\(\int \frac {(A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(a g+b g x)^3 (c i+d i x)} \, dx\) [192]

Optimal result

Integrand size = 45, antiderivative size = 369 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^3 (c i+d i x)} \, dx=\frac {4 b B^2 d n^2 (c+d x)}{(b c-a d)^3 g^3 i (a+b x)}-\frac {b^2 B^2 n^2 (c+d x)^2}{4 (b c-a d)^3 g^3 i (a+b x)^2}+\frac {4 b B d n (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 g^3 i (a+b x)}-\frac {b^2 B n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^3 g^3 i (a+b x)^2}+\frac {2 b d (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b c-a d)^3 g^3 i (a+b x)}-\frac {b^2 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 (b c-a d)^3 g^3 i (a+b x)^2}+\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{3 B (b c-a d)^3 g^3 i n} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(975\) vs. \(2(369)=738\).

Time = 1.23 (sec) , antiderivative size = 975, normalized size of antiderivative = 2.64 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^3 (c i+d i x)} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.74, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2961, 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.

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

\(\Big \downarrow \) 2961

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

((4*b*B^2*d*n^2*(c + d*x))/(a + b*x) - (b^2*B^2*n^2*(c + d*x)^2)/(4*(a + b 
*x)^2) + (4*b*B*d*n*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + 
 b*x) - (b^2*B*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(a 
 + b*x)^2) + (2*b*d*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(a 
 + b*x) - (b^2*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*(a 
 + b*x)^2) + (d^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3)/(3*B*n))/((b*c 
 - a*d)^3*g^3*i)
 

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 2961

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( 
B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol 
] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q   Subst[Int[x^m*((A + B*L 
og[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[ 
b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IntegersQ[m, q]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1235\) vs. \(2(361)=722\).

Time = 13.40 (sec) , antiderivative size = 1236, normalized size of antiderivative = 3.35

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(1236\)

Input:

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

Output:

-1/12*(4*B^2*ln(e*((b*x+a)/(d*x+c))^n)^3*a^6*c^2*d^2+12*A^2*ln(e*((b*x+a)/ 
(d*x+c))^n)*a^6*c^2*d^2+36*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)*a^4*b^2*c^3*d*n 
^2+24*A*B*x*ln(e*((b*x+a)/(d*x+c))^n)^2*a^5*b*c^2*d^2+48*A*B*x*a^5*b*c^2*d 
^2*n^2-60*A*B*x*a^4*b^2*c^3*d*n^2+48*A*B*ln(e*((b*x+a)/(d*x+c))^n)*a^5*b*c 
^3*d*n+4*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)^3*a^4*b^2*c^2*d^2+45*B^2*x^2*a^ 
4*b^2*c^2*d^2*n^3-48*B^2*x^2*a^3*b^3*c^3*d*n^3+6*A*B*x^2*a^2*b^4*c^4*n^2+8 
*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)^3*a^5*b*c^2*d^2+48*B^2*x*a^5*b*c^2*d^2*n^ 
3-54*B^2*x*a^4*b^2*c^3*d*n^3-36*A^2*x*a^4*b^2*c^3*d*n-12*A*B*ln(e*((b*x+a) 
/(d*x+c))^n)*a^4*b^2*c^4*n+12*A^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^4*b^2*c^ 
2*d^2+18*A^2*x^2*a^4*b^2*c^2*d^2*n-24*A^2*x^2*a^3*b^3*c^3*d*n+12*A*B*x*a^3 
*b^3*c^4*n^2+24*B^2*ln(e*((b*x+a)/(d*x+c))^n)^2*a^5*b*c^3*d*n+48*B^2*ln(e* 
((b*x+a)/(d*x+c))^n)*a^5*b*c^3*d*n^2+24*A^2*x*ln(e*((b*x+a)/(d*x+c))^n)*a^ 
5*b*c^2*d^2+24*A^2*x*a^5*b*c^2*d^2*n+18*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)^ 
2*a^4*b^2*c^2*d^2*n+42*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^4*b^2*c^2*d^2*n 
^2+12*A*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)^2*a^4*b^2*c^2*d^2+42*A*B*x^2*a^4*b 
^2*c^2*d^2*n^2-48*A*B*x^2*a^3*b^3*c^3*d*n^2+24*B^2*x*ln(e*((b*x+a)/(d*x+c) 
)^n)^2*a^5*b*c^2*d^2*n+12*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)^2*a^4*b^2*c^3*d* 
n+48*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)*a^5*b*c^2*d^2*n^2+3*B^2*x^2*a^2*b^4*c 
^4*n^3+6*B^2*x*a^3*b^3*c^4*n^3+6*A^2*x^2*a^2*b^4*c^4*n-6*B^2*ln(e*((b*x+a) 
/(d*x+c))^n)^2*a^4*b^2*c^4*n+36*A*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^4*b...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1068 vs. \(2 (361) = 722\).

Time = 0.10 (sec) , antiderivative size = 1068, normalized size of antiderivative = 2.89 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^3 (c i+d i x)} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/12*(6*A^2*b^2*c^2 - 24*A^2*a*b*c*d + 18*A^2*a^2*d^2 - 4*(B^2*b^2*d^2*n^ 
2*x^2 + 2*B^2*a*b*d^2*n^2*x + B^2*a^2*d^2*n^2)*log((b*x + a)/(d*x + c))^3 
+ 3*(B^2*b^2*c^2 - 16*B^2*a*b*c*d + 15*B^2*a^2*d^2)*n^2 + 6*(B^2*b^2*c^2 - 
 4*B^2*a*b*c*d + 3*B^2*a^2*d^2 - 2*(B^2*b^2*c*d - B^2*a*b*d^2)*x - 2*(B^2* 
b^2*d^2*x^2 + 2*B^2*a*b*d^2*x + B^2*a^2*d^2)*log((b*x + a)/(d*x + c)))*log 
(e)^2 - 6*(2*A*B*a^2*d^2*n - (B^2*b^2*c^2 - 4*B^2*a*b*c*d)*n^2 + (3*B^2*b^ 
2*d^2*n^2 + 2*A*B*b^2*d^2*n)*x^2 + 2*(2*A*B*a*b*d^2*n + (B^2*b^2*c*d + 2*B 
^2*a*b*d^2)*n^2)*x)*log((b*x + a)/(d*x + c))^2 + 6*(A*B*b^2*c^2 - 8*A*B*a* 
b*c*d + 7*A*B*a^2*d^2)*n - 6*(2*A^2*b^2*c*d - 2*A^2*a*b*d^2 + 7*(B^2*b^2*c 
*d - B^2*a*b*d^2)*n^2 + 6*(A*B*b^2*c*d - A*B*a*b*d^2)*n)*x + 6*(2*A*B*b^2* 
c^2 - 8*A*B*a*b*c*d + 6*A*B*a^2*d^2 - 2*(B^2*b^2*d^2*n*x^2 + 2*B^2*a*b*d^2 
*n*x + B^2*a^2*d^2*n)*log((b*x + a)/(d*x + c))^2 + (B^2*b^2*c^2 - 8*B^2*a* 
b*c*d + 7*B^2*a^2*d^2)*n - 2*(2*A*B*b^2*c*d - 2*A*B*a*b*d^2 + 3*(B^2*b^2*c 
*d - B^2*a*b*d^2)*n)*x - 2*(2*A*B*a^2*d^2 + (3*B^2*b^2*d^2*n + 2*A*B*b^2*d 
^2)*x^2 - (B^2*b^2*c^2 - 4*B^2*a*b*c*d)*n + 2*(2*A*B*a*b*d^2 + (B^2*b^2*c* 
d + 2*B^2*a*b*d^2)*n)*x)*log((b*x + a)/(d*x + c)))*log(e) - 6*(2*A^2*a^2*d 
^2 - (B^2*b^2*c^2 - 8*B^2*a*b*c*d)*n^2 + (7*B^2*b^2*d^2*n^2 + 6*A*B*b^2*d^ 
2*n + 2*A^2*b^2*d^2)*x^2 - 2*(A*B*b^2*c^2 - 4*A*B*a*b*c*d)*n + 2*(2*A^2*a* 
b*d^2 + (3*B^2*b^2*c*d + 4*B^2*a*b*d^2)*n^2 + 2*(A*B*b^2*c*d + 2*A*B*a*b*d 
^2)*n)*x)*log((b*x + a)/(d*x + c)))/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2126 vs. \(2 (361) = 722\).

Time = 0.19 (sec) , antiderivative size = 2126, normalized size of antiderivative = 5.76 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^3 (c i+d i x)} \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*B^2*((2*b*d*x - b*c + 3*a*d)/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*g^ 
3*i*x^2 + 2*(a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2)*g^3*i*x + (a^2*b^2*c^2 
 - 2*a^3*b*c*d + a^4*d^2)*g^3*i) + 2*d^2*log(b*x + a)/((b^3*c^3 - 3*a*b^2* 
c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^3*i) - 2*d^2*log(d*x + c)/((b^3*c^3 - 3 
*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^3*i))*log(e*(b*x/(d*x + c) + a/( 
d*x + c))^n)^2 + A*B*((2*b*d*x - b*c + 3*a*d)/((b^4*c^2 - 2*a*b^3*c*d + a^ 
2*b^2*d^2)*g^3*i*x^2 + 2*(a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2)*g^3*i*x + 
 (a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2)*g^3*i) + 2*d^2*log(b*x + a)/((b^3*c 
^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^3*i) - 2*d^2*log(d*x + c)/ 
((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^3*i))*log(e*(b*x/(d 
*x + c) + a/(d*x + c))^n) - 1/12*((3*b^2*c^2 - 48*a*b*c*d + 45*a^2*d^2 - 4 
*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(b*x + a)^3 + 4*(b^2*d^2*x^2 + 2 
*a*b*d^2*x + a^2*d^2)*log(d*x + c)^3 + 18*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2 
*d^2)*log(b*x + a)^2 + 6*(3*b^2*d^2*x^2 + 6*a*b*d^2*x + 3*a^2*d^2 - 2*(b^2 
*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(b*x + a))*log(d*x + c)^2 - 42*(b^2*c 
*d - a*b*d^2)*x - 42*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(b*x + a) + 
6*(7*b^2*d^2*x^2 + 14*a*b*d^2*x + 7*a^2*d^2 + 2*(b^2*d^2*x^2 + 2*a*b*d^2*x 
 + a^2*d^2)*log(b*x + a)^2 - 6*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(b 
*x + a))*log(d*x + c))*n^2/(a^2*b^3*c^3*g^3*i - 3*a^3*b^2*c^2*d*g^3*i + 3* 
a^4*b*c*d^2*g^3*i - a^5*d^3*g^3*i + (b^5*c^3*g^3*i - 3*a*b^4*c^2*d*g^3*...
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 29.93 (sec) , antiderivative size = 1011, normalized size of antiderivative = 2.74 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^3 (c i+d i x)} \, dx =\text {Too large to display} \] Input:

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

Output:

log(e*((a + b*x)/(c + d*x))^n)*((B^2*n)/(x^2*(b^3*c*g^3*i - a*b^2*d*g^3*i) 
 + x*(2*a*b^2*c*g^3*i - 2*a^2*b*d*g^3*i) - a^3*d*g^3*i + a^2*b*c*g^3*i) - 
(d^2*(3*B^2*n + 2*A*B)*((a*g^3*i*n*(a*d - b*c)^2)/(2*d) + (g^3*i*n*(a*d - 
b*c)^2*(2*a*d - b*c))/(2*d^2) + (b*g^3*i*n*x*(a*d - b*c)^2)/d))/(g^3*i*n*( 
a*d - b*c)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)*(x^2*(b^3*c*g^3*i - a*b^2*d*g^3 
*i) + x*(2*a*b^2*c*g^3*i - 2*a^2*b*d*g^3*i) - a^3*d*g^3*i + a^2*b*c*g^3*i) 
)) - log(e*((a + b*x)/(c + d*x))^n)^2*((d^2*(3*B^2*n + 2*A*B))/(2*g^3*i*n* 
(a*d - b*c)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) - (B^2*d^2*((g^3*i*n*(a*d - b 
*c)*(2*a*d - b*c))/(2*d^2) + (a*g^3*i*n*(a*d - b*c))/(2*d) + (b*g^3*i*n*x* 
(a*d - b*c))/d))/(g^3*i*n*(a*d - b*c)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)*(a^2 
*g^3*i + b^2*g^3*i*x^2 + 2*a*b*g^3*i*x))) - ((6*A^2*a*d - 2*A^2*b*c + 15*B 
^2*a*d*n^2 - B^2*b*c*n^2 + 14*A*B*a*d*n - 2*A*B*b*c*n)/(2*(a*d - b*c)) + ( 
x*(2*A^2*b*d + 7*B^2*b*d*n^2 + 6*A*B*b*d*n))/(a*d - b*c))/(x^2*(2*b^3*c*g^ 
3*i - 2*a*b^2*d*g^3*i) + x*(4*a*b^2*c*g^3*i - 4*a^2*b*d*g^3*i) - 2*a^3*d*g 
^3*i + 2*a^2*b*c*g^3*i) + (d^2*atan((d^2*((a^3*d^3*g^3*i + b^3*c^3*g^3*i - 
 a*b^2*c^2*d*g^3*i - a^2*b*c*d^2*g^3*i)/(a^2*d^2*g^3*i + b^2*c^2*g^3*i - 2 
*a*b*c*d*g^3*i) + 2*b*d*x)*(A^2 + (7*B^2*n^2)/2 + 3*A*B*n)*(a^2*d^2*g^3*i 
+ b^2*c^2*g^3*i - 2*a*b*c*d*g^3*i)*2i)/(g^3*i*(a*d - b*c)^3*(2*A^2*d^2 + 7 
*B^2*d^2*n^2 + 6*A*B*d^2*n)))*(A^2 + (7*B^2*n^2)/2 + 3*A*B*n)*2i)/(g^3*i*( 
a*d - b*c)^3) - (B^2*d^2*log(e*((a + b*x)/(c + d*x))^n)^3)/(3*g^3*i*n*(...
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 1609, normalized size of antiderivative = 4.36 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^3 (c i+d i x)} \, dx =\text {Too large to display} \] Input:

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

Output:

(i*(12*log(a + b*x)*a**5*d**2*n + 24*log(a + b*x)*a**4*b*d**2*n**2 + 24*lo 
g(a + b*x)*a**4*b*d**2*n*x + 12*log(a + b*x)*a**3*b**2*c*d*n**2 + 24*log(a 
 + b*x)*a**3*b**2*d**2*n**3 + 48*log(a + b*x)*a**3*b**2*d**2*n**2*x + 12*l 
og(a + b*x)*a**3*b**2*d**2*n*x**2 + 18*log(a + b*x)*a**2*b**3*c*d*n**3 + 2 
4*log(a + b*x)*a**2*b**3*c*d*n**2*x + 48*log(a + b*x)*a**2*b**3*d**2*n**3* 
x + 24*log(a + b*x)*a**2*b**3*d**2*n**2*x**2 + 36*log(a + b*x)*a*b**4*c*d* 
n**3*x + 12*log(a + b*x)*a*b**4*c*d*n**2*x**2 + 24*log(a + b*x)*a*b**4*d** 
2*n**3*x**2 + 18*log(a + b*x)*b**5*c*d*n**3*x**2 - 12*log(c + d*x)*a**5*d* 
*2*n - 24*log(c + d*x)*a**4*b*d**2*n**2 - 24*log(c + d*x)*a**4*b*d**2*n*x 
- 12*log(c + d*x)*a**3*b**2*c*d*n**2 - 24*log(c + d*x)*a**3*b**2*d**2*n**3 
 - 48*log(c + d*x)*a**3*b**2*d**2*n**2*x - 12*log(c + d*x)*a**3*b**2*d**2* 
n*x**2 - 18*log(c + d*x)*a**2*b**3*c*d*n**3 - 24*log(c + d*x)*a**2*b**3*c* 
d*n**2*x - 48*log(c + d*x)*a**2*b**3*d**2*n**3*x - 24*log(c + d*x)*a**2*b* 
*3*d**2*n**2*x**2 - 36*log(c + d*x)*a*b**4*c*d*n**3*x - 12*log(c + d*x)*a* 
b**4*c*d*n**2*x**2 - 24*log(c + d*x)*a*b**4*d**2*n**3*x**2 - 18*log(c + d* 
x)*b**5*c*d*n**3*x**2 + 4*log(((a + b*x)**n*e)/(c + d*x)**n)**3*a**3*b**2* 
d**2 + 8*log(((a + b*x)**n*e)/(c + d*x)**n)**3*a**2*b**3*d**2*x + 4*log((( 
a + b*x)**n*e)/(c + d*x)**n)**3*a*b**4*d**2*x**2 + 12*log(((a + b*x)**n*e) 
/(c + d*x)**n)**2*a**4*b*d**2 + 24*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a 
**3*b**2*d**2*x + 24*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a**2*b**3*c*...