\(\int (a+b x)^2 (A+B \log (e (a+b x)^n (c+d x)^{-n}))^3 \, dx\) [165]

Optimal result

Integrand size = 33, antiderivative size = 614 \[ \int (a+b x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=-\frac {B^3 (b c-a d)^3 n^3 \log (c+d x)}{b d^3}+\frac {B^2 (b c-a d)^2 n^2 (a+b x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{b d^2}+\frac {4 B^2 (b c-a d)^3 n^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{b d^3}+\frac {2 B (b c-a d)^2 n (a+b x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{b d^2}-\frac {b B (b c-a d) n (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{2 d^3}+\frac {B (b c-a d)^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{b d^3}+\frac {(a+b x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{3 b}-\frac {B^2 (b c-a d)^3 n^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b d^3}+\frac {4 B^3 (b c-a d)^3 n^3 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{b d^3}+\frac {2 B^2 (b c-a d)^3 n^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{b d^3}+\frac {B^3 (b c-a d)^3 n^3 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b d^3}-\frac {2 B^3 (b c-a d)^3 n^3 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{b d^3} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(4802\) vs. \(2(614)=1228\).

Time = 2.51 (sec) , antiderivative size = 4802, normalized size of antiderivative = 7.82 \[ \int (a+b x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=\text {Result too large to show} \] Input:

Integrate[(a + b*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3,x]
 

Output:

(-6*a^3*A*B^2*n^2)/b - (2*a*A*b*B^2*c^2*n^2)/d^2 + (4*a^2*A*B^2*c*n^2)/d - 
 (4*a^3*B^3*n^3)/b - (3*a*b*B^3*c^2*n^3)/d^2 + (7*a^2*B^3*c*n^3)/d + a^2*A 
^3*x + 2*a^2*A^2*B*n*x + (A^2*b^2*B*c^2*n*x)/d^2 - (3*a*A^2*b*B*c*n*x)/d + 
 a^2*A*B^2*n^2*x + (A*b^2*B^2*c^2*n^2*x)/d^2 - (2*a*A*b*B^2*c*n^2*x)/d + a 
*A^3*b*x^2 + (a*A^2*b*B*n*x^2)/2 - (A^2*b^2*B*c*n*x^2)/(2*d) + (A^3*b^2*x^ 
3)/3 + (a^3*A^2*B*n*Log[a + b*x])/b + (3*a^3*A*B^2*n^2*Log[a + b*x])/b + ( 
2*a*A*b*B^2*c^2*n^2*Log[a + b*x])/d^2 - (5*a^2*A*B^2*c*n^2*Log[a + b*x])/d 
 + (7*a^3*B^3*n^3*Log[a + b*x])/b + (3*a*b*B^3*c^2*n^3*Log[a + b*x])/d^2 - 
 (6*a^2*B^3*c*n^3*Log[a + b*x])/d - (a^3*A*B^2*n^2*Log[a + b*x]^2)/b - (3* 
a^3*B^3*n^3*Log[a + b*x]^2)/(2*b) - (a*b*B^3*c^2*n^3*Log[a + b*x]^2)/d^2 + 
 (5*a^2*B^3*c*n^3*Log[a + b*x]^2)/(2*d) + (a^3*B^3*n^3*Log[a + b*x]^3)/(3* 
b) - (A^2*b^2*B*c^3*n*Log[c + d*x])/d^3 + (3*a*A^2*b*B*c^2*n*Log[c + d*x]) 
/d^2 - (3*a^2*A^2*B*c*n*Log[c + d*x])/d - (3*A*b^2*B^2*c^3*n^2*Log[c + d*x 
])/d^3 + (7*a*A*b*B^2*c^2*n^2*Log[c + d*x])/d^2 - (4*a^2*A*B^2*c*n^2*Log[c 
 + d*x])/d - (6*a^3*B^3*n^3*Log[c + d*x])/b - (b^2*B^3*c^3*n^3*Log[c + d*x 
])/d^3 + (3*a^2*B^3*c*n^3*Log[c + d*x])/d + (2*a^3*A*B^2*n^2*Log[a + b*x]* 
Log[c + d*x])/b + (2*A*b^2*B^2*c^3*n^2*Log[a + b*x]*Log[c + d*x])/d^3 - (6 
*a*A*b*B^2*c^2*n^2*Log[a + b*x]*Log[c + d*x])/d^2 + (6*a^2*A*B^2*c*n^2*Log 
[a + b*x]*Log[c + d*x])/d + (3*b^2*B^3*c^3*n^3*Log[a + b*x]*Log[c + d*x])/ 
d^3 - (7*a*b*B^3*c^2*n^3*Log[a + b*x]*Log[c + d*x])/d^2 + (4*a^2*B^3*c*...
 

Rubi [A] (warning: unable to verify)

Time = 1.03 (sec) , antiderivative size = 573, normalized size of antiderivative = 0.93, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {2973, 2949, 2781, 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.

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

\(\Big \downarrow \) 2973

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

\(\Big \downarrow \) 2949

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

\(\Big \downarrow \) 2781

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

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

Input:

Int[(a + b*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3,x]
 

Output:

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

Defintions of rubi rules used

rule 2009

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

rule 2781

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

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 [F]

\[\int \left (b x +a \right )^{2} {\left (A +B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )\right )}^{3}d x\]

Input:

int((b*x+a)^2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^3,x)
 

Output:

int((b*x+a)^2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^3,x)
 

Fricas [F]

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

integrate((b*x+a)^2*(A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^3,x, algorithm="fri 
cas")
 

Output:

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

Sympy [F(-2)]

Exception generated. \[ \int (a+b x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

integrate((b*x+a)**2*(A+B*ln(e*(b*x+a)**n/((d*x+c)**n)))**3,x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

integrate((b*x+a)^2*(A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^3,x, algorithm="max 
ima")
 

Output:

A^2*B*b^2*x^3*log((b*x + a)^n*e/(d*x + c)^n) + 1/3*A^3*b^2*x^3 + 3*A^2*B*a 
*b*x^2*log((b*x + a)^n*e/(d*x + c)^n) + A^3*a*b*x^2 + 3*A^2*B*a^2*x*log((b 
*x + a)^n*e/(d*x + c)^n) + A^3*a^2*x + 3*(a*e*n*log(b*x + a)/b - c*e*n*log 
(d*x + c)/d)*A^2*B*a^2/e - 3*(a^2*e*n*log(b*x + a)/b^2 - c^2*e*n*log(d*x + 
 c)/d^2 + (b*c*e*n - a*d*e*n)*x/(b*d))*A^2*B*a*b/e + 1/2*(2*a^3*e*n*log(b* 
x + a)/b^3 - 2*c^3*e*n*log(d*x + c)/d^3 - ((b^2*c*d*e*n - a*b*d^2*e*n)*x^2 
 - 2*(b^2*c^2*e*n - a^2*d^2*e*n)*x)/(b^2*d^2))*A^2*B*b^2/e - 1/6*(2*(B^3*b 
^3*d^3*x^3 + 3*B^3*a*b^2*d^3*x^2 + 3*B^3*a^2*b*d^3*x)*log((d*x + c)^n)^3 - 
 3*(2*B^3*a^3*d^3*n*log(b*x + a) - 2*(b^3*c^3*n - 3*a*b^2*c^2*d*n + 3*a^2* 
b*c*d^2*n)*B^3*log(d*x + c) + 2*(B^3*b^3*d^3*log(e) + A*B^2*b^3*d^3)*x^3 + 
 (6*A*B^2*a*b^2*d^3 + (a*b^2*d^3*(n + 6*log(e)) - b^3*c*d^2*n)*B^3)*x^2 + 
2*(3*A*B^2*a^2*b*d^3 + (a^2*b*d^3*(2*n + 3*log(e)) + b^3*c^2*d*n - 3*a*b^2 
*c*d^2*n)*B^3)*x + 2*(B^3*b^3*d^3*x^3 + 3*B^3*a*b^2*d^3*x^2 + 3*B^3*a^2*b* 
d^3*x)*log((b*x + a)^n))*log((d*x + c)^n)^2)/(b*d^3) - integrate(-(B^3*a^2 
*b*c*d^2*log(e)^3 + 3*A*B^2*a^2*b*c*d^2*log(e)^2 + (B^3*b^3*d^3*log(e)^3 + 
 3*A*B^2*b^3*d^3*log(e)^2)*x^3 + (B^3*b^3*d^3*x^3 + B^3*a^2*b*c*d^2 + (b^3 
*c*d^2 + 2*a*b^2*d^3)*B^3*x^2 + (2*a*b^2*c*d^2 + a^2*b*d^3)*B^3*x)*log((b* 
x + a)^n)^3 + (3*(b^3*c*d^2*log(e)^2 + 2*a*b^2*d^3*log(e)^2)*A*B^2 + (b^3* 
c*d^2*log(e)^3 + 2*a*b^2*d^3*log(e)^3)*B^3)*x^2 + 3*(B^3*a^2*b*c*d^2*log(e 
) + A*B^2*a^2*b*c*d^2 + (B^3*b^3*d^3*log(e) + A*B^2*b^3*d^3)*x^3 + ((b^...
 

Giac [F]

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

integrate((b*x+a)^2*(A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^3,x, algorithm="gia 
c")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (a+b x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=\text {too large to display} \] Input:

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

Output:

(6*int((log(((a + b*x)**n*e)/(c + d*x)**n)**2*x)/(a*c + a*d*x + b*c*x + b* 
d*x**2),x)*a**3*b**3*d**4*n - 18*int((log(((a + b*x)**n*e)/(c + d*x)**n)** 
2*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*a**2*b**4*c*d**3*n + 18*int((log( 
((a + b*x)**n*e)/(c + d*x)**n)**2*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*a 
*b**5*c**2*d**2*n - 6*int((log(((a + b*x)**n*e)/(c + d*x)**n)**2*x)/(a*c + 
 a*d*x + b*c*x + b*d*x**2),x)*b**6*c**3*d*n + 12*int((log(((a + b*x)**n*e) 
/(c + d*x)**n)*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*a**4*b**2*d**4*n - 3 
6*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x)/(a*c + a*d*x + b*c*x + b*d*x* 
*2),x)*a**3*b**3*c*d**3*n + 18*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x)/ 
(a*c + a*d*x + b*c*x + b*d*x**2),x)*a**3*b**3*d**4*n**2 + 36*int((log(((a 
+ b*x)**n*e)/(c + d*x)**n)*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*a**2*b** 
4*c**2*d**2*n - 54*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x)/(a*c + a*d*x 
 + b*c*x + b*d*x**2),x)*a**2*b**4*c*d**3*n**2 - 12*int((log(((a + b*x)**n* 
e)/(c + d*x)**n)*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*a*b**5*c**3*d*n + 
54*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x)/(a*c + a*d*x + b*c*x + b*d*x 
**2),x)*a*b**5*c**2*d**2*n**2 - 18*int((log(((a + b*x)**n*e)/(c + d*x)**n) 
*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*b**6*c**3*d*n**2 + 6*log(c + d*x)* 
a**5*d**3*n - 18*log(c + d*x)*a**4*b*c*d**2*n + 18*log(c + d*x)*a**4*b*d** 
3*n**2 + 18*log(c + d*x)*a**3*b**2*c**2*d*n - 54*log(c + d*x)*a**3*b**2*c* 
d**2*n**2 + 6*log(c + d*x)*a**3*b**2*d**3*n**3 - 6*log(c + d*x)*a**2*b*...