\(\int \frac {(A+B \log (e (a+b x)^n (c+d x)^{-n}))^2}{(a+b x)^4} \, dx\) [162]

Optimal result

Integrand size = 33, antiderivative size = 427 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(a+b x)^4} \, dx=-\frac {2 B^2 d^2 n^2 (c+d x)}{(b c-a d)^3 (a+b x)}+\frac {b B^2 d n^2 (c+d x)^2}{2 (b c-a d)^3 (a+b x)^2}-\frac {2 b^2 B^2 n^2 (c+d x)^3}{27 (b c-a d)^3 (a+b x)^3}-\frac {2 B d^2 n (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{(b c-a d)^3 (a+b x)}+\frac {b B d n (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{(b c-a d)^3 (a+b x)^2}-\frac {2 b^2 B n (c+d x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{9 (b c-a d)^3 (a+b x)^3}-\frac {d^2 (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(b c-a d)^3 (a+b x)}+\frac {b d (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(b c-a d)^3 (a+b x)^2}-\frac {b^2 (c+d x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{3 (b c-a d)^3 (a+b x)^3} \] Output:

-2*B^2*d^2*n^2*(d*x+c)/(-a*d+b*c)^3/(b*x+a)+1/2*b*B^2*d*n^2*(d*x+c)^2/(-a* 
d+b*c)^3/(b*x+a)^2-2/27*b^2*B^2*n^2*(d*x+c)^3/(-a*d+b*c)^3/(b*x+a)^3-2*B*d 
^2*n*(d*x+c)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/(-a*d+b*c)^3/(b*x+a)+b*B*d* 
n*(d*x+c)^2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/(-a*d+b*c)^3/(b*x+a)^2-2/9*b 
^2*B*n*(d*x+c)^3*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/(-a*d+b*c)^3/(b*x+a)^3- 
d^2*(d*x+c)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2/(-a*d+b*c)^3/(b*x+a)+b*d*( 
d*x+c)^2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2/(-a*d+b*c)^3/(b*x+a)^2-1/3*b^ 
2*(d*x+c)^3*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2/(-a*d+b*c)^3/(b*x+a)^3
 

Mathematica [A] (verified)

Time = 0.81 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.01 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(a+b x)^4} \, dx=\frac {18 B^2 d^3 n^2 (a+b x)^3 \log ^2(a+b x)+18 B^2 d^3 n^2 (a+b x)^3 \log ^2(c+d x)+6 B d^3 n (a+b x)^3 \log (c+d x) \left (6 A+11 B n+6 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )-6 B d^3 n (a+b x)^3 \log (a+b x) \left (6 A+11 B n+6 B n \log (c+d x)+6 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )-(b c-a d) \left (18 A^2 (b c-a d)^2+6 A B n \left (11 a^2 d^2+a b d (-7 c+15 d x)+b^2 \left (2 c^2-3 c d x+6 d^2 x^2\right )\right )+B^2 n^2 \left (85 a^2 d^2+a b d (-23 c+147 d x)+b^2 \left (4 c^2-15 c d x+66 d^2 x^2\right )\right )+6 B \left (6 A (b c-a d)^2+B n \left (11 a^2 d^2+a b d (-7 c+15 d x)+b^2 \left (2 c^2-3 c d x+6 d^2 x^2\right )\right )\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+18 B^2 (b c-a d)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{54 b (b c-a d)^3 (a+b x)^3} \] Input:

Integrate[(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2/(a + b*x)^4,x]
 

Output:

(18*B^2*d^3*n^2*(a + b*x)^3*Log[a + b*x]^2 + 18*B^2*d^3*n^2*(a + b*x)^3*Lo 
g[c + d*x]^2 + 6*B*d^3*n*(a + b*x)^3*Log[c + d*x]*(6*A + 11*B*n + 6*B*Log[ 
(e*(a + b*x)^n)/(c + d*x)^n]) - 6*B*d^3*n*(a + b*x)^3*Log[a + b*x]*(6*A + 
11*B*n + 6*B*n*Log[c + d*x] + 6*B*Log[(e*(a + b*x)^n)/(c + d*x)^n]) - (b*c 
 - a*d)*(18*A^2*(b*c - a*d)^2 + 6*A*B*n*(11*a^2*d^2 + a*b*d*(-7*c + 15*d*x 
) + b^2*(2*c^2 - 3*c*d*x + 6*d^2*x^2)) + B^2*n^2*(85*a^2*d^2 + a*b*d*(-23* 
c + 147*d*x) + b^2*(4*c^2 - 15*c*d*x + 66*d^2*x^2)) + 6*B*(6*A*(b*c - a*d) 
^2 + B*n*(11*a^2*d^2 + a*b*d*(-7*c + 15*d*x) + b^2*(2*c^2 - 3*c*d*x + 6*d^ 
2*x^2)))*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 18*B^2*(b*c - a*d)^2*Log[(e*(a 
 + b*x)^n)/(c + d*x)^n]^2))/(54*b*(b*c - a*d)^3*(a + b*x)^3)
 

Rubi [A] (warning: unable to verify)

Time = 0.64 (sec) , antiderivative size = 342, normalized size of antiderivative = 0.80, 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.

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

\(\Big \downarrow \) 2973

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

\(\Big \downarrow \) 2949

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {b^2 (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 (a+b x)^3}-\frac {2 b^2 B n (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{9 (a+b x)^3}-\frac {d^2 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{a+b x}-\frac {2 B d^2 n (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}+\frac {b d (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{(a+b x)^2}+\frac {b B d n (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{(a+b x)^2}-\frac {2 b^2 B^2 n^2 (c+d x)^3}{27 (a+b x)^3}-\frac {2 B^2 d^2 n^2 (c+d x)}{a+b x}+\frac {b B^2 d n^2 (c+d x)^2}{2 (a+b x)^2}}{(b c-a d)^3}\)

Input:

Int[(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2/(a + b*x)^4,x]
 

Output:

((-2*B^2*d^2*n^2*(c + d*x))/(a + b*x) + (b*B^2*d*n^2*(c + d*x)^2)/(2*(a + 
b*x)^2) - (2*b^2*B^2*n^2*(c + d*x)^3)/(27*(a + b*x)^3) - (2*B*d^2*n*(c + d 
*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x) + (b*B*d*n*(c + d*x) 
^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x)^2 - (2*b^2*B*n*(c + d 
*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(9*(a + b*x)^3) - (d^2*(c + 
d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(a + b*x) + (b*d*(c + d*x)^ 
2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(a + b*x)^2 - (b^2*(c + d*x)^3 
*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(3*(a + b*x)^3))/(b*c - a*d)^3
 

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. \(1399\) vs. \(2(419)=838\).

Time = 39.47 (sec) , antiderivative size = 1400, normalized size of antiderivative = 3.28

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(1400\)
risch	\(\text {Expression too large to display}\)	\(25057\)

Input:

int((A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2/(b*x+a)^4,x,method=_RETURNVERBOSE)
 

Output:

-1/54*(85*B^2*a^3*b^4*d^4*n^2-4*B^2*b^7*c^3*d*n^2-108*A*B*ln(b*x+a)*x*a^2* 
b^5*d^4*n+108*A*B*ln(d*x+c)*x*a^2*b^5*d^4*n+36*B^2*x^2*ln(e*(b*x+a)^n/((d* 
x+c)^n))*a*b^6*d^4*n-36*B^2*x^2*ln(e*(b*x+a)^n/((d*x+c)^n))*b^7*c*d^3*n+36 
*A*B*x^2*a*b^6*d^4*n-36*A*B*x^2*b^7*c*d^3*n+90*B^2*x*ln(e*(b*x+a)^n/((d*x+ 
c)^n))*a^2*b^5*d^4*n+18*B^2*x*ln(e*(b*x+a)^n/((d*x+c)^n))*b^7*c^2*d^2*n-16 
2*B^2*x*a*b^6*c*d^3*n^2+90*A*B*x*a^2*b^5*d^4*n+18*A*B*x*b^7*c^2*d^2*n-108* 
B^2*ln(e*(b*x+a)^n/((d*x+c)^n))*a^2*b^5*c*d^3*n+54*B^2*ln(e*(b*x+a)^n/((d* 
x+c)^n))*a*b^6*c^2*d^2*n-108*A*B*ln(e*(b*x+a)^n/((d*x+c)^n))*a^2*b^5*c*d^3 
+108*A*B*ln(e*(b*x+a)^n/((d*x+c)^n))*a*b^6*c^2*d^2-36*A*B*ln(b*x+a)*x^3*b^ 
7*d^4*n-54*A^2*a^2*b^5*c*d^3+54*A^2*a*b^6*c^2*d^2-18*B^2*x^3*ln(e*(b*x+a)^ 
n/((d*x+c)^n))^2*b^7*d^4-18*B^2*ln(e*(b*x+a)^n/((d*x+c)^n))^2*b^7*c^3*d+18 
*A^2*a^3*b^4*d^4-18*A^2*b^7*c^3*d-108*B^2*a^2*b^5*c*d^3*n^2+27*B^2*a*b^6*c 
^2*d^2*n^2+66*A*B*a^3*b^4*d^4*n-12*A*B*b^7*c^3*d*n+66*B^2*ln(d*x+c)*a^3*b^ 
4*d^4*n^2+108*A*B*ln(d*x+c)*x^2*a*b^6*d^4*n-66*B^2*ln(b*x+a)*x^3*b^7*d^4*n 
^2+66*B^2*ln(d*x+c)*x^3*b^7*d^4*n^2-66*B^2*ln(b*x+a)*a^3*b^4*d^4*n^2-54*B^ 
2*x^2*ln(e*(b*x+a)^n/((d*x+c)^n))^2*a*b^6*d^4+66*B^2*x^2*a*b^6*d^4*n^2-66* 
B^2*x^2*b^7*c*d^3*n^2-54*B^2*x*ln(e*(b*x+a)^n/((d*x+c)^n))^2*a^2*b^5*d^4+1 
47*B^2*x*a^2*b^5*d^4*n^2+15*B^2*x*b^7*c^2*d^2*n^2-54*B^2*ln(e*(b*x+a)^n/(( 
d*x+c)^n))^2*a^2*b^5*c*d^3+54*B^2*ln(e*(b*x+a)^n/((d*x+c)^n))^2*a*b^6*c^2* 
d^2+66*B^2*ln(e*(b*x+a)^n/((d*x+c)^n))*a^3*b^4*d^4*n-12*B^2*ln(e*(b*x+a...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1635 vs. \(2 (419) = 838\).

Time = 0.13 (sec) , antiderivative size = 1635, normalized size of antiderivative = 3.83 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(a+b x)^4} \, dx=\text {Too large to display} \] Input:

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^2/(b*x+a)^4,x, algorithm="fri 
cas")
 

Output:

-1/54*(18*A^2*b^3*c^3 - 54*A^2*a*b^2*c^2*d + 54*A^2*a^2*b*c*d^2 - 18*A^2*a 
^3*d^3 + (4*B^2*b^3*c^3 - 27*B^2*a*b^2*c^2*d + 108*B^2*a^2*b*c*d^2 - 85*B^ 
2*a^3*d^3)*n^2 + 6*(11*(B^2*b^3*c*d^2 - B^2*a*b^2*d^3)*n^2 + 6*(A*B*b^3*c* 
d^2 - A*B*a*b^2*d^3)*n)*x^2 + 18*(B^2*b^3*d^3*n^2*x^3 + 3*B^2*a*b^2*d^3*n^ 
2*x^2 + 3*B^2*a^2*b*d^3*n^2*x + (B^2*b^3*c^3 - 3*B^2*a*b^2*c^2*d + 3*B^2*a 
^2*b*c*d^2)*n^2)*log(b*x + a)^2 + 18*(B^2*b^3*d^3*n^2*x^3 + 3*B^2*a*b^2*d^ 
3*n^2*x^2 + 3*B^2*a^2*b*d^3*n^2*x + (B^2*b^3*c^3 - 3*B^2*a*b^2*c^2*d + 3*B 
^2*a^2*b*c*d^2)*n^2)*log(d*x + c)^2 + 18*(B^2*b^3*c^3 - 3*B^2*a*b^2*c^2*d 
+ 3*B^2*a^2*b*c*d^2 - B^2*a^3*d^3)*log(e)^2 + 6*(2*A*B*b^3*c^3 - 9*A*B*a*b 
^2*c^2*d + 18*A*B*a^2*b*c*d^2 - 11*A*B*a^3*d^3)*n - 3*((5*B^2*b^3*c^2*d - 
54*B^2*a*b^2*c*d^2 + 49*B^2*a^2*b*d^3)*n^2 + 6*(A*B*b^3*c^2*d - 6*A*B*a*b^ 
2*c*d^2 + 5*A*B*a^2*b*d^3)*n)*x + 6*((11*B^2*b^3*d^3*n^2 + 6*A*B*b^3*d^3*n 
)*x^3 + (2*B^2*b^3*c^3 - 9*B^2*a*b^2*c^2*d + 18*B^2*a^2*b*c*d^2)*n^2 + 3*( 
6*A*B*a*b^2*d^3*n + (2*B^2*b^3*c*d^2 + 9*B^2*a*b^2*d^3)*n^2)*x^2 + 6*(A*B* 
b^3*c^3 - 3*A*B*a*b^2*c^2*d + 3*A*B*a^2*b*c*d^2)*n + 3*(6*A*B*a^2*b*d^3*n 
- (B^2*b^3*c^2*d - 6*B^2*a*b^2*c*d^2 - 6*B^2*a^2*b*d^3)*n^2)*x + 6*(B^2*b^ 
3*d^3*n*x^3 + 3*B^2*a*b^2*d^3*n*x^2 + 3*B^2*a^2*b*d^3*n*x + (B^2*b^3*c^3 - 
 3*B^2*a*b^2*c^2*d + 3*B^2*a^2*b*c*d^2)*n)*log(e))*log(b*x + a) - 6*((11*B 
^2*b^3*d^3*n^2 + 6*A*B*b^3*d^3*n)*x^3 + (2*B^2*b^3*c^3 - 9*B^2*a*b^2*c^2*d 
 + 18*B^2*a^2*b*c*d^2)*n^2 + 3*(6*A*B*a*b^2*d^3*n + (2*B^2*b^3*c*d^2 + ...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(a+b x)^4} \, dx=\text {Timed out} \] Input:

integrate((A+B*ln(e*(b*x+a)**n/((d*x+c)**n)))**2/(b*x+a)**4,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1500 vs. \(2 (419) = 838\).

Time = 0.13 (sec) , antiderivative size = 1500, normalized size of antiderivative = 3.51 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(a+b x)^4} \, dx=\text {Too large to display} \] Input:

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^2/(b*x+a)^4,x, algorithm="max 
ima")
 

Output:

-1/54*B^2*(6*(6*d^3*e*n*log(b*x + a)/(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2* 
c*d^2 - a^3*b*d^3) - 6*d^3*e*n*log(d*x + c)/(b^4*c^3 - 3*a*b^3*c^2*d + 3*a 
^2*b^2*c*d^2 - a^3*b*d^3) + (6*b^2*d^2*e*n*x^2 + 2*b^2*c^2*e*n - 7*a*b*c*d 
*e*n + 11*a^2*d^2*e*n - 3*(b^2*c*d*e*n - 5*a*b*d^2*e*n)*x)/(a^3*b^3*c^2 - 
2*a^4*b^2*c*d + a^5*b*d^2 + (b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*x^3 + 3* 
(a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*x^2 + 3*(a^2*b^4*c^2 - 2*a^3*b^3 
*c*d + a^4*b^2*d^2)*x))*log((b*x + a)^n*e/(d*x + c)^n)/e + (4*b^3*c^3*e^2* 
n^2 - 27*a*b^2*c^2*d*e^2*n^2 + 108*a^2*b*c*d^2*e^2*n^2 - 85*a^3*d^3*e^2*n^ 
2 + 66*(b^3*c*d^2*e^2*n^2 - a*b^2*d^3*e^2*n^2)*x^2 - 18*(b^3*d^3*e^2*n^2*x 
^3 + 3*a*b^2*d^3*e^2*n^2*x^2 + 3*a^2*b*d^3*e^2*n^2*x + a^3*d^3*e^2*n^2)*lo 
g(b*x + a)^2 - 18*(b^3*d^3*e^2*n^2*x^3 + 3*a*b^2*d^3*e^2*n^2*x^2 + 3*a^2*b 
*d^3*e^2*n^2*x + a^3*d^3*e^2*n^2)*log(d*x + c)^2 - 3*(5*b^3*c^2*d*e^2*n^2 
- 54*a*b^2*c*d^2*e^2*n^2 + 49*a^2*b*d^3*e^2*n^2)*x + 66*(b^3*d^3*e^2*n^2*x 
^3 + 3*a*b^2*d^3*e^2*n^2*x^2 + 3*a^2*b*d^3*e^2*n^2*x + a^3*d^3*e^2*n^2)*lo 
g(b*x + a) - 6*(11*b^3*d^3*e^2*n^2*x^3 + 33*a*b^2*d^3*e^2*n^2*x^2 + 33*a^2 
*b*d^3*e^2*n^2*x + 11*a^3*d^3*e^2*n^2 - 6*(b^3*d^3*e^2*n^2*x^3 + 3*a*b^2*d 
^3*e^2*n^2*x^2 + 3*a^2*b*d^3*e^2*n^2*x + a^3*d^3*e^2*n^2)*log(b*x + a))*lo 
g(d*x + c))/((a^3*b^4*c^3 - 3*a^4*b^3*c^2*d + 3*a^5*b^2*c*d^2 - a^6*b*d^3 
+ (b^7*c^3 - 3*a*b^6*c^2*d + 3*a^2*b^5*c*d^2 - a^3*b^4*d^3)*x^3 + 3*(a*b^6 
*c^3 - 3*a^2*b^5*c^2*d + 3*a^3*b^4*c*d^2 - a^4*b^3*d^3)*x^2 + 3*(a^2*b^...
 

Giac [F]

\[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(a+b x)^4} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{2}}{{\left (b x + a\right )}^{4}} \,d x } \] Input:

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^2/(b*x+a)^4,x, algorithm="gia 
c")
 

Output:

integrate((B*log((b*x + a)^n*e/(d*x + c)^n) + A)^2/(b*x + a)^4, x)
 

Mupad [B] (verification not implemented)

Time = 28.76 (sec) , antiderivative size = 911, normalized size of antiderivative = 2.13 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(a+b x)^4} \, dx =\text {Too large to display} \] Input:

int((A + B*log((e*(a + b*x)^n)/(c + d*x)^n))^2/(a + b*x)^4,x)
 

Output:

((18*A^2*a^2*d^2 + 18*A^2*b^2*c^2 + 85*B^2*a^2*d^2*n^2 + 4*B^2*b^2*c^2*n^2 
 - 36*A^2*a*b*c*d + 66*A*B*a^2*d^2*n + 12*A*B*b^2*c^2*n - 23*B^2*a*b*c*d*n 
^2 - 42*A*B*a*b*c*d*n)/(6*(a*d - b*c)) + (x*(49*B^2*a*b*d^2*n^2 - 5*B^2*b^ 
2*c*d*n^2 + 30*A*B*a*b*d^2*n - 6*A*B*b^2*c*d*n))/(2*(a*d - b*c)) + (d*x^2* 
(11*B^2*b^2*d*n^2 + 6*A*B*b^2*d*n))/(a*d - b*c))/(x^3*(9*b^5*c - 9*a*b^4*d 
) + x*(27*a^2*b^3*c - 27*a^3*b^2*d) - x^2*(27*a^2*b^3*d - 27*a*b^4*c) + 9* 
a^3*b^2*c - 9*a^4*b*d) - log((e*(a + b*x)^n)/(c + d*x)^n)^2*(B^2/(3*b*(a^3 
 + b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x)) - (B^2*d^3)/(3*b*(a^3*d^3 - b^3*c^3 
 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))) - log((e*(a + b*x)^n)/(c + d*x)^n)*((2 
*A*B)/(3*(a^3*b + b^4*x^3 + 3*a^2*b^2*x + 3*a*b^3*x^2)) + (2*B^2*d^3*(a*(( 
b*n*(a*d - b*c)*(3*a*d - b*c))/(2*d^2) + (a*b*n*(a*d - b*c))/d) + x*(b*((b 
*n*(a*d - b*c)*(3*a*d - b*c))/(2*d^2) + (a*b*n*(a*d - b*c))/d) + (2*a*b^2* 
n*(a*d - b*c))/d + (b^2*n*(a*d - b*c)*(3*a*d - b*c))/d^2) + (b*n*(a*d - b* 
c)*(3*a^2*d^2 + b^2*c^2 - 3*a*b*c*d))/d^3 + (3*b^3*n*x^2*(a*d - b*c))/d))/ 
(9*b*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)*(a^3*b + b^4*x^3 
+ 3*a^2*b^2*x + 3*a*b^3*x^2))) - (B*d^3*n*atan((B*d^3*n*(6*A + 11*B*n)*((b 
^4*c^3 + a^3*b*d^3 - a^2*b^2*c*d^2 - a*b^3*c^2*d)/(b^3*c^2 + a^2*b*d^2 - 2 
*a*b^2*c*d) + 2*b*d*x)*(b^3*c^2 + a^2*b*d^2 - 2*a*b^2*c*d)*1i)/(b*(11*B^2* 
d^3*n^2 + 6*A*B*d^3*n)*(a*d - b*c)^3))*(6*A + 11*B*n)*2i)/(9*b*(a*d - b*c) 
^3)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 1595, normalized size of antiderivative = 3.74 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(a+b x)^4} \, dx =\text {Too large to display} \] Input:

int((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^2/(b*x+a)^4,x)
 

Output:

(36*log(a + b*x)*a**5*b*d**3*n + 54*log(a + b*x)*a**4*b**2*d**3*n**2 + 108 
*log(a + b*x)*a**4*b**2*d**3*n*x + 12*log(a + b*x)*a**3*b**3*c*d**2*n**2 + 
 162*log(a + b*x)*a**3*b**3*d**3*n**2*x + 108*log(a + b*x)*a**3*b**3*d**3* 
n*x**2 + 36*log(a + b*x)*a**2*b**4*c*d**2*n**2*x + 162*log(a + b*x)*a**2*b 
**4*d**3*n**2*x**2 + 36*log(a + b*x)*a**2*b**4*d**3*n*x**3 + 36*log(a + b* 
x)*a*b**5*c*d**2*n**2*x**2 + 54*log(a + b*x)*a*b**5*d**3*n**2*x**3 + 12*lo 
g(a + b*x)*b**6*c*d**2*n**2*x**3 - 36*log(c + d*x)*a**5*b*d**3*n - 54*log( 
c + d*x)*a**4*b**2*d**3*n**2 - 108*log(c + d*x)*a**4*b**2*d**3*n*x - 12*lo 
g(c + d*x)*a**3*b**3*c*d**2*n**2 - 162*log(c + d*x)*a**3*b**3*d**3*n**2*x 
- 108*log(c + d*x)*a**3*b**3*d**3*n*x**2 - 36*log(c + d*x)*a**2*b**4*c*d** 
2*n**2*x - 162*log(c + d*x)*a**2*b**4*d**3*n**2*x**2 - 36*log(c + d*x)*a** 
2*b**4*d**3*n*x**3 - 36*log(c + d*x)*a*b**5*c*d**2*n**2*x**2 - 54*log(c + 
d*x)*a*b**5*d**3*n**2*x**3 - 12*log(c + d*x)*b**6*c*d**2*n**2*x**3 + 54*lo 
g(((a + b*x)**n*e)/(c + d*x)**n)**2*a**3*b**3*c*d**2 + 54*log(((a + b*x)** 
n*e)/(c + d*x)**n)**2*a**3*b**3*d**3*x - 54*log(((a + b*x)**n*e)/(c + d*x) 
**n)**2*a**2*b**4*c**2*d + 54*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a**2*b 
**4*d**3*x**2 + 18*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a*b**5*c**3 + 18* 
log(((a + b*x)**n*e)/(c + d*x)**n)**2*a*b**5*d**3*x**3 - 36*log(((a + b*x) 
**n*e)/(c + d*x)**n)*a**5*b*d**3 + 108*log(((a + b*x)**n*e)/(c + d*x)**n)* 
a**4*b**2*c*d**2 - 54*log(((a + b*x)**n*e)/(c + d*x)**n)*a**4*b**2*d**3...