\(\int \frac {(A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(a g+b g x)^4 (c i+d i x)^2} \, dx\) [99]

Optimal result

Integrand size = 42, antiderivative size = 682 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4 (c i+d i x)^2} \, dx=-\frac {2 A B d^4 (a+b x)}{(b c-a d)^5 g^4 i^2 (c+d x)}+\frac {2 B^2 d^4 (a+b x)}{(b c-a d)^5 g^4 i^2 (c+d x)}-\frac {12 b^2 B^2 d^2 (c+d x)}{(b c-a d)^5 g^4 i^2 (a+b x)}+\frac {b^3 B^2 d (c+d x)^2}{(b c-a d)^5 g^4 i^2 (a+b x)^2}-\frac {2 b^4 B^2 (c+d x)^3}{27 (b c-a d)^5 g^4 i^2 (a+b x)^3}-\frac {2 B^2 d^4 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(b c-a d)^5 g^4 i^2 (c+d x)}-\frac {12 b^2 B d^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^5 g^4 i^2 (a+b x)}+\frac {2 b^3 B d (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^5 g^4 i^2 (a+b x)^2}-\frac {2 b^4 B (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9 (b c-a d)^5 g^4 i^2 (a+b x)^3}+\frac {d^4 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^5 g^4 i^2 (c+d x)}-\frac {6 b^2 d^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^5 g^4 i^2 (a+b x)}+\frac {2 b^3 d (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^5 g^4 i^2 (a+b x)^2}-\frac {b^4 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 (b c-a d)^5 g^4 i^2 (a+b x)^3}-\frac {4 b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^3}{3 B (b c-a d)^5 g^4 i^2} \] Output:

-2*A*B*d^4*(b*x+a)/(-a*d+b*c)^5/g^4/i^2/(d*x+c)+2*B^2*d^4*(b*x+a)/(-a*d+b* 
c)^5/g^4/i^2/(d*x+c)-12*b^2*B^2*d^2*(d*x+c)/(-a*d+b*c)^5/g^4/i^2/(b*x+a)+b 
^3*B^2*d*(d*x+c)^2/(-a*d+b*c)^5/g^4/i^2/(b*x+a)^2-2/27*b^4*B^2*(d*x+c)^3/( 
-a*d+b*c)^5/g^4/i^2/(b*x+a)^3-2*B^2*d^4*(b*x+a)*ln(e*(b*x+a)/(d*x+c))/(-a* 
d+b*c)^5/g^4/i^2/(d*x+c)-12*b^2*B*d^2*(d*x+c)*(A+B*ln(e*(b*x+a)/(d*x+c)))/ 
(-a*d+b*c)^5/g^4/i^2/(b*x+a)+2*b^3*B*d*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c) 
))/(-a*d+b*c)^5/g^4/i^2/(b*x+a)^2-2/9*b^4*B*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d 
*x+c)))/(-a*d+b*c)^5/g^4/i^2/(b*x+a)^3+d^4*(b*x+a)*(A+B*ln(e*(b*x+a)/(d*x+ 
c)))^2/(-a*d+b*c)^5/g^4/i^2/(d*x+c)-6*b^2*d^2*(d*x+c)*(A+B*ln(e*(b*x+a)/(d 
*x+c)))^2/(-a*d+b*c)^5/g^4/i^2/(b*x+a)+2*b^3*d*(d*x+c)^2*(A+B*ln(e*(b*x+a) 
/(d*x+c)))^2/(-a*d+b*c)^5/g^4/i^2/(b*x+a)^2-1/3*b^4*(d*x+c)^3*(A+B*ln(e*(b 
*x+a)/(d*x+c)))^2/(-a*d+b*c)^5/g^4/i^2/(b*x+a)^3-4/3*b*d^3*(A+B*ln(e*(b*x+ 
a)/(d*x+c)))^3/B/(-a*d+b*c)^5/g^4/i^2
 

Mathematica [A] (verified)

Time = 1.85 (sec) , antiderivative size = 613, normalized size of antiderivative = 0.90 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4 (c i+d i x)^2} \, dx=-\frac {-27 \left (A^2-2 A B+2 B^2\right ) d^3 (-b c+a d) (a+b x)^3+b \left (9 A^2+6 A B+2 B^2\right ) (b c-a d)^3 (c+d x)-3 b \left (9 A^2+12 A B+7 B^2\right ) d (b c-a d)^2 (a+b x) (c+d x)+3 b \left (27 A^2+78 A B+92 B^2\right ) d^2 (b c-a d) (a+b x)^2 (c+d x)+6 b \left (18 A^2+30 A B+55 B^2\right ) d^3 (a+b x)^3 (c+d x) \log (a+b x)+6 B (b c-a d) \left (9 (A-B) d^3 (a+b x)^3+b (3 A+B) (b c-a d)^2 (c+d x)-3 b (3 A+2 B) d (b c-a d) (a+b x) (c+d x)+3 b (9 A+13 B) d^2 (a+b x)^2 (c+d x)\right ) \log \left (\frac {e (a+b x)}{c+d x}\right )+9 B \left (-3 a^4 B d^4+12 a^3 b d^3 (-B d x+A (c+d x))+18 a^2 b^2 d^2 (2 A d x (c+d x)+B c (c+2 d x))+6 a b^3 d \left (6 A d^2 x^2 (c+d x)+B \left (-c^3+3 c^2 d x+9 c d^2 x^2+3 d^3 x^3\right )\right )+b^4 \left (12 A d^3 x^3 (c+d x)+B \left (c^4-2 c^3 d x+6 c^2 d^2 x^2+22 c d^3 x^3+10 d^4 x^4\right )\right )\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )+36 b B^2 d^3 (a+b x)^3 (c+d x) \log ^3\left (\frac {e (a+b x)}{c+d x}\right )-6 b \left (18 A^2+30 A B+55 B^2\right ) d^3 (a+b x)^3 (c+d x) \log (c+d x)}{27 (b c-a d)^5 g^4 i^2 (a+b x)^3 (c+d x)} \] Input:

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

Output:

-1/27*(-27*(A^2 - 2*A*B + 2*B^2)*d^3*(-(b*c) + a*d)*(a + b*x)^3 + b*(9*A^2 
 + 6*A*B + 2*B^2)*(b*c - a*d)^3*(c + d*x) - 3*b*(9*A^2 + 12*A*B + 7*B^2)*d 
*(b*c - a*d)^2*(a + b*x)*(c + d*x) + 3*b*(27*A^2 + 78*A*B + 92*B^2)*d^2*(b 
*c - a*d)*(a + b*x)^2*(c + d*x) + 6*b*(18*A^2 + 30*A*B + 55*B^2)*d^3*(a + 
b*x)^3*(c + d*x)*Log[a + b*x] + 6*B*(b*c - a*d)*(9*(A - B)*d^3*(a + b*x)^3 
 + b*(3*A + B)*(b*c - a*d)^2*(c + d*x) - 3*b*(3*A + 2*B)*d*(b*c - a*d)*(a 
+ b*x)*(c + d*x) + 3*b*(9*A + 13*B)*d^2*(a + b*x)^2*(c + d*x))*Log[(e*(a + 
 b*x))/(c + d*x)] + 9*B*(-3*a^4*B*d^4 + 12*a^3*b*d^3*(-(B*d*x) + A*(c + d* 
x)) + 18*a^2*b^2*d^2*(2*A*d*x*(c + d*x) + B*c*(c + 2*d*x)) + 6*a*b^3*d*(6* 
A*d^2*x^2*(c + d*x) + B*(-c^3 + 3*c^2*d*x + 9*c*d^2*x^2 + 3*d^3*x^3)) + b^ 
4*(12*A*d^3*x^3*(c + d*x) + B*(c^4 - 2*c^3*d*x + 6*c^2*d^2*x^2 + 22*c*d^3* 
x^3 + 10*d^4*x^4)))*Log[(e*(a + b*x))/(c + d*x)]^2 + 36*b*B^2*d^3*(a + b*x 
)^3*(c + d*x)*Log[(e*(a + b*x))/(c + d*x)]^3 - 6*b*(18*A^2 + 30*A*B + 55*B 
^2)*d^3*(a + b*x)^3*(c + d*x)*Log[c + d*x])/((b*c - a*d)^5*g^4*i^2*(a + b* 
x)^3*(c + d*x))
 

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 475, normalized size of antiderivative = 0.70, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2962, 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.

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

\(\Big \downarrow \) 2962

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {b^4 (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{3 (a+b x)^3}-\frac {2 b^4 B (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{9 (a+b x)^3}+\frac {2 b^3 d (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{(a+b x)^2}+\frac {2 b^3 B d (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{(a+b x)^2}-\frac {6 b^2 d^2 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{a+b x}-\frac {12 b^2 B d^2 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}+\frac {d^4 (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{c+d x}-\frac {2 A B d^4 (a+b x)}{c+d x}-\frac {4 b d^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^3}{3 B}-\frac {2 b^4 B^2 (c+d x)^3}{27 (a+b x)^3}+\frac {b^3 B^2 d (c+d x)^2}{(a+b x)^2}-\frac {12 b^2 B^2 d^2 (c+d x)}{a+b x}-\frac {2 B^2 d^4 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x}+\frac {2 B^2 d^4 (a+b x)}{c+d x}}{g^4 i^2 (b c-a d)^5}\)

Input:

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

Output:

((-2*A*B*d^4*(a + b*x))/(c + d*x) + (2*B^2*d^4*(a + b*x))/(c + d*x) - (12* 
b^2*B^2*d^2*(c + d*x))/(a + b*x) + (b^3*B^2*d*(c + d*x)^2)/(a + b*x)^2 - ( 
2*b^4*B^2*(c + d*x)^3)/(27*(a + b*x)^3) - (2*B^2*d^4*(a + b*x)*Log[(e*(a + 
 b*x))/(c + d*x)])/(c + d*x) - (12*b^2*B*d^2*(c + d*x)*(A + B*Log[(e*(a + 
b*x))/(c + d*x)]))/(a + b*x) + (2*b^3*B*d*(c + d*x)^2*(A + B*Log[(e*(a + b 
*x))/(c + d*x)]))/(a + b*x)^2 - (2*b^4*B*(c + d*x)^3*(A + B*Log[(e*(a + b* 
x))/(c + d*x)]))/(9*(a + b*x)^3) + (d^4*(a + b*x)*(A + B*Log[(e*(a + b*x)) 
/(c + d*x)])^2)/(c + d*x) - (6*b^2*d^2*(c + d*x)*(A + B*Log[(e*(a + b*x))/ 
(c + d*x)])^2)/(a + b*x) + (2*b^3*d*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/( 
c + d*x)])^2)/(a + b*x)^2 - (b^4*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + 
 d*x)])^2)/(3*(a + b*x)^3) - (4*b*d^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]) 
^3)/(3*B))/((b*c - a*d)^5*g^4*i^2)
 

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 2962

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ 
)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Sy 
mbol] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q   Subst[Int[x^m*((A + 
 B*Log[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; 
 FreeQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && EqQ[n + mn, 0] && IGt 
Q[n, 0] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && I 
ntegersQ[m, q]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1385\) vs. \(2(674)=1348\).

Time = 3.59 (sec) , antiderivative size = 1386, normalized size of antiderivative = 2.03

method	result	size
parts	\(\text {Expression too large to display}\)	\(1386\)
derivativedivides	\(\text {Expression too large to display}\)	\(1588\)
default	\(\text {Expression too large to display}\)	\(1588\)
risch	\(\text {Expression too large to display}\)	\(2173\)
parallelrisch	\(\text {Expression too large to display}\)	\(2680\)
norman	\(\text {Expression too large to display}\)	\(2748\)

Input:

int((A+B*ln(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^4/(d*i*x+c*i)^2,x,method=_RE 
TURNVERBOSE)
 

Output:

A^2/g^4/i^2*(-d^3/(a*d-b*c)^4/(d*x+c)-4*d^3/(a*d-b*c)^5*b*ln(d*x+c)-1/3*b/ 
(a*d-b*c)^2/(b*x+a)^3+4*d^3/(a*d-b*c)^5*b*ln(b*x+a)-3*b/(a*d-b*c)^4*d^2/(b 
*x+a)-b/(a*d-b*c)^3*d/(b*x+a)^2)-B^2/g^4/i^2/(a*d-b*c)/e*(d^4/(a*d-b*c)^4* 
((b*e/d+(a*d-b*c)*e/d/(d*x+c))*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2-2*(b*e/d+ 
(a*d-b*c)*e/d/(d*x+c))*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))+2*(a*d-b*c)*e/d/(d* 
x+c)+2*b*e/d)-4/3/(a*d-b*c)^4*b*d^3*e*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3+6/ 
(a*d-b*c)^4*b^2*d^2*e^2*(-1/(b*e/d+(a*d-b*c)*e/d/(d*x+c))*ln(b*e/d+(a*d-b* 
c)*e/d/(d*x+c))^2-2/(b*e/d+(a*d-b*c)*e/d/(d*x+c))*ln(b*e/d+(a*d-b*c)*e/d/( 
d*x+c))-2/(b*e/d+(a*d-b*c)*e/d/(d*x+c)))-4/(a*d-b*c)^4*b^3*d*e^3*(-1/2/(b* 
e/d+(a*d-b*c)*e/d/(d*x+c))^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2-1/2/(b*e/d+ 
(a*d-b*c)*e/d/(d*x+c))^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/4/(b*e/d+(a*d-b 
*c)*e/d/(d*x+c))^2)+1/(a*d-b*c)^4*b^4*e^4*(-1/3/(b*e/d+(a*d-b*c)*e/d/(d*x+ 
c))^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2-2/9/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^ 
3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-2/27/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3))-2 
*A*B/g^4/i^2/(a*d-b*c)/e*(d^4/(a*d-b*c)^4*((b*e/d+(a*d-b*c)*e/d/(d*x+c))*l 
n(b*e/d+(a*d-b*c)*e/d/(d*x+c))-(a*d-b*c)*e/d/(d*x+c)-b*e/d)-2/(a*d-b*c)^4* 
b*d^3*e*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2+6/(a*d-b*c)^4*b^2*d^2*e^2*(-1/(b 
*e/d+(a*d-b*c)*e/d/(d*x+c))*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/(b*e/d+(a*d- 
b*c)*e/d/(d*x+c)))-4/(a*d-b*c)^4*b^3*d*e^3*(-1/2/(b*e/d+(a*d-b*c)*e/d/(d*x 
+c))^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/4/(b*e/d+(a*d-b*c)*e/d/(d*x+c)...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1534 vs. \(2 (674) = 1348\).

Time = 0.12 (sec) , antiderivative size = 1534, normalized size of antiderivative = 2.25 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4 (c i+d i x)^2} \, dx=\text {Too large to display} \] Input:

integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^4/(d*i*x+c*i)^2,x, al 
gorithm="fricas")
 

Output:

-1/27*((9*A^2 + 6*A*B + 2*B^2)*b^4*c^4 - 27*(2*A^2 + 2*A*B + B^2)*a*b^3*c^ 
3*d + 162*(A^2 + 2*A*B + 2*B^2)*a^2*b^2*c^2*d^2 - 5*(18*A^2 + 66*A*B + 49* 
B^2)*a^3*b*c*d^3 - 27*(A^2 - 2*A*B + 2*B^2)*a^4*d^4 + 6*((18*A^2 + 30*A*B 
+ 55*B^2)*b^4*c*d^3 - (18*A^2 + 30*A*B + 55*B^2)*a*b^3*d^4)*x^3 + 36*(B^2* 
b^4*d^4*x^4 + B^2*a^3*b*c*d^3 + (B^2*b^4*c*d^3 + 3*B^2*a*b^3*d^4)*x^3 + 3* 
(B^2*a*b^3*c*d^3 + B^2*a^2*b^2*d^4)*x^2 + (3*B^2*a^2*b^2*c*d^3 + B^2*a^3*b 
*d^4)*x)*log((b*e*x + a*e)/(d*x + c))^3 + 3*((18*A^2 + 66*A*B + 85*B^2)*b^ 
4*c^2*d^2 + 8*(9*A^2 + 6*A*B + 20*B^2)*a*b^3*c*d^3 - (90*A^2 + 114*A*B + 2 
45*B^2)*a^2*b^2*d^4)*x^2 + 9*(2*(6*A*B + 5*B^2)*b^4*d^4*x^4 + B^2*b^4*c^4 
- 6*B^2*a*b^3*c^3*d + 18*B^2*a^2*b^2*c^2*d^2 + 12*A*B*a^3*b*c*d^3 - 3*B^2* 
a^4*d^4 + 2*((6*A*B + 11*B^2)*b^4*c*d^3 + 9*(2*A*B + B^2)*a*b^3*d^4)*x^3 + 
 6*(B^2*b^4*c^2*d^2 + 6*A*B*a^2*b^2*d^4 + 3*(2*A*B + 3*B^2)*a*b^3*c*d^3)*x 
^2 - 2*(B^2*b^4*c^3*d - 9*B^2*a*b^3*c^2*d^2 - 18*(A*B + B^2)*a^2*b^2*c*d^3 
 - 6*(A*B - B^2)*a^3*b*d^4)*x)*log((b*e*x + a*e)/(d*x + c))^2 - ((18*A^2 + 
 30*A*B + 19*B^2)*b^4*c^3*d - 81*(2*A^2 + 6*A*B + 7*B^2)*a*b^3*c^2*d^2 - 3 
*(18*A^2 - 114*A*B - 29*B^2)*a^2*b^2*c*d^3 + (198*A^2 + 114*A*B + 461*B^2) 
*a^3*b*d^4)*x + 6*((18*A^2 + 30*A*B + 55*B^2)*b^4*d^4*x^4 + 18*A^2*a^3*b*c 
*d^3 + (3*A*B + B^2)*b^4*c^4 - 9*(2*A*B + B^2)*a*b^3*c^3*d + 54*(A*B + B^2 
)*a^2*b^2*c^2*d^2 - 9*(A*B - B^2)*a^4*d^4 + ((18*A^2 + 66*A*B + 85*B^2)*b^ 
4*c*d^3 + 27*(2*A^2 + 2*A*B + 5*B^2)*a*b^3*d^4)*x^3 + 3*((6*A*B + 11*B^...
                                                                                    
                                                                                    
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6160 vs. \(2 (674) = 1348\).

Time = 0.57 (sec) , antiderivative size = 6160, normalized size of antiderivative = 9.03 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4 (c i+d i x)^2} \, dx=\text {Too large to display} \] Input:

integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^4/(d*i*x+c*i)^2,x, al 
gorithm="maxima")
 

Output:

-1/3*B^2*((12*b^3*d^3*x^3 + b^3*c^3 - 5*a*b^2*c^2*d + 13*a^2*b*c*d^2 + 3*a 
^3*d^3 + 6*(b^3*c*d^2 + 5*a*b^2*d^3)*x^2 - 2*(b^3*c^2*d - 8*a*b^2*c*d^2 - 
11*a^2*b*d^3)*x)/((b^7*c^4*d - 4*a*b^6*c^3*d^2 + 6*a^2*b^5*c^2*d^3 - 4*a^3 
*b^4*c*d^4 + a^4*b^3*d^5)*g^4*i^2*x^4 + (b^7*c^5 - a*b^6*c^4*d - 6*a^2*b^5 
*c^3*d^2 + 14*a^3*b^4*c^2*d^3 - 11*a^4*b^3*c*d^4 + 3*a^5*b^2*d^5)*g^4*i^2* 
x^3 + 3*(a*b^6*c^5 - 3*a^2*b^5*c^4*d + 2*a^3*b^4*c^3*d^2 + 2*a^4*b^3*c^2*d 
^3 - 3*a^5*b^2*c*d^4 + a^6*b*d^5)*g^4*i^2*x^2 + (3*a^2*b^5*c^5 - 11*a^3*b^ 
4*c^4*d + 14*a^4*b^3*c^3*d^2 - 6*a^5*b^2*c^2*d^3 - a^6*b*c*d^4 + a^7*d^5)* 
g^4*i^2*x + (a^3*b^4*c^5 - 4*a^4*b^3*c^4*d + 6*a^5*b^2*c^3*d^2 - 4*a^6*b*c 
^2*d^3 + a^7*c*d^4)*g^4*i^2) + 12*b*d^3*log(b*x + a)/((b^5*c^5 - 5*a*b^4*c 
^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)* 
g^4*i^2) - 12*b*d^3*log(d*x + c)/((b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^ 
3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*g^4*i^2))*log(b*e*x/ 
(d*x + c) + a*e/(d*x + c))^2 - 2/3*A*B*((12*b^3*d^3*x^3 + b^3*c^3 - 5*a*b^ 
2*c^2*d + 13*a^2*b*c*d^2 + 3*a^3*d^3 + 6*(b^3*c*d^2 + 5*a*b^2*d^3)*x^2 - 2 
*(b^3*c^2*d - 8*a*b^2*c*d^2 - 11*a^2*b*d^3)*x)/((b^7*c^4*d - 4*a*b^6*c^3*d 
^2 + 6*a^2*b^5*c^2*d^3 - 4*a^3*b^4*c*d^4 + a^4*b^3*d^5)*g^4*i^2*x^4 + (b^7 
*c^5 - a*b^6*c^4*d - 6*a^2*b^5*c^3*d^2 + 14*a^3*b^4*c^2*d^3 - 11*a^4*b^3*c 
*d^4 + 3*a^5*b^2*d^5)*g^4*i^2*x^3 + 3*(a*b^6*c^5 - 3*a^2*b^5*c^4*d + 2*a^3 
*b^4*c^3*d^2 + 2*a^4*b^3*c^2*d^3 - 3*a^5*b^2*c*d^4 + a^6*b*d^5)*g^4*i^2...
 

Giac [A] (verification not implemented)

Time = 86.56 (sec) , antiderivative size = 754, normalized size of antiderivative = 1.11 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4 (c i+d i x)^2} \, dx =\text {Too large to display} \] Input:

integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^4/(d*i*x+c*i)^2,x, al 
gorithm="giac")
 

Output:

-1/54*(18*(B^2*b^2*e^4 - 3*(b*e*x + a*e)*B^2*b*d*e^3/(d*x + c) + 3*(b*e*x 
+ a*e)^2*B^2*d^2*e^2/(d*x + c)^2)*log((b*e*x + a*e)/(d*x + c))^2/((b*e*x + 
 a*e)^3*b^2*c^2*g^4*i^2/(d*x + c)^3 - 2*(b*e*x + a*e)^3*a*b*c*d*g^4*i^2/(d 
*x + c)^3 + (b*e*x + a*e)^3*a^2*d^2*g^4*i^2/(d*x + c)^3) + 6*(6*A*B*b^2*e^ 
4 + 2*B^2*b^2*e^4 - 18*(b*e*x + a*e)*A*B*b*d*e^3/(d*x + c) - 9*(b*e*x + a* 
e)*B^2*b*d*e^3/(d*x + c) + 18*(b*e*x + a*e)^2*A*B*d^2*e^2/(d*x + c)^2 + 18 
*(b*e*x + a*e)^2*B^2*d^2*e^2/(d*x + c)^2)*log((b*e*x + a*e)/(d*x + c))/((b 
*e*x + a*e)^3*b^2*c^2*g^4*i^2/(d*x + c)^3 - 2*(b*e*x + a*e)^3*a*b*c*d*g^4* 
i^2/(d*x + c)^3 + (b*e*x + a*e)^3*a^2*d^2*g^4*i^2/(d*x + c)^3) + (18*A^2*b 
^2*e^4 + 12*A*B*b^2*e^4 + 4*B^2*b^2*e^4 - 54*(b*e*x + a*e)*A^2*b*d*e^3/(d* 
x + c) - 54*(b*e*x + a*e)*A*B*b*d*e^3/(d*x + c) - 27*(b*e*x + a*e)*B^2*b*d 
*e^3/(d*x + c) + 54*(b*e*x + a*e)^2*A^2*d^2*e^2/(d*x + c)^2 + 108*(b*e*x + 
 a*e)^2*A*B*d^2*e^2/(d*x + c)^2 + 108*(b*e*x + a*e)^2*B^2*d^2*e^2/(d*x + c 
)^2)/((b*e*x + a*e)^3*b^2*c^2*g^4*i^2/(d*x + c)^3 - 2*(b*e*x + a*e)^3*a*b* 
c*d*g^4*i^2/(d*x + c)^3 + (b*e*x + a*e)^3*a^2*d^2*g^4*i^2/(d*x + c)^3))*(b 
*c/((b*c*e - a*d*e)*(b*c - a*d)) - a*d/((b*c*e - a*d*e)*(b*c - a*d)))^2
 

Mupad [B] (verification not implemented)

Time = 35.65 (sec) , antiderivative size = 2701, normalized size of antiderivative = 3.96 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4 (c i+d i x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

(log((e*(a + b*x))/(c + d*x))*(x^2*((4*B^2*b*d)/(g^4*i^2*(a*d - b*c)^3) - 
(4*b*d^3*(b*d*((2*a^2*d^2 + b^2*c^2 - 3*a*b*c*d)/(2*b*d^3) + (a*(a*d - b*c 
))/(2*b*d^2)) + ((a*d + b*c)*(a*d - b*c))/d^2)*(5*B^2 + 6*A*B))/(3*g^4*i^2 
*(a*d - b*c)^2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))) + x*( 
(8*(2*B^2 - 3*A*B))/(9*g^4*i^2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (4*B^2*( 
a*d + b*c))/(g^4*i^2*(a*d - b*c)^3) - (4*b*d^3*(((2*a^2*d^2 + b^2*c^2 - 3* 
a*b*c*d)/(2*b*d^3) + (a*(a*d - b*c))/(2*b*d^2))*(a*d + b*c) + (a*c*(a*d - 
b*c))/d^2)*(5*B^2 + 6*A*B))/(3*g^4*i^2*(a*d - b*c)^2*(a^3*d^3 - b^3*c^3 + 
3*a*b^2*c^2*d - 3*a^2*b*c*d^2))) - (2*(B^2*b*c - 9*B^2*a*d + 9*A*B*a*d + 3 
*A*B*b*c))/(9*g^4*i^2*(a^2*b*d^3 + b^3*c^2*d - 2*a*b^2*c*d^2)) + (4*B^2*a* 
c)/(g^4*i^2*(a*d - b*c)^3) - (4*b^2*d^2*x^3*(5*B^2 + 6*A*B))/(3*g^4*i^2*(a 
*d - b*c)*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) - (4*a*b*c* 
d^3*((2*a^2*d^2 + b^2*c^2 - 3*a*b*c*d)/(2*b*d^3) + (a*(a*d - b*c))/(2*b*d^ 
2))*(5*B^2 + 6*A*B))/(3*g^4*i^2*(a*d - b*c)^2*(a^3*d^3 - b^3*c^3 + 3*a*b^2 
*c^2*d - 3*a^2*b*c*d^2))))/(b^2*x^4 + (a^3*c)/(b*d) + (x*(a^3*d + 3*a^2*b* 
c))/(b*d) + (x^3*(b^3*c + 3*a*b^2*d))/(b*d) + (x^2*(3*a*b^2*c + 3*a^2*b*d) 
)/(b*d)) - ((27*A^2*a^3*d^3 + 9*A^2*b^3*c^3 + 54*B^2*a^3*d^3 + 2*B^2*b^3*c 
^3 - 54*A*B*a^3*d^3 + 6*A*B*b^3*c^3 - 45*A^2*a*b^2*c^2*d + 117*A^2*a^2*b*c 
*d^2 - 25*B^2*a*b^2*c^2*d + 299*B^2*a^2*b*c*d^2 - 48*A*B*a*b^2*c^2*d + 276 
*A*B*a^2*b*c*d^2)/(3*(a*d - b*c)) + (2*x^3*(18*A^2*b^3*d^3 + 55*B^2*b^3...
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 5157, normalized size of antiderivative = 7.56 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4 (c i+d i x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 324*log(a + b*x)*a**6*b*c*d**4 - 324*log(a + b*x)*a**6*b*d**5*x - 108* 
log(a + b*x)*a**5*b**2*c**2*d**3 - 1080*log(a + b*x)*a**5*b**2*c*d**4*x - 
324*log(a + b*x)*a**5*b**2*c*d**4 - 972*log(a + b*x)*a**5*b**2*d**5*x**2 - 
 324*log(a + b*x)*a**5*b**2*d**5*x - 324*log(a + b*x)*a**4*b**3*c**2*d**3* 
x - 396*log(a + b*x)*a**4*b**3*c**2*d**3 - 1296*log(a + b*x)*a**4*b**3*c*d 
**4*x**2 - 1368*log(a + b*x)*a**4*b**3*c*d**4*x - 810*log(a + b*x)*a**4*b* 
*3*c*d**4 - 972*log(a + b*x)*a**4*b**3*d**5*x**3 - 972*log(a + b*x)*a**4*b 
**3*d**5*x**2 - 810*log(a + b*x)*a**4*b**3*d**5*x - 324*log(a + b*x)*a**3* 
b**4*c**2*d**3*x**2 - 1188*log(a + b*x)*a**3*b**4*c**2*d**3*x - 510*log(a 
+ b*x)*a**3*b**4*c**2*d**3 - 648*log(a + b*x)*a**3*b**4*c*d**4*x**3 - 2160 
*log(a + b*x)*a**3*b**4*c*d**4*x**2 - 2940*log(a + b*x)*a**3*b**4*c*d**4*x 
 - 324*log(a + b*x)*a**3*b**4*d**5*x**4 - 972*log(a + b*x)*a**3*b**4*d**5* 
x**3 - 2430*log(a + b*x)*a**3*b**4*d**5*x**2 - 108*log(a + b*x)*a**2*b**5* 
c**2*d**3*x**3 - 1188*log(a + b*x)*a**2*b**5*c**2*d**3*x**2 - 1530*log(a + 
 b*x)*a**2*b**5*c**2*d**3*x - 108*log(a + b*x)*a**2*b**5*c*d**4*x**4 - 151 
2*log(a + b*x)*a**2*b**5*c*d**4*x**3 - 3960*log(a + b*x)*a**2*b**5*c*d**4* 
x**2 - 324*log(a + b*x)*a**2*b**5*d**5*x**4 - 2430*log(a + b*x)*a**2*b**5* 
d**5*x**3 - 396*log(a + b*x)*a*b**6*c**2*d**3*x**3 - 1530*log(a + b*x)*a*b 
**6*c**2*d**3*x**2 - 396*log(a + b*x)*a*b**6*c*d**4*x**4 - 2340*log(a + b* 
x)*a*b**6*c*d**4*x**3 - 810*log(a + b*x)*a*b**6*d**5*x**4 - 510*log(a +...