\(\int (a+b x)^3 (A+B \log (e (a+b x)^n (c+d x)^{-n}))^2 \, dx\) [156]

Optimal result

Integrand size = 33, antiderivative size = 322 \[ \int (a+b x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 \, dx=-\frac {B (b c-a d) n (a+b x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{6 b d}+\frac {(a+b x)^4 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{4 b}+\frac {B (b c-a d)^2 n (a+b x)^2 \left (3 A+B n+3 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{12 b d^2}-\frac {B (b c-a d)^3 n (a+b x) \left (6 A+5 B n+6 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{12 b d^3}-\frac {B (b c-a d)^4 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (6 A+11 B n+6 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{12 b d^4}-\frac {B^2 (b c-a d)^4 n^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{2 b d^4} \] Output:

-1/6*B*(-a*d+b*c)*n*(b*x+a)^3*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/b/d+1/4*(b 
*x+a)^4*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2/b+1/12*B*(-a*d+b*c)^2*n*(b*x+a 
)^2*(3*A+B*n+3*B*ln(e*(b*x+a)^n/((d*x+c)^n)))/b/d^2-1/12*B*(-a*d+b*c)^3*n* 
(b*x+a)*(6*A+5*B*n+6*B*ln(e*(b*x+a)^n/((d*x+c)^n)))/b/d^3-1/12*B*(-a*d+b*c 
)^4*n*ln((-a*d+b*c)/b/(d*x+c))*(6*A+11*B*n+6*B*ln(e*(b*x+a)^n/((d*x+c)^n)) 
)/b/d^4-1/2*B^2*(-a*d+b*c)^4*n^2*polylog(2,d*(b*x+a)/b/(d*x+c))/b/d^4
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1709\) vs. \(2(322)=644\).

Time = 1.72 (sec) , antiderivative size = 1709, normalized size of antiderivative = 5.31 \[ \int (a+b x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 \, dx =\text {Too large to display} \] Input:

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

Output:

(-24*a^4*A*B*d^4*n + 6*a*b^3*B^2*c^3*d*n^2 - 24*a^2*b^2*B^2*c^2*d^2*n^2 + 
36*a^3*b*B^2*c*d^3*n^2 - 24*a^4*B^2*d^4*n^2 + 12*a^3*A^2*b*d^4*x - 6*A*b^4 
*B*c^3*d*n*x + 24*a*A*b^3*B*c^2*d^2*n*x - 36*a^2*A*b^2*B*c*d^3*n*x + 18*a^ 
3*A*b*B*d^4*n*x - 5*b^4*B^2*c^3*d*n^2*x + 17*a*b^3*B^2*c^2*d^2*n^2*x - 19* 
a^2*b^2*B^2*c*d^3*n^2*x + 7*a^3*b*B^2*d^4*n^2*x + 18*a^2*A^2*b^2*d^4*x^2 + 
 3*A*b^4*B*c^2*d^2*n*x^2 - 12*a*A*b^3*B*c*d^3*n*x^2 + 9*a^2*A*b^2*B*d^4*n* 
x^2 + b^4*B^2*c^2*d^2*n^2*x^2 - 2*a*b^3*B^2*c*d^3*n^2*x^2 + a^2*b^2*B^2*d^ 
4*n^2*x^2 + 12*a*A^2*b^3*d^4*x^3 - 2*A*b^4*B*c*d^3*n*x^3 + 2*a*A*b^3*B*d^4 
*n*x^3 + 3*A^2*b^4*d^4*x^4 - 3*a^4*B^2*d^4*n^2*Log[a + b*x]^2 + 6*A*b^4*B* 
c^4*n*Log[c + d*x] - 24*a*A*b^3*B*c^3*d*n*Log[c + d*x] + 36*a^2*A*b^2*B*c^ 
2*d^2*n*Log[c + d*x] - 24*a^3*A*b*B*c*d^3*n*Log[c + d*x] + 11*b^4*B^2*c^4* 
n^2*Log[c + d*x] - 38*a*b^3*B^2*c^3*d*n^2*Log[c + d*x] + 45*a^2*b^2*B^2*c^ 
2*d^2*n^2*Log[c + d*x] - 18*a^3*b*B^2*c*d^3*n^2*Log[c + d*x] - 24*a^4*B^2* 
d^4*n^2*Log[c + d*x] + 3*b^4*B^2*c^4*n^2*Log[c + d*x]^2 - 12*a*b^3*B^2*c^3 
*d*n^2*Log[c + d*x]^2 + 18*a^2*b^2*B^2*c^2*d^2*n^2*Log[c + d*x]^2 - 12*a^3 
*b*B^2*c*d^3*n^2*Log[c + d*x]^2 - 24*a^4*B^2*d^4*n*Log[(e*(a + b*x)^n)/(c 
+ d*x)^n] + 24*a^3*A*b*B*d^4*x*Log[(e*(a + b*x)^n)/(c + d*x)^n] - 6*b^4*B^ 
2*c^3*d*n*x*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 24*a*b^3*B^2*c^2*d^2*n*x*Lo 
g[(e*(a + b*x)^n)/(c + d*x)^n] - 36*a^2*b^2*B^2*c*d^3*n*x*Log[(e*(a + b*x) 
^n)/(c + d*x)^n] + 18*a^3*b*B^2*d^4*n*x*Log[(e*(a + b*x)^n)/(c + d*x)^n...
 

Rubi [A] (warning: unable to verify)

Time = 1.00 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.19, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {2973, 2949, 2781, 2784, 2784, 2784, 2754, 2838}

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.

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

\(\Big \downarrow \) 2973

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

\(\Big \downarrow \) 2949

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

\(\Big \downarrow \) 2781

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

\(\Big \downarrow \) 2784

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

\(\Big \downarrow \) 2784

\(\displaystyle (b c-a d)^4 \left (\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {\frac {(a+b x)^2 \left (3 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+3 A+B n\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\int \frac {(a+b x) \left (6 A+5 B n+6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{2 d}}{3 d}\right )}{2 b}\right )\)

\(\Big \downarrow \) 2784

\(\displaystyle (b c-a d)^4 \left (\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {\frac {(a+b x)^2 \left (3 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+3 A+B n\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\frac {(a+b x) \left (6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+6 A+5 B n\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\int \frac {6 A+11 B n+6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b-\frac {d (a+b x)}{c+d x}}d\frac {a+b x}{c+d x}}{d}}{2 d}}{3 d}\right )}{2 b}\right )\)

\(\Big \downarrow \) 2754

\(\displaystyle (b c-a d)^4 \left (\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {\frac {(a+b x)^2 \left (3 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+3 A+B n\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\frac {(a+b x) \left (6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+6 A+5 B n\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\frac {6 B n \int \frac {(c+d x) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right )}{a+b x}d\frac {a+b x}{c+d x}}{d}-\frac {\log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+6 A+11 B n\right )}{d}}{d}}{2 d}}{3 d}\right )}{2 b}\right )\)

\(\Big \downarrow \) 2838

\(\displaystyle (b c-a d)^4 \left (\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {\frac {(a+b x)^2 \left (3 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+3 A+B n\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\frac {(a+b x) \left (6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+6 A+5 B n\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {-\frac {\log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+6 A+11 B n\right )}{d}-\frac {6 B n \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d}}{d}}{2 d}}{3 d}\right )}{2 b}\right )\)

Input:

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

Output:

(b*c - a*d)^4*(((a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(4*b 
*(c + d*x)^4*(b - (d*(a + b*x))/(c + d*x))^4) - (B*n*(((a + b*x)^3*(A + B* 
Log[e*((a + b*x)/(c + d*x))^n]))/(3*d*(c + d*x)^3*(b - (d*(a + b*x))/(c + 
d*x))^3) - (((a + b*x)^2*(3*A + B*n + 3*B*Log[e*((a + b*x)/(c + d*x))^n])) 
/(2*d*(c + d*x)^2*(b - (d*(a + b*x))/(c + d*x))^2) - (((a + b*x)*(6*A + 5* 
B*n + 6*B*Log[e*((a + b*x)/(c + d*x))^n]))/(d*(c + d*x)*(b - (d*(a + b*x)) 
/(c + d*x))) - (-(((6*A + 11*B*n + 6*B*Log[e*((a + b*x)/(c + d*x))^n])*Log 
[1 - (d*(a + b*x))/(b*(c + d*x))])/d) - (6*B*n*PolyLog[2, (d*(a + b*x))/(b 
*(c + d*x))])/d)/d)/(2*d))/(3*d)))/(2*b))
 

Defintions of rubi rules used

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 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 2784

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

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 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 [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 230.62 (sec) , antiderivative size = 6465, normalized size of antiderivative = 20.08

method	result	size
risch	\(\text {Expression too large to display}\)	\(6465\)

Input:

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

Output:

result too large to display
 

Fricas [F]

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

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

Output:

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

Sympy [F(-2)]

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

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

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1871 vs. \(2 (309) = 618\).

Time = 0.61 (sec) , antiderivative size = 1871, normalized size of antiderivative = 5.81 \[ \int (a+b x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*A*B*b^3*x^4*log((b*x + a)^n*e/(d*x + c)^n) + 1/4*A^2*b^3*x^4 + 2*A*B*a 
*b^2*x^3*log((b*x + a)^n*e/(d*x + c)^n) + A^2*a*b^2*x^3 + 3*A*B*a^2*b*x^2* 
log((b*x + a)^n*e/(d*x + c)^n) + 3/2*A^2*a^2*b*x^2 + 2*A*B*a^3*x*log((b*x 
+ a)^n*e/(d*x + c)^n) + A^2*a^3*x + 2*(a*e*n*log(b*x + a)/b - c*e*n*log(d* 
x + c)/d)*A*B*a^3/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*B*a^2*b/e + (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*B*a*b^2/e - 1/12*(6*a^4*e*n*log(b* 
x + a)/b^4 - 6*c^4*e*n*log(d*x + c)/d^4 + (2*(b^3*c*d^2*e*n - a*b^2*d^3*e* 
n)*x^3 - 3*(b^3*c^2*d*e*n - a^2*b*d^3*e*n)*x^2 + 6*(b^3*c^3*e*n - a^3*d^3* 
e*n)*x)/(b^3*d^3))*A*B*b^3/e + 1/12*((11*n^2 + 6*n*log(e))*b^3*c^4 - 2*(19 
*n^2 + 12*n*log(e))*a*b^2*c^3*d + 9*(5*n^2 + 4*n*log(e))*a^2*b*c^2*d^2 - 6 
*(3*n^2 + 4*n*log(e))*a^3*c*d^3)*B^2*log(d*x + c)/d^4 + 1/2*(b^4*c^4*n^2 - 
 4*a*b^3*c^3*d*n^2 + 6*a^2*b^2*c^2*d^2*n^2 - 4*a^3*b*c*d^3*n^2 + a^4*d^4*n 
^2)*(log(b*x + a)*log((b*d*x + a*d)/(b*c - a*d) + 1) + dilog(-(b*d*x + a*d 
)/(b*c - a*d)))*B^2/(b*d^4) + 1/12*(3*B^2*b^4*d^4*x^4*log(e)^2 - 3*B^2*a^4 
*d^4*n^2*log(b*x + a)^2 - 2*(b^4*c*d^3*n*log(e) - (n*log(e) + 6*log(e)^2)* 
a*b^3*d^4)*B^2*x^3 + ((n^2 + 3*n*log(e))*b^4*c^2*d^2 - 2*(n^2 + 6*n*log(e) 
)*a*b^3*c*d^3 + (n^2 + 9*n*log(e) + 18*log(e)^2)*a^2*b^2*d^4)*B^2*x^2 - 6* 
(b^4*c^4*n^2 - 4*a*b^3*c^3*d*n^2 + 6*a^2*b^2*c^2*d^2*n^2 - 4*a^3*b*c*d^...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(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**4*b**2*d**5*n - 24*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**4*n + 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**3*n - 24*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**2*n + 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)*b**6*c**4*d*n + 6*log(c 
+ d*x)*a**5*d**4*n - 24*log(c + d*x)*a**4*b*c*d**3*n + 11*log(c + d*x)*a** 
4*b*d**4*n**2 + 36*log(c + d*x)*a**3*b**2*c**2*d**2*n - 44*log(c + d*x)*a* 
*3*b**2*c*d**3*n**2 - 24*log(c + d*x)*a**2*b**3*c**3*d*n + 66*log(c + d*x) 
*a**2*b**3*c**2*d**2*n**2 + 6*log(c + d*x)*a*b**4*c**4*n - 44*log(c + d*x) 
*a*b**4*c**3*d*n**2 + 11*log(c + d*x)*b**5*c**4*n**2 + 9*log(((a + b*x)**n 
*e)/(c + d*x)**n)**2*a**3*b**2*c*d**3 + 12*log(((a + b*x)**n*e)/(c + d*x)* 
*n)**2*a**3*b**2*d**4*x - 9*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a**2*b** 
3*c**2*d**2 + 18*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a**2*b**3*d**4*x**2 
 + 3*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a*b**4*c**3*d + 12*log(((a + b* 
x)**n*e)/(c + d*x)**n)**2*a*b**4*d**4*x**3 + 3*log(((a + b*x)**n*e)/(c + d 
*x)**n)**2*b**5*d**4*x**4 + 6*log(((a + b*x)**n*e)/(c + d*x)**n)*a**5*d**4 
 + 11*log(((a + b*x)**n*e)/(c + d*x)**n)*a**4*b*d**4*n + 24*log(((a + b*x) 
**n*e)/(c + d*x)**n)*a**4*b*d**4*x - 26*log(((a + b*x)**n*e)/(c + d*x)*...