*x+c)*(A+B 
*ln(e*(b*x+a)^n/((d*x+c)^n)))/(-a*d+b*c)^4/(b*x+a)-9/4*b*B^2*d^2*n^2*(d*x+ 
c)^2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/(-a*d+b*c)^4/(b*x+a)^2+2/3*b^2*B^2* 
d*n^2*(d*x+c)^3*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/(-a*d+b*c)^4/(b*x+a)^3-3 
/32*b^3*B^2*n^2*(d*x+c)^4*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/(-a*d+b*c)^4/( 
b*x+a)^4+3*B*d^3*n*(d*x+c)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2/(-a*d+b*c)^ 
4/(b*x+a)-9/4*b*B*d^2*n*(d*x+c)^2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2/(-a* 
d+b*c)^4/(b*x+a)^2+b^2*B*d*n*(d*x+c)^3*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2 
/(-a*d+b*c)^4/(b*x+a)^3-3/16*b^3*B*n*(d*x+c)^4*(A+B*ln(e*(b*x+a)^n/((d*x+c 
)^n)))^2/(-a*d+b*c)^4/(b*x+a)^4+d^3*(d*x+c)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n 
)))^3/(-a*d+b*c)^4/(b*x+a)-3/2*b*d^2*(d*x+c)^2*(A+B*ln(e*(b*x+a)^n/((d*x+c 
)^n)))^3/(-a*d+b*c)^4/(b*x+a)^2+b^2*d*(d*x+c)^3*(A+B*ln(e*(b*x+a)^n/((d*x+ 
c)^n)))^3/(-a*d+b*c)^4/(b*x+a)^3-1/4*b^3*(d*x+c)^4*(A+B*ln(e*(b*x+a)^n/((d 
*x+c)^n)))^3/(-a*d+b*c)^4/(b*x+a)^4
 

Mathematica [A] (verified)

Time = 2.06 (sec) , antiderivative size = 1370, normalized size of antiderivative = 1.65 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^5} \, dx =\text {Too large to display} \] Input:

Integrate[(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3/(a + b*x)^5,x]
 

Output:

-1/1152*(-288*B^3*d^4*n^3*(a + b*x)^4*Log[a + b*x]^3 + 288*B^3*d^4*n^3*(a 
+ b*x)^4*Log[c + d*x]^3 + 72*B^2*d^4*n^2*(a + b*x)^4*Log[c + d*x]^2*(12*A 
+ 25*B*n + 12*B*Log[(e*(a + b*x)^n)/(c + d*x)^n]) + 72*B^2*d^4*n^2*(a + b* 
x)^4*Log[a + b*x]^2*(12*A + 25*B*n + 12*B*n*Log[c + d*x] + 12*B*Log[(e*(a 
+ b*x)^n)/(c + d*x)^n]) + 12*B*d^4*n*(a + b*x)^4*Log[c + d*x]*(72*A^2 + 30 
0*A*B*n + 415*B^2*n^2 + 12*B*(12*A + 25*B*n)*Log[(e*(a + b*x)^n)/(c + d*x) 
^n] + 72*B^2*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2) + (b*c - a*d)*(288*A^3*b^ 
3*c^3 - 864*a*A^3*b^2*c^2*d + 864*a^2*A^3*b*c*d^2 - 288*a^3*A^3*d^3 + 216* 
A^2*b^3*B*c^3*n - 936*a*A^2*b^2*B*c^2*d*n + 1656*a^2*A^2*b*B*c*d^2*n - 180 
0*a^3*A^2*B*d^3*n + 108*A*b^3*B^2*c^3*n^2 - 660*a*A*b^2*B^2*c^2*d*n^2 + 19 
32*a^2*A*b*B^2*c*d^2*n^2 - 4980*a^3*A*B^2*d^3*n^2 + 27*b^3*B^3*c^3*n^3 - 2 
29*a*b^2*B^3*c^2*d*n^3 + 1067*a^2*b*B^3*c*d^2*n^3 - 5845*a^3*B^3*d^3*n^3 - 
 288*A^2*b^3*B*c^2*d*n*x + 1440*a*A^2*b^2*B*c*d^2*n*x - 3744*a^2*A^2*b*B*d 
^3*n*x - 336*A*b^3*B^2*c^2*d*n^2*x + 2544*a*A*b^2*B^2*c*d^2*n^2*x - 13008* 
a^2*A*b*B^2*d^3*n^2*x - 148*b^3*B^3*c^2*d*n^3*x + 1676*a*b^2*B^3*c*d^2*n^3 
*x - 16468*a^2*b*B^3*d^3*n^3*x + 432*A^2*b^3*B*c*d^2*n*x^2 - 3024*a*A^2*b^ 
2*B*d^3*n*x^2 + 936*A*b^3*B^2*c*d^2*n^2*x^2 - 11736*a*A*b^2*B^2*d^3*n^2*x^ 
2 + 690*b^3*B^3*c*d^2*n^3*x^2 - 15630*a*b^2*B^3*d^3*n^3*x^2 - 864*A^2*b^3* 
B*d^3*n*x^3 - 3600*A*b^3*B^2*d^3*n^2*x^3 - 4980*b^3*B^3*d^3*n^3*x^3 + 12*B 
*(72*A^2*(b*c - a*d)^3 + B^2*n^2*(-415*a^3*d^3 + a^2*b*d^2*(161*c - 108...
 

Rubi [A] (warning: unable to verify)

Time = 0.87 (sec) , antiderivative size = 669, normalized size of antiderivative = 0.81, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {2973, 2949, 2795, 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.

\(\displaystyle \int \frac {\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{(a+b x)^5} \, dx\)

\(\Big \downarrow \) 2973

\(\displaystyle \int \frac {\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{(a+b x)^5}dx\)

\(\Big \downarrow \) 2949

\(\displaystyle \frac {\int \frac {(c+d x)^5 \left (b-\frac {d (a+b x)}{c+d x}\right )^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{(a+b x)^5}d\frac {a+b x}{c+d x}}{(b c-a d)^4}\)

\(\Big \downarrow \) 2795

\(\displaystyle \frac {\int \left (\frac {b^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3 (c+d x)^5}{(a+b x)^5}-\frac {3 b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3 (c+d x)^4}{(a+b x)^4}+\frac {3 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3 (c+d x)^3}{(a+b x)^3}-\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3 (c+d x)^2}{(a+b x)^2}\right )d\frac {a+b x}{c+d x}}{(b c-a d)^4}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {3 b^3 B^2 n^2 (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{32 (a+b x)^4}-\frac {b^3 (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{4 (a+b x)^4}-\frac {3 b^3 B n (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{16 (a+b x)^4}+\frac {2 b^2 B^2 d n^2 (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 (a+b x)^3}+\frac {b^2 d (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{(a+b x)^3}+\frac {b^2 B d n (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{(a+b x)^3}+\frac {6 B^2 d^3 n^2 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}-\frac {9 b B^2 d^2 n^2 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 (a+b x)^2}+\frac {d^3 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{a+b x}+\frac {3 B d^3 n (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{a+b x}-\frac {3 b d^2 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{2 (a+b x)^2}-\frac {9 b B d^2 n (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 (a+b x)^2}-\frac {3 b^3 B^3 n^3 (c+d x)^4}{128 (a+b x)^4}+\frac {2 b^2 B^3 d n^3 (c+d x)^3}{9 (a+b x)^3}+\frac {6 B^3 d^3 n^3 (c+d x)}{a+b x}-\frac {9 b B^3 d^2 n^3 (c+d x)^2}{8 (a+b x)^2}}{(b c-a d)^4}\)

Input:

Int[(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3/(a + b*x)^5,x]
 

Output:

((6*B^3*d^3*n^3*(c + d*x))/(a + b*x) - (9*b*B^3*d^2*n^3*(c + d*x)^2)/(8*(a 
 + b*x)^2) + (2*b^2*B^3*d*n^3*(c + d*x)^3)/(9*(a + b*x)^3) - (3*b^3*B^3*n^ 
3*(c + d*x)^4)/(128*(a + b*x)^4) + (6*B^2*d^3*n^2*(c + d*x)*(A + B*Log[e*( 
(a + b*x)/(c + d*x))^n]))/(a + b*x) - (9*b*B^2*d^2*n^2*(c + d*x)^2*(A + B* 
Log[e*((a + b*x)/(c + d*x))^n]))/(4*(a + b*x)^2) + (2*b^2*B^2*d*n^2*(c + d 
*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*(a + b*x)^3) - (3*b^3*B^2 
*n^2*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(32*(a + b*x)^4) 
+ (3*B*d^3*n*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(a + b*x) 
 - (9*b*B*d^2*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(4*( 
a + b*x)^2) + (b^2*B*d*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n] 
)^2)/(a + b*x)^3 - (3*b^3*B*n*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x 
))^n])^2)/(16*(a + b*x)^4) + (d^3*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d 
*x))^n])^3)/(a + b*x) - (3*b*d^2*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + 
d*x))^n])^3)/(2*(a + b*x)^2) + (b^2*d*(c + d*x)^3*(A + B*Log[e*((a + b*x)/ 
(c + d*x))^n])^3)/(a + b*x)^3 - (b^3*(c + d*x)^4*(A + B*Log[e*((a + b*x)/( 
c + d*x))^n])^3)/(4*(a + b*x)^4))/(b*c - a*d)^4
 

Defintions of rubi rules used

rule 2009

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

rule 2795

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

rule 2949

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( 
B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(b*c - a*d)^(m + 
1)*(g/b)^m   Subst[Int[x^m*((A + B*Log[e*x^n])^p/(b - d*x)^(m + 2)), x], x, 
 (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] && Ne 
Q[b*c - a*d, 0] && IntegersQ[m, p] && EqQ[b*f - a*g, 0] && (GtQ[p, 0] || Lt 
Q[m, -1])
 

rule 2973

Int[((A_.) + Log[(e_.)*(u_)^(n_.)*(v_)^(mn_)]*(B_.))^(p_.)*(w_.), x_Symbol] 
 :> Subst[Int[w*(A + B*Log[e*(u/v)^n])^p, x], e*(u/v)^n, e*(u^n/v^n)] /; Fr 
eeQ[{e, A, B, n, p}, x] && EqQ[n + mn, 0] && LinearQ[{u, v}, x] &&  !Intege 
rQ[n]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(8291\) vs. \(2(810)=1620\).

Time = 130.01 (sec) , antiderivative size = 8292, normalized size of antiderivative = 9.99

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(8292\)
risch	\(\text {Expression too large to display}\)	\(236754\)

Input:

int((A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^3/(b*x+a)^5,x,method=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6057 vs. \(2 (810) = 1620\).

Time = 0.33 (sec) , antiderivative size = 6057, normalized size of antiderivative = 7.30 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^5} \, dx=\text {Too large to display} \] Input:

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^3/(b*x+a)^5,x, algorithm="fri 
cas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^5} \, dx=\text {Timed out} \] Input:

integrate((A+B*ln(e*(b*x+a)**n/((d*x+c)**n)))**3/(b*x+a)**5,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5280 vs. \(2 (810) = 1620\).

Time = 0.41 (sec) , antiderivative size = 5280, normalized size of antiderivative = 6.36 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^5} \, dx=\text {Too large to display} \] Input:

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^3/(b*x+a)^5,x, algorithm="max 
ima")
 

Output:

-1/4*B^3*log((b*x + a)^n*e/(d*x + c)^n)^3/(b^5*x^4 + 4*a*b^4*x^3 + 6*a^2*b 
^3*x^2 + 4*a^3*b^2*x + a^4*b) + 1/1152*(72*(12*d^4*e*n*log(b*x + a)/(b^5*c 
^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4) - 12 
*d^4*e*n*log(d*x + c)/(b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3 
*b^2*c*d^3 + a^4*b*d^4) + (12*b^3*d^3*e*n*x^3 - 3*b^3*c^3*e*n + 13*a*b^2*c 
^2*d*e*n - 23*a^2*b*c*d^2*e*n + 25*a^3*d^3*e*n - 6*(b^3*c*d^2*e*n - 7*a*b^ 
2*d^3*e*n)*x^2 + 4*(b^3*c^2*d*e*n - 5*a*b^2*c*d^2*e*n + 13*a^2*b*d^3*e*n)* 
x)/(a^4*b^4*c^3 - 3*a^5*b^3*c^2*d + 3*a^6*b^2*c*d^2 - a^7*b*d^3 + (b^8*c^3 
 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a^3*b^5*d^3)*x^4 + 4*(a*b^7*c^3 - 3*a 
^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a^4*b^4*d^3)*x^3 + 6*(a^2*b^6*c^3 - 3*a^3 
*b^5*c^2*d + 3*a^4*b^4*c*d^2 - a^5*b^3*d^3)*x^2 + 4*(a^3*b^5*c^3 - 3*a^4*b 
^4*c^2*d + 3*a^5*b^3*c*d^2 - a^6*b^2*d^3)*x))*log((b*x + a)^n*e/(d*x + c)^ 
n)^2/e - (12*(9*b^4*c^4*e^2*n^2 - 64*a*b^3*c^3*d*e^2*n^2 + 216*a^2*b^2*c^2 
*d^2*e^2*n^2 - 576*a^3*b*c*d^3*e^2*n^2 + 415*a^4*d^4*e^2*n^2 - 300*(b^4*c* 
d^3*e^2*n^2 - a*b^3*d^4*e^2*n^2)*x^3 + 6*(13*b^4*c^2*d^2*e^2*n^2 - 176*a*b 
^3*c*d^3*e^2*n^2 + 163*a^2*b^2*d^4*e^2*n^2)*x^2 + 72*(b^4*d^4*e^2*n^2*x^4 
+ 4*a*b^3*d^4*e^2*n^2*x^3 + 6*a^2*b^2*d^4*e^2*n^2*x^2 + 4*a^3*b*d^4*e^2*n^ 
2*x + a^4*d^4*e^2*n^2)*log(b*x + a)^2 + 72*(b^4*d^4*e^2*n^2*x^4 + 4*a*b^3* 
d^4*e^2*n^2*x^3 + 6*a^2*b^2*d^4*e^2*n^2*x^2 + 4*a^3*b*d^4*e^2*n^2*x + a^4* 
d^4*e^2*n^2)*log(d*x + c)^2 - 4*(7*b^4*c^3*d*e^2*n^2 - 60*a*b^3*c^2*d^2...
 

Giac [F]

\[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^5} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{3}}{{\left (b x + a\right )}^{5}} \,d x } \] Input:

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^3/(b*x+a)^5,x, algorithm="gia 
c")
 

Output:

integrate((B*log((b*x + a)^n*e/(d*x + c)^n) + A)^3/(b*x + a)^5, x)
 

Mupad [B] (verification not implemented)

Time = 32.66 (sec) , antiderivative size = 4257, normalized size of antiderivative = 5.13 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^5} \, dx=\text {Too large to display} \] Input:

int((A + B*log((e*(a + b*x)^n)/(c + d*x)^n))^3/(a + b*x)^5,x)
 

Output:

log((e*(a + b*x)^n)/(c + d*x)^n)*((x*((a*d + b*c)*(a*((9*B^3*a*d^2*n^2)/2 
- (3*B^3*b*c*d*n^2)/2) + 13*B^3*a^2*d^2*n^2 + (11*B^3*b^2*c^2*n^2)/2 - 6*A 
^2*B*a^2*d^2 - 6*A^2*B*b^2*c^2 - (31*B^3*a*b*c*d*n^2)/2 + 12*A^2*B*a*b*c*d 
) + a*c*(b*((9*B^3*a*d^2*n^2)/2 - (3*B^3*b*c*d*n^2)/2) + (27*B^3*a*b*d^2*n 
^2)/2 - (9*B^3*b^2*c*d*n^2)/2)) + x^2*((a*d + b*c)*(b*((9*B^3*a*d^2*n^2)/2 
 - (3*B^3*b*c*d*n^2)/2) + (27*B^3*a*b*d^2*n^2)/2 - (9*B^3*b^2*c*d*n^2)/2) 
+ b*d*(a*((9*B^3*a*d^2*n^2)/2 - (3*B^3*b*c*d*n^2)/2) + 13*B^3*a^2*d^2*n^2 
+ (11*B^3*b^2*c^2*n^2)/2 - 6*A^2*B*a^2*d^2 - 6*A^2*B*b^2*c^2 - (31*B^3*a*b 
*c*d*n^2)/2 + 12*A^2*B*a*b*c*d) + 6*B^3*a*b^2*c*d^2*n^2) + x^3*(b*d*(b*((9 
*B^3*a*d^2*n^2)/2 - (3*B^3*b*c*d*n^2)/2) + (27*B^3*a*b*d^2*n^2)/2 - (9*B^3 
*b^2*c*d*n^2)/2) + 6*B^3*b^2*d^2*n^2*(a*d + b*c)) + a*c*(a*((9*B^3*a*d^2*n 
^2)/2 - (3*B^3*b*c*d*n^2)/2) + 13*B^3*a^2*d^2*n^2 + (11*B^3*b^2*c^2*n^2)/2 
 - 6*A^2*B*a^2*d^2 - 6*A^2*B*b^2*c^2 - (31*B^3*a*b*c*d*n^2)/2 + 12*A^2*B*a 
*b*c*d) + 6*B^3*b^3*d^3*n^2*x^4)/(8*b*(a*d - b*c)^2*(a + b*x)^5*(c + d*x)) 
 - (d^4*(12*A*B^2 + 25*B^3*n)*(x^3*((a*d + b*c)*(b*(b*((2*a*b*n*(a*d - b*c 
)^3)/d + (2*b*n*(a*d - b*c)^3*(4*a*d - b*c))/(3*d^2)) + (4*b^2*n*(a*d - b* 
c)^3*(4*a*d - b*c))/(3*d^2) + (4*a*b^2*n*(a*d - b*c)^3)/d) + (2*b^3*n*(a*d 
 - b*c)^3*(4*a*d - b*c))/d^2 + (6*a*b^3*n*(a*d - b*c)^3)/d) + b*d*(b*(a*(( 
2*a*b*n*(a*d - b*c)^3)/d + (2*b*n*(a*d - b*c)^3*(4*a*d - b*c))/(3*d^2)) + 
(2*b*n*(a*d - b*c)^3*(6*a^2*d^2 + b^2*c^2 - 4*a*b*c*d))/(3*d^3)) + a*(b...
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 4255, normalized size of antiderivative = 5.13 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^5} \, dx =\text {Too large to display} \] Input:

int((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^3/(b*x+a)^5,x)
 

Output:

(864*log(a + b*x)*a**7*b*d**4*n + 3168*log(a + b*x)*a**6*b**2*d**4*n**2 + 
3456*log(a + b*x)*a**6*b**2*d**4*n*x + 432*log(a + b*x)*a**5*b**3*c*d**3*n 
**2 + 4080*log(a + b*x)*a**5*b**3*d**4*n**3 + 12672*log(a + b*x)*a**5*b**3 
*d**4*n**2*x + 5184*log(a + b*x)*a**5*b**3*d**4*n*x**2 + 900*log(a + b*x)* 
a**4*b**4*c*d**3*n**3 + 1728*log(a + b*x)*a**4*b**4*c*d**3*n**2*x + 16320* 
log(a + b*x)*a**4*b**4*d**4*n**3*x + 19008*log(a + b*x)*a**4*b**4*d**4*n** 
2*x**2 + 3456*log(a + b*x)*a**4*b**4*d**4*n*x**3 + 3600*log(a + b*x)*a**3* 
b**5*c*d**3*n**3*x + 2592*log(a + b*x)*a**3*b**5*c*d**3*n**2*x**2 + 24480* 
log(a + b*x)*a**3*b**5*d**4*n**3*x**2 + 12672*log(a + b*x)*a**3*b**5*d**4* 
n**2*x**3 + 864*log(a + b*x)*a**3*b**5*d**4*n*x**4 + 5400*log(a + b*x)*a** 
2*b**6*c*d**3*n**3*x**2 + 1728*log(a + b*x)*a**2*b**6*c*d**3*n**2*x**3 + 1 
6320*log(a + b*x)*a**2*b**6*d**4*n**3*x**3 + 3168*log(a + b*x)*a**2*b**6*d 
**4*n**2*x**4 + 3600*log(a + b*x)*a*b**7*c*d**3*n**3*x**3 + 432*log(a + b* 
x)*a*b**7*c*d**3*n**2*x**4 + 4080*log(a + b*x)*a*b**7*d**4*n**3*x**4 + 900 
*log(a + b*x)*b**8*c*d**3*n**3*x**4 - 864*log(c + d*x)*a**7*b*d**4*n - 316 
8*log(c + d*x)*a**6*b**2*d**4*n**2 - 3456*log(c + d*x)*a**6*b**2*d**4*n*x 
- 432*log(c + d*x)*a**5*b**3*c*d**3*n**2 - 4080*log(c + d*x)*a**5*b**3*d** 
4*n**3 - 12672*log(c + d*x)*a**5*b**3*d**4*n**2*x - 5184*log(c + d*x)*a**5 
*b**3*d**4*n*x**2 - 900*log(c + d*x)*a**4*b**4*c*d**3*n**3 - 1728*log(c + 
d*x)*a**4*b**4*c*d**3*n**2*x - 16320*log(c + d*x)*a**4*b**4*d**4*n**3*x...