\(\int \frac {(A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(a g+b g x)^3 (c i+d i x)^2} \, dx\) [98]

Optimal result

Integrand size = 42, antiderivative size = 523 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3 (c i+d i x)^2} \, dx=\frac {2 A B d^3 (a+b x)}{(b c-a d)^4 g^3 i^2 (c+d x)}-\frac {2 B^2 d^3 (a+b x)}{(b c-a d)^4 g^3 i^2 (c+d x)}+\frac {6 b^2 B^2 d (c+d x)}{(b c-a d)^4 g^3 i^2 (a+b x)}-\frac {b^3 B^2 (c+d x)^2}{4 (b c-a d)^4 g^3 i^2 (a+b x)^2}+\frac {2 B^2 d^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(b c-a d)^4 g^3 i^2 (c+d x)}+\frac {6 b^2 B d (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 g^3 i^2 (a+b x)}-\frac {b^3 B (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^4 g^3 i^2 (a+b x)^2}-\frac {d^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^4 g^3 i^2 (c+d x)}+\frac {3 b^2 d (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^4 g^3 i^2 (a+b x)}-\frac {b^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 (b c-a d)^4 g^3 i^2 (a+b x)^2}+\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^3}{B (b c-a d)^4 g^3 i^2} \] Output:

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

Mathematica [A] (verified)

Time = 1.28 (sec) , antiderivative size = 466, normalized size of antiderivative = 0.89 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3 (c i+d i x)^2} \, dx=\frac {4 \left (A^2-2 A B+2 B^2\right ) d^2 (b c-a d) (a+b x)^2-b \left (2 A^2+2 A B+B^2\right ) (b c-a d)^2 (c+d x)+2 b \left (4 A^2+10 A B+11 B^2\right ) d (b c-a d) (a+b x) (c+d x)+6 b \left (2 A^2+2 A B+5 B^2\right ) d^2 (a+b x)^2 (c+d x) \log (a+b x)+2 B (b c-a d) \left (4 (A-B) d^2 (a+b x)^2-b (2 A+B) (b c-a d) (c+d x)+2 b (4 A+5 B) d (a+b x) (c+d x)\right ) \log \left (\frac {e (a+b x)}{c+d x}\right )-2 B \left (2 a^3 B d^3-6 a^2 b d^2 (-B d x+A (c+d x))-6 a b^2 d (2 A d x (c+d x)+B c (c+2 d x))+b^3 \left (-6 A d^2 x^2 (c+d x)+B \left (c^3-3 c^2 d x-9 c d^2 x^2-3 d^3 x^3\right )\right )\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )+4 b B^2 d^2 (a+b x)^2 (c+d x) \log ^3\left (\frac {e (a+b x)}{c+d x}\right )-6 b \left (2 A^2+2 A B+5 B^2\right ) d^2 (a+b x)^2 (c+d x) \log (c+d x)}{4 (b c-a d)^4 g^3 i^2 (a+b x)^2 (c+d x)} \] Input:

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

Output:

(4*(A^2 - 2*A*B + 2*B^2)*d^2*(b*c - a*d)*(a + b*x)^2 - b*(2*A^2 + 2*A*B + 
B^2)*(b*c - a*d)^2*(c + d*x) + 2*b*(4*A^2 + 10*A*B + 11*B^2)*d*(b*c - a*d) 
*(a + b*x)*(c + d*x) + 6*b*(2*A^2 + 2*A*B + 5*B^2)*d^2*(a + b*x)^2*(c + d* 
x)*Log[a + b*x] + 2*B*(b*c - a*d)*(4*(A - B)*d^2*(a + b*x)^2 - b*(2*A + B) 
*(b*c - a*d)*(c + d*x) + 2*b*(4*A + 5*B)*d*(a + b*x)*(c + d*x))*Log[(e*(a 
+ b*x))/(c + d*x)] - 2*B*(2*a^3*B*d^3 - 6*a^2*b*d^2*(-(B*d*x) + A*(c + d*x 
)) - 6*a*b^2*d*(2*A*d*x*(c + d*x) + B*c*(c + 2*d*x)) + b^3*(-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)))*Log[(e*(a + b*x 
))/(c + d*x)]^2 + 4*b*B^2*d^2*(a + b*x)^2*(c + d*x)*Log[(e*(a + b*x))/(c + 
 d*x)]^3 - 6*b*(2*A^2 + 2*A*B + 5*B^2)*d^2*(a + b*x)^2*(c + d*x)*Log[c + d 
*x])/(4*(b*c - a*d)^4*g^3*i^2*(a + b*x)^2*(c + d*x))
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 364, 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.

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

\(\Big \downarrow \) 2962

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

((2*A*B*d^3*(a + b*x))/(c + d*x) - (2*B^2*d^3*(a + b*x))/(c + d*x) + (6*b^ 
2*B^2*d*(c + d*x))/(a + b*x) - (b^3*B^2*(c + d*x)^2)/(4*(a + b*x)^2) + (2* 
B^2*d^3*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)])/(c + d*x) + (6*b^2*B*d*(c 
+ d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a + b*x) - (b^3*B*(c + d*x)^ 
2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(a + b*x)^2) - (d^3*(a + b*x)*( 
A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(c + d*x) + (3*b^2*d*(c + d*x)*(A + 
 B*Log[(e*(a + b*x))/(c + d*x)])^2)/(a + b*x) - (b^3*(c + d*x)^2*(A + B*Lo 
g[(e*(a + b*x))/(c + d*x)])^2)/(2*(a + b*x)^2) + (b*d^2*(A + B*Log[(e*(a + 
 b*x))/(c + d*x)])^3)/B)/((b*c - a*d)^4*g^3*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. \(1079\) vs. \(2(517)=1034\).

Time = 3.48 (sec) , antiderivative size = 1080, normalized size of antiderivative = 2.07

method	result	size
parts	\(\text {Expression too large to display}\)	\(1080\)
derivativedivides	\(\text {Expression too large to display}\)	\(1228\)
default	\(\text {Expression too large to display}\)	\(1228\)
risch	\(\text {Expression too large to display}\)	\(1736\)
parallelrisch	\(\text {Expression too large to display}\)	\(1748\)
norman	\(\text {Expression too large to display}\)	\(1849\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 1005, normalized size of antiderivative = 1.92 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3 (c i+d i x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2683 vs. \(2 (483) = 966\).

Time = 27.33 (sec) , antiderivative size = 2683, normalized size of antiderivative = 5.13 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3 (c i+d i x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

B**2*b*d**2*log(e*(a + b*x)/(c + d*x))**3/(a**4*d**4*g**3*i**2 - 4*a**3*b* 
c*d**3*g**3*i**2 + 6*a**2*b**2*c**2*d**2*g**3*i**2 - 4*a*b**3*c**3*d*g**3* 
i**2 + b**4*c**4*g**3*i**2) - 3*b*d**2*(2*A**2 + 2*A*B + 5*B**2)*log(x + ( 
6*A**2*a*b*d**3 + 6*A**2*b**2*c*d**2 + 6*A*B*a*b*d**3 + 6*A*B*b**2*c*d**2 
+ 15*B**2*a*b*d**3 + 15*B**2*b**2*c*d**2 - 3*a**5*b*d**7*(2*A**2 + 2*A*B + 
 5*B**2)/(a*d - b*c)**4 + 15*a**4*b**2*c*d**6*(2*A**2 + 2*A*B + 5*B**2)/(a 
*d - b*c)**4 - 30*a**3*b**3*c**2*d**5*(2*A**2 + 2*A*B + 5*B**2)/(a*d - b*c 
)**4 + 30*a**2*b**4*c**3*d**4*(2*A**2 + 2*A*B + 5*B**2)/(a*d - b*c)**4 - 1 
5*a*b**5*c**4*d**3*(2*A**2 + 2*A*B + 5*B**2)/(a*d - b*c)**4 + 3*b**6*c**5* 
d**2*(2*A**2 + 2*A*B + 5*B**2)/(a*d - b*c)**4)/(12*A**2*b**2*d**3 + 12*A*B 
*b**2*d**3 + 30*B**2*b**2*d**3))/(2*g**3*i**2*(a*d - b*c)**4) + 3*b*d**2*( 
2*A**2 + 2*A*B + 5*B**2)*log(x + (6*A**2*a*b*d**3 + 6*A**2*b**2*c*d**2 + 6 
*A*B*a*b*d**3 + 6*A*B*b**2*c*d**2 + 15*B**2*a*b*d**3 + 15*B**2*b**2*c*d**2 
 + 3*a**5*b*d**7*(2*A**2 + 2*A*B + 5*B**2)/(a*d - b*c)**4 - 15*a**4*b**2*c 
*d**6*(2*A**2 + 2*A*B + 5*B**2)/(a*d - b*c)**4 + 30*a**3*b**3*c**2*d**5*(2 
*A**2 + 2*A*B + 5*B**2)/(a*d - b*c)**4 - 30*a**2*b**4*c**3*d**4*(2*A**2 + 
2*A*B + 5*B**2)/(a*d - b*c)**4 + 15*a*b**5*c**4*d**3*(2*A**2 + 2*A*B + 5*B 
**2)/(a*d - b*c)**4 - 3*b**6*c**5*d**2*(2*A**2 + 2*A*B + 5*B**2)/(a*d - b* 
c)**4)/(12*A**2*b**2*d**3 + 12*A*B*b**2*d**3 + 30*B**2*b**2*d**3))/(2*g**3 
*i**2*(a*d - b*c)**4) + (-4*A*B*a**2*d**2 - 10*A*B*a*b*c*d - 18*A*B*a*b...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4187 vs. \(2 (517) = 1034\).

Time = 0.39 (sec) , antiderivative size = 4187, normalized size of antiderivative = 8.01 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3 (c i+d i x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*B^2*((6*b^2*d^2*x^2 - b^2*c^2 + 5*a*b*c*d + 2*a^2*d^2 + 3*(b^2*c*d + 3 
*a*b*d^2)*x)/((b^5*c^3*d - 3*a*b^4*c^2*d^2 + 3*a^2*b^3*c*d^3 - a^3*b^2*d^4 
)*g^3*i^2*x^3 + (b^5*c^4 - a*b^4*c^3*d - 3*a^2*b^3*c^2*d^2 + 5*a^3*b^2*c*d 
^3 - 2*a^4*b*d^4)*g^3*i^2*x^2 + (2*a*b^4*c^4 - 5*a^2*b^3*c^3*d + 3*a^3*b^2 
*c^2*d^2 + a^4*b*c*d^3 - a^5*d^4)*g^3*i^2*x + (a^2*b^3*c^4 - 3*a^3*b^2*c^3 
*d + 3*a^4*b*c^2*d^2 - a^5*c*d^3)*g^3*i^2) + 6*b*d^2*log(b*x + a)/((b^4*c^ 
4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*g^3*i^2) 
- 6*b*d^2*log(d*x + c)/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a 
^3*b*c*d^3 + a^4*d^4)*g^3*i^2))*log(b*e*x/(d*x + c) + a*e/(d*x + c))^2 + A 
*B*((6*b^2*d^2*x^2 - b^2*c^2 + 5*a*b*c*d + 2*a^2*d^2 + 3*(b^2*c*d + 3*a*b* 
d^2)*x)/((b^5*c^3*d - 3*a*b^4*c^2*d^2 + 3*a^2*b^3*c*d^3 - a^3*b^2*d^4)*g^3 
*i^2*x^3 + (b^5*c^4 - a*b^4*c^3*d - 3*a^2*b^3*c^2*d^2 + 5*a^3*b^2*c*d^3 - 
2*a^4*b*d^4)*g^3*i^2*x^2 + (2*a*b^4*c^4 - 5*a^2*b^3*c^3*d + 3*a^3*b^2*c^2* 
d^2 + a^4*b*c*d^3 - a^5*d^4)*g^3*i^2*x + (a^2*b^3*c^4 - 3*a^3*b^2*c^3*d + 
3*a^4*b*c^2*d^2 - a^5*c*d^3)*g^3*i^2) + 6*b*d^2*log(b*x + a)/((b^4*c^4 - 4 
*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*g^3*i^2) - 6*b 
*d^2*log(d*x + c)/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b* 
c*d^3 + a^4*d^4)*g^3*i^2))*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - 1/4*B^2* 
(2*(b^3*c^3 - 12*a*b^2*c^2*d + 15*a^2*b*c*d^2 - 4*a^3*d^3 - 6*(b^3*c*d^2 - 
 a*b^2*d^3)*x^2 + 6*(b^3*d^3*x^3 + a^2*b*c*d^2 + (b^3*c*d^2 + 2*a*b^2*d...
 

Giac [A] (verification not implemented)

Time = 71.57 (sec) , antiderivative size = 460, normalized size of antiderivative = 0.88 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3 (c i+d i x)^2} \, dx=-\frac {1}{4} \, {\left (\frac {2 \, {\left (B^{2} b e^{3} - \frac {2 \, {\left (b e x + a e\right )} B^{2} d e^{2}}{d x + c}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2}}{\frac {{\left (b e x + a e\right )}^{2} b c g^{3} i^{2}}{{\left (d x + c\right )}^{2}} - \frac {{\left (b e x + a e\right )}^{2} a d g^{3} i^{2}}{{\left (d x + c\right )}^{2}}} + \frac {2 \, {\left (2 \, A B b e^{3} + B^{2} b e^{3} - \frac {4 \, {\left (b e x + a e\right )} A B d e^{2}}{d x + c} - \frac {4 \, {\left (b e x + a e\right )} B^{2} d e^{2}}{d x + c}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{\frac {{\left (b e x + a e\right )}^{2} b c g^{3} i^{2}}{{\left (d x + c\right )}^{2}} - \frac {{\left (b e x + a e\right )}^{2} a d g^{3} i^{2}}{{\left (d x + c\right )}^{2}}} + \frac {2 \, A^{2} b e^{3} + 2 \, A B b e^{3} + B^{2} b e^{3} - \frac {4 \, {\left (b e x + a e\right )} A^{2} d e^{2}}{d x + c} - \frac {8 \, {\left (b e x + a e\right )} A B d e^{2}}{d x + c} - \frac {8 \, {\left (b e x + a e\right )} B^{2} d e^{2}}{d x + c}}{\frac {{\left (b e x + a e\right )}^{2} b c g^{3} i^{2}}{{\left (d x + c\right )}^{2}} - \frac {{\left (b e x + a e\right )}^{2} a d g^{3} i^{2}}{{\left (d x + c\right )}^{2}}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}^{2} \] Input:

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

Output:

-1/4*(2*(B^2*b*e^3 - 2*(b*e*x + a*e)*B^2*d*e^2/(d*x + c))*log((b*e*x + a*e 
)/(d*x + c))^2/((b*e*x + a*e)^2*b*c*g^3*i^2/(d*x + c)^2 - (b*e*x + a*e)^2* 
a*d*g^3*i^2/(d*x + c)^2) + 2*(2*A*B*b*e^3 + B^2*b*e^3 - 4*(b*e*x + a*e)*A* 
B*d*e^2/(d*x + c) - 4*(b*e*x + a*e)*B^2*d*e^2/(d*x + c))*log((b*e*x + a*e) 
/(d*x + c))/((b*e*x + a*e)^2*b*c*g^3*i^2/(d*x + c)^2 - (b*e*x + a*e)^2*a*d 
*g^3*i^2/(d*x + c)^2) + (2*A^2*b*e^3 + 2*A*B*b*e^3 + B^2*b*e^3 - 4*(b*e*x 
+ a*e)*A^2*d*e^2/(d*x + c) - 8*(b*e*x + a*e)*A*B*d*e^2/(d*x + c) - 8*(b*e* 
x + a*e)*B^2*d*e^2/(d*x + c))/((b*e*x + a*e)^2*b*c*g^3*i^2/(d*x + c)^2 - ( 
b*e*x + a*e)^2*a*d*g^3*i^2/(d*x + c)^2))*(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 = 32.86 (sec) , antiderivative size = 1497, normalized size of antiderivative = 2.86 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3 (c i+d i x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

(B^2*b*d^2*log((e*(a + b*x))/(c + d*x))^3)/(g^3*i^2*(a*d - b*c)^2*(a^2*d^2 
 + b^2*c^2 - 2*a*b*c*d)) - ((4*A^2*a^2*d^2 - 2*A^2*b^2*c^2 + 8*B^2*a^2*d^2 
 - B^2*b^2*c^2 - 8*A*B*a^2*d^2 - 2*A*B*b^2*c^2 + 10*A^2*a*b*c*d + 23*B^2*a 
*b*c*d + 22*A*B*a*b*c*d)/(2*(a*d - b*c)) + (3*x^2*(2*A^2*b^2*d^2 + 5*B^2*b 
^2*d^2 + 2*A*B*b^2*d^2))/(a*d - b*c) + (3*x*(6*A^2*a*b*d^2 + 13*B^2*a*b*d^ 
2 + 2*A^2*b^2*c*d + 7*B^2*b^2*c*d + 2*A*B*a*b*d^2 + 6*A*B*b^2*c*d))/(2*(a* 
d - b*c)))/(x*(2*a^4*d^3*g^3*i^2 + 4*a*b^3*c^3*g^3*i^2 - 6*a^2*b^2*c^2*d*g 
^3*i^2) + x^2*(2*b^4*c^3*g^3*i^2 + 4*a^3*b*d^3*g^3*i^2 - 6*a^2*b^2*c*d^2*g 
^3*i^2) + x^3*(2*a^2*b^2*d^3*g^3*i^2 + 2*b^4*c^2*d*g^3*i^2 - 4*a*b^3*c*d^2 
*g^3*i^2) + 2*a^2*b^2*c^3*g^3*i^2 + 2*a^4*c*d^2*g^3*i^2 - 4*a^3*b*c^2*d*g^ 
3*i^2) - (log((e*(a + b*x))/(c + d*x))*((B^2*b*c - 4*B^2*a*d + 4*A*B*a*d + 
 2*A*B*b*c)/(2*g^3*i^2*(a^2*b*d^3 + b^3*c^2*d - 2*a*b^2*c*d^2)) - x*((3*(B 
^2 - 2*A*B))/(2*g^3*i^2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) - (3*B*(2*A + B)* 
(a*d + b*c))/(g^3*i^2*(a*d - b*c)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))) + (3*B 
*a*c*(2*A + B))/(g^3*i^2*(a*d - b*c)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (3 
*B*b*d*x^2*(2*A + B))/(g^3*i^2*(a*d - b*c)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d) 
)))/(b*x^3 + (a^2*c)/(b*d) + (x^2*(b^2*c + 2*a*b*d))/(b*d) + (x*(a^2*d + 2 
*a*b*c))/(b*d)) - (b*d^2*atan((b*d^2*(2*A^2 + 5*B^2 + 2*A*B)*(2*a^4*d^4*g^ 
3*i^2 - 2*b^4*c^4*g^3*i^2 + 4*a*b^3*c^3*d*g^3*i^2 - 4*a^3*b*c*d^3*g^3*i^2) 
*3i)/(2*g^3*i^2*(a*d - b*c)^4*(6*A^2*b*d^2 + 15*B^2*b*d^2 + 6*A*B*b*d^2...
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 3290, normalized size of antiderivative = 6.29 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3 (c i+d i x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 24*log(a + b*x)*a**5*b*c*d**3 - 24*log(a + b*x)*a**5*b*d**4*x - 12*log 
(a + b*x)*a**4*b**2*c**2*d**2 - 60*log(a + b*x)*a**4*b**2*c*d**3*x - 48*lo 
g(a + b*x)*a**4*b**2*d**4*x**2 - 24*log(a + b*x)*a**3*b**3*c**2*d**2*x - 3 
6*log(a + b*x)*a**3*b**3*c**2*d**2 - 48*log(a + b*x)*a**3*b**3*c*d**3*x**2 
 - 36*log(a + b*x)*a**3*b**3*c*d**3*x - 48*log(a + b*x)*a**3*b**3*c*d**3 - 
 24*log(a + b*x)*a**3*b**3*d**4*x**3 - 48*log(a + b*x)*a**3*b**3*d**4*x - 
12*log(a + b*x)*a**2*b**4*c**2*d**2*x**2 - 72*log(a + b*x)*a**2*b**4*c**2* 
d**2*x - 42*log(a + b*x)*a**2*b**4*c**2*d**2 - 12*log(a + b*x)*a**2*b**4*c 
*d**3*x**3 - 72*log(a + b*x)*a**2*b**4*c*d**3*x**2 - 138*log(a + b*x)*a**2 
*b**4*c*d**3*x - 96*log(a + b*x)*a**2*b**4*d**4*x**2 - 36*log(a + b*x)*a*b 
**5*c**2*d**2*x**2 - 84*log(a + b*x)*a*b**5*c**2*d**2*x - 36*log(a + b*x)* 
a*b**5*c*d**3*x**3 - 132*log(a + b*x)*a*b**5*c*d**3*x**2 - 48*log(a + b*x) 
*a*b**5*d**4*x**3 - 42*log(a + b*x)*b**6*c**2*d**2*x**2 - 42*log(a + b*x)* 
b**6*c*d**3*x**3 + 24*log(c + d*x)*a**5*b*c*d**3 + 24*log(c + d*x)*a**5*b* 
d**4*x + 12*log(c + d*x)*a**4*b**2*c**2*d**2 + 60*log(c + d*x)*a**4*b**2*c 
*d**3*x + 48*log(c + d*x)*a**4*b**2*d**4*x**2 + 24*log(c + d*x)*a**3*b**3* 
c**2*d**2*x + 36*log(c + d*x)*a**3*b**3*c**2*d**2 + 48*log(c + d*x)*a**3*b 
**3*c*d**3*x**2 + 36*log(c + d*x)*a**3*b**3*c*d**3*x + 48*log(c + d*x)*a** 
3*b**3*c*d**3 + 24*log(c + d*x)*a**3*b**3*d**4*x**3 + 48*log(c + d*x)*a**3 
*b**3*d**4*x + 12*log(c + d*x)*a**2*b**4*c**2*d**2*x**2 + 72*log(c + d*...