\(\int \frac {(A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(a g+b g x)^4 (c i+d i x)} \, dx\) [91]

Optimal result

Integrand size = 42, antiderivative size = 507 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4 (c i+d i x)} \, dx=-\frac {6 b B^2 d^2 (c+d x)}{(b c-a d)^4 g^4 i (a+b x)}+\frac {3 b^2 B^2 d (c+d x)^2}{4 (b c-a d)^4 g^4 i (a+b x)^2}-\frac {2 b^3 B^2 (c+d x)^3}{27 (b c-a d)^4 g^4 i (a+b x)^3}-\frac {6 b B d^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 g^4 i (a+b x)}+\frac {3 b^2 B d (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^4 g^4 i (a+b x)^2}-\frac {2 b^3 B (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9 (b c-a d)^4 g^4 i (a+b x)^3}-\frac {3 b d^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^4 g^4 i (a+b x)}+\frac {3 b^2 d (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 (b c-a d)^4 g^4 i (a+b x)^2}-\frac {b^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 (b c-a d)^4 g^4 i (a+b x)^3}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^3}{3 B (b c-a d)^4 g^4 i} \] Output:

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

Mathematica [A] (verified)

Time = 1.44 (sec) , antiderivative size = 442, normalized size of antiderivative = 0.87 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4 (c i+d i x)} \, dx=-\frac {4 \left (9 A^2+6 A B+2 B^2\right ) (b c-a d)^3-3 \left (18 A^2+30 A B+19 B^2\right ) d (b c-a d)^2 (a+b x)-6 \left (18 A^2+66 A B+85 B^2\right ) d^2 (-b c+a d) (a+b x)^2+6 \left (18 A^2+66 A B+85 B^2\right ) d^3 (a+b x)^3 \log (a+b x)+6 B (b c-a d) \left (4 (3 A+B) (b c-a d)^2+3 (6 A+5 B) d (-b c+a d) (a+b x)+6 (6 A+11 B) d^2 (a+b x)^2\right ) \log \left (\frac {e (a+b x)}{c+d x}\right )+18 B \left (6 a^3 A d^3+18 a^2 b d^2 (A d x+B (c+d x))+9 a b^2 d \left (2 A d^2 x^2+B \left (-c^2+2 c d x+3 d^2 x^2\right )\right )+b^3 \left (6 A d^3 x^3+B \left (2 c^3-3 c^2 d x+6 c d^2 x^2+11 d^3 x^3\right )\right )\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )+36 B^2 d^3 (a+b x)^3 \log ^3\left (\frac {e (a+b x)}{c+d x}\right )-6 \left (18 A^2+66 A B+85 B^2\right ) d^3 (a+b x)^3 \log (c+d x)}{108 (b c-a d)^4 g^4 i (a+b x)^3} \] Input:

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

Output:

-1/108*(4*(9*A^2 + 6*A*B + 2*B^2)*(b*c - a*d)^3 - 3*(18*A^2 + 30*A*B + 19* 
B^2)*d*(b*c - a*d)^2*(a + b*x) - 6*(18*A^2 + 66*A*B + 85*B^2)*d^2*(-(b*c) 
+ a*d)*(a + b*x)^2 + 6*(18*A^2 + 66*A*B + 85*B^2)*d^3*(a + b*x)^3*Log[a + 
b*x] + 6*B*(b*c - a*d)*(4*(3*A + B)*(b*c - a*d)^2 + 3*(6*A + 5*B)*d*(-(b*c 
) + a*d)*(a + b*x) + 6*(6*A + 11*B)*d^2*(a + b*x)^2)*Log[(e*(a + b*x))/(c 
+ d*x)] + 18*B*(6*a^3*A*d^3 + 18*a^2*b*d^2*(A*d*x + B*(c + d*x)) + 9*a*b^2 
*d*(2*A*d^2*x^2 + B*(-c^2 + 2*c*d*x + 3*d^2*x^2)) + b^3*(6*A*d^3*x^3 + B*( 
2*c^3 - 3*c^2*d*x + 6*c*d^2*x^2 + 11*d^3*x^3)))*Log[(e*(a + b*x))/(c + d*x 
)]^2 + 36*B^2*d^3*(a + b*x)^3*Log[(e*(a + b*x))/(c + d*x)]^3 - 6*(18*A^2 + 
 66*A*B + 85*B^2)*d^3*(a + b*x)^3*Log[c + d*x])/((b*c - a*d)^4*g^4*i*(a + 
b*x)^3)
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 364, normalized size of antiderivative = 0.72, 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.

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

\(\Big \downarrow \) 2962

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

((-6*b*B^2*d^2*(c + d*x))/(a + b*x) + (3*b^2*B^2*d*(c + d*x)^2)/(4*(a + b* 
x)^2) - (2*b^3*B^2*(c + d*x)^3)/(27*(a + b*x)^3) - (6*b*B*d^2*(c + d*x)*(A 
 + B*Log[(e*(a + b*x))/(c + d*x)]))/(a + b*x) + (3*b^2*B*d*(c + d*x)^2*(A 
+ B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(a + b*x)^2) - (2*b^3*B*(c + d*x)^3* 
(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(9*(a + b*x)^3) - (3*b*d^2*(c + d*x) 
*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(a + b*x) + (3*b^2*d*(c + d*x)^2* 
(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*(a + b*x)^2) - (b^3*(c + d*x)^3 
*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(3*(a + b*x)^3) - (d^3*(A + B*Log 
[(e*(a + b*x))/(c + d*x)])^3)/(3*B))/((b*c - a*d)^4*g^4*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 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. \(1075\) vs. \(2(493)=986\).

Time = 2.64 (sec) , antiderivative size = 1076, normalized size of antiderivative = 2.12

method	result	size
parts	\(\text {Expression too large to display}\)	\(1076\)
derivativedivides	\(\text {Expression too large to display}\)	\(1244\)
default	\(\text {Expression too large to display}\)	\(1244\)
risch	\(\text {Expression too large to display}\)	\(1422\)
norman	\(\text {Expression too large to display}\)	\(1693\)
parallelrisch	\(\text {Expression too large to display}\)	\(1967\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 940, normalized size of antiderivative = 1.85 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4 (c i+d i x)} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/108*(4*(9*A^2 + 6*A*B + 2*B^2)*b^3*c^3 - 81*(2*A^2 + 2*A*B + B^2)*a*b^2 
*c^2*d + 324*(A^2 + 2*A*B + 2*B^2)*a^2*b*c*d^2 - (198*A^2 + 510*A*B + 575* 
B^2)*a^3*d^3 + 36*(B^2*b^3*d^3*x^3 + 3*B^2*a*b^2*d^3*x^2 + 3*B^2*a^2*b*d^3 
*x + B^2*a^3*d^3)*log((b*e*x + a*e)/(d*x + c))^3 + 6*((18*A^2 + 66*A*B + 8 
5*B^2)*b^3*c*d^2 - (18*A^2 + 66*A*B + 85*B^2)*a*b^2*d^3)*x^2 + 18*((6*A*B 
+ 11*B^2)*b^3*d^3*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 + 6*A*B*a^3*d^3 + 3*(2*B^2*b^3*c*d^2 + 3*(2*A*B + 3*B^2)*a*b^2*d^3)*x 
^2 - 3*(B^2*b^3*c^2*d - 6*B^2*a*b^2*c*d^2 - 6*(A*B + B^2)*a^2*b*d^3)*x)*lo 
g((b*e*x + a*e)/(d*x + c))^2 - 3*((18*A^2 + 30*A*B + 19*B^2)*b^3*c^2*d - 5 
4*(2*A^2 + 6*A*B + 7*B^2)*a*b^2*c*d^2 + (90*A^2 + 294*A*B + 359*B^2)*a^2*b 
*d^3)*x + 6*((18*A^2 + 66*A*B + 85*B^2)*b^3*d^3*x^3 + 18*A^2*a^3*d^3 + 4*( 
3*A*B + B^2)*b^3*c^3 - 27*(2*A*B + B^2)*a*b^2*c^2*d + 108*(A*B + B^2)*a^2* 
b*c*d^2 + 3*(2*(6*A*B + 11*B^2)*b^3*c*d^2 + 9*(2*A^2 + 6*A*B + 7*B^2)*a*b^ 
2*d^3)*x^2 - 3*((6*A*B + 5*B^2)*b^3*c^2*d - 18*(2*A*B + 3*B^2)*a*b^2*c*d^2 
 - 18*(A^2 + 2*A*B + 2*B^2)*a^2*b*d^3)*x)*log((b*e*x + a*e)/(d*x + c)))/(( 
b^7*c^4 - 4*a*b^6*c^3*d + 6*a^2*b^5*c^2*d^2 - 4*a^3*b^4*c*d^3 + a^4*b^3*d^ 
4)*g^4*i*x^3 + 3*(a*b^6*c^4 - 4*a^2*b^5*c^3*d + 6*a^3*b^4*c^2*d^2 - 4*a^4* 
b^3*c*d^3 + a^5*b^2*d^4)*g^4*i*x^2 + 3*(a^2*b^5*c^4 - 4*a^3*b^4*c^3*d + 6* 
a^4*b^3*c^2*d^2 - 4*a^5*b^2*c*d^3 + a^6*b*d^4)*g^4*i*x + (a^3*b^4*c^4 - 4* 
a^4*b^3*c^3*d + 6*a^5*b^2*c^2*d^2 - 4*a^6*b*c*d^3 + a^7*d^4)*g^4*i)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2388 vs. \(2 (459) = 918\).

Time = 19.09 (sec) , antiderivative size = 2388, normalized size of antiderivative = 4.71 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4 (c i+d i x)} \, dx=\text {Too large to display} \] Input:

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

Output:

-B**2*d**3*log(e*(a + b*x)/(c + d*x))**3/(3*a**4*d**4*g**4*i - 12*a**3*b*c 
*d**3*g**4*i + 18*a**2*b**2*c**2*d**2*g**4*i - 12*a*b**3*c**3*d*g**4*i + 3 
*b**4*c**4*g**4*i) + d**3*(18*A**2 + 66*A*B + 85*B**2)*log(x + (18*A**2*a* 
d**4 + 18*A**2*b*c*d**3 + 66*A*B*a*d**4 + 66*A*B*b*c*d**3 + 85*B**2*a*d**4 
 + 85*B**2*b*c*d**3 - a**5*d**8*(18*A**2 + 66*A*B + 85*B**2)/(a*d - b*c)** 
4 + 5*a**4*b*c*d**7*(18*A**2 + 66*A*B + 85*B**2)/(a*d - b*c)**4 - 10*a**3* 
b**2*c**2*d**6*(18*A**2 + 66*A*B + 85*B**2)/(a*d - b*c)**4 + 10*a**2*b**3* 
c**3*d**5*(18*A**2 + 66*A*B + 85*B**2)/(a*d - b*c)**4 - 5*a*b**4*c**4*d**4 
*(18*A**2 + 66*A*B + 85*B**2)/(a*d - b*c)**4 + b**5*c**5*d**3*(18*A**2 + 6 
6*A*B + 85*B**2)/(a*d - b*c)**4)/(36*A**2*b*d**4 + 132*A*B*b*d**4 + 170*B* 
*2*b*d**4))/(18*g**4*i*(a*d - b*c)**4) - d**3*(18*A**2 + 66*A*B + 85*B**2) 
*log(x + (18*A**2*a*d**4 + 18*A**2*b*c*d**3 + 66*A*B*a*d**4 + 66*A*B*b*c*d 
**3 + 85*B**2*a*d**4 + 85*B**2*b*c*d**3 + a**5*d**8*(18*A**2 + 66*A*B + 85 
*B**2)/(a*d - b*c)**4 - 5*a**4*b*c*d**7*(18*A**2 + 66*A*B + 85*B**2)/(a*d 
- b*c)**4 + 10*a**3*b**2*c**2*d**6*(18*A**2 + 66*A*B + 85*B**2)/(a*d - b*c 
)**4 - 10*a**2*b**3*c**3*d**5*(18*A**2 + 66*A*B + 85*B**2)/(a*d - b*c)**4 
+ 5*a*b**4*c**4*d**4*(18*A**2 + 66*A*B + 85*B**2)/(a*d - b*c)**4 - b**5*c* 
*5*d**3*(18*A**2 + 66*A*B + 85*B**2)/(a*d - b*c)**4)/(36*A**2*b*d**4 + 132 
*A*B*b*d**4 + 170*B**2*b*d**4))/(18*g**4*i*(a*d - b*c)**4) + (66*A*B*a**2* 
d**2 - 42*A*B*a*b*c*d + 90*A*B*a*b*d**2*x + 12*A*B*b**2*c**2 - 18*A*B*b...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3434 vs. \(2 (493) = 986\).

Time = 0.33 (sec) , antiderivative size = 3434, normalized size of antiderivative = 6.77 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4 (c i+d i x)} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 72.17 (sec) , antiderivative size = 451, normalized size of antiderivative = 0.89 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4 (c i+d i x)} \, dx=-\frac {1}{108} \, {\left (\frac {18 \, {\left (2 \, B^{2} b e^{4} - \frac {3 \, {\left (b e x + a e\right )} B^{2} d e^{3}}{d x + c}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2}}{\frac {{\left (b e x + a e\right )}^{3} b c g^{4} i}{{\left (d x + c\right )}^{3}} - \frac {{\left (b e x + a e\right )}^{3} a d g^{4} i}{{\left (d x + c\right )}^{3}}} + \frac {6 \, {\left (12 \, A B b e^{4} + 4 \, B^{2} b e^{4} - \frac {18 \, {\left (b e x + a e\right )} A B d e^{3}}{d x + c} - \frac {9 \, {\left (b e x + a e\right )} B^{2} d e^{3}}{d x + c}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{\frac {{\left (b e x + a e\right )}^{3} b c g^{4} i}{{\left (d x + c\right )}^{3}} - \frac {{\left (b e x + a e\right )}^{3} a d g^{4} i}{{\left (d x + c\right )}^{3}}} + \frac {36 \, A^{2} b e^{4} + 24 \, A B b e^{4} + 8 \, B^{2} b e^{4} - \frac {54 \, {\left (b e x + a e\right )} A^{2} d e^{3}}{d x + c} - \frac {54 \, {\left (b e x + a e\right )} A B d e^{3}}{d x + c} - \frac {27 \, {\left (b e x + a e\right )} B^{2} d e^{3}}{d x + c}}{\frac {{\left (b e x + a e\right )}^{3} b c g^{4} i}{{\left (d x + c\right )}^{3}} - \frac {{\left (b e x + a e\right )}^{3} a d g^{4} i}{{\left (d x + c\right )}^{3}}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}^{2} \] Input:

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

Output:

-1/108*(18*(2*B^2*b*e^4 - 3*(b*e*x + a*e)*B^2*d*e^3/(d*x + c))*log((b*e*x 
+ a*e)/(d*x + c))^2/((b*e*x + a*e)^3*b*c*g^4*i/(d*x + c)^3 - (b*e*x + a*e) 
^3*a*d*g^4*i/(d*x + c)^3) + 6*(12*A*B*b*e^4 + 4*B^2*b*e^4 - 18*(b*e*x + a* 
e)*A*B*d*e^3/(d*x + c) - 9*(b*e*x + a*e)*B^2*d*e^3/(d*x + c))*log((b*e*x + 
 a*e)/(d*x + c))/((b*e*x + a*e)^3*b*c*g^4*i/(d*x + c)^3 - (b*e*x + a*e)^3* 
a*d*g^4*i/(d*x + c)^3) + (36*A^2*b*e^4 + 24*A*B*b*e^4 + 8*B^2*b*e^4 - 54*( 
b*e*x + a*e)*A^2*d*e^3/(d*x + c) - 54*(b*e*x + a*e)*A*B*d*e^3/(d*x + c) - 
27*(b*e*x + a*e)*B^2*d*e^3/(d*x + c))/((b*e*x + a*e)^3*b*c*g^4*i/(d*x + c) 
^3 - (b*e*x + a*e)^3*a*d*g^4*i/(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 = 33.83 (sec) , antiderivative size = 1882, normalized size of antiderivative = 3.71 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4 (c i+d i x)} \, dx=\text {Too large to display} \] Input:

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

Output:

log((e*(a + b*x))/(c + d*x))^2*(((B^2*d^3*(a*((3*a^2*d^2 + b^2*c^2 - 4*a*b 
*c*d)/(6*b*d^3) + (a*(a*d - b*c))/(3*b*d^2)) + (3*a^3*d^3 - b^3*c^3 + 4*a* 
b^2*c^2*d - 6*a^2*b*c*d^2)/(3*b*d^4)))/(g^4*i*(a^4*d^4 + b^4*c^4 + 6*a^2*b 
^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) - (B^2*d^3*x^2*((b^2*c - a*b* 
d)/(3*d^2) - (2*b*(a*d - b*c))/(3*d^2)))/(g^4*i*(a^4*d^4 + b^4*c^4 + 6*a^2 
*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) + (B^2*d^3*x*(b*((3*a^2*d^2 
 + b^2*c^2 - 4*a*b*c*d)/(6*b*d^3) + (a*(a*d - b*c))/(3*b*d^2)) + (3*a^2*d^ 
2 + b^2*c^2 - 4*a*b*c*d)/(3*d^3) + (2*a*(a*d - b*c))/(3*d^2)))/(g^4*i*(a^4 
*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)))/((3* 
a^2*x)/d + a^3/(b*d) + (b^2*x^3)/d + (3*a*b*x^2)/d) - (d^3*(11*B^2 + 6*A*B 
))/(6*g^4*i*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3 
*b*c*d^3))) + ((198*A^2*a^2*d^2 + 36*A^2*b^2*c^2 + 575*B^2*a^2*d^2 + 8*B^2 
*b^2*c^2 + 510*A*B*a^2*d^2 + 24*A*B*b^2*c^2 - 126*A^2*a*b*c*d - 73*B^2*a*b 
*c*d - 138*A*B*a*b*c*d)/(6*(a*d - b*c)) + (x*(90*A^2*a*b*d^2 + 359*B^2*a*b 
*d^2 - 18*A^2*b^2*c*d - 19*B^2*b^2*c*d + 294*A*B*a*b*d^2 - 30*A*B*b^2*c*d) 
)/(2*(a*d - b*c)) + (d*x^2*(18*A^2*b^2*d + 85*B^2*b^2*d + 66*A*B*b^2*d))/( 
a*d - b*c))/(x*(54*a^4*b*d^2*g^4*i + 54*a^2*b^3*c^2*g^4*i - 108*a^3*b^2*c* 
d*g^4*i) + x^2*(54*a*b^4*c^2*g^4*i + 54*a^3*b^2*d^2*g^4*i - 108*a^2*b^3*c* 
d*g^4*i) + x^3*(18*b^5*c^2*g^4*i + 18*a^2*b^3*d^2*g^4*i - 36*a*b^4*c*d*g^4 
*i) + 18*a^5*d^2*g^4*i + 18*a^3*b^2*c^2*g^4*i - 36*a^4*b*c*d*g^4*i) - (...
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 2371, normalized size of antiderivative = 4.68 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4 (c i+d i x)} \, dx =\text {Too large to display} \] Input:

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

Output:

(i*(108*log(a + b*x)*a**6*d**3 + 324*log(a + b*x)*a**5*b*d**3*x + 324*log( 
a + b*x)*a**5*b*d**3 + 72*log(a + b*x)*a**4*b**2*c*d**2 + 324*log(a + b*x) 
*a**4*b**2*d**3*x**2 + 972*log(a + b*x)*a**4*b**2*d**3*x + 378*log(a + b*x 
)*a**4*b**2*d**3 + 216*log(a + b*x)*a**3*b**3*c*d**2*x + 132*log(a + b*x)* 
a**3*b**3*c*d**2 + 108*log(a + b*x)*a**3*b**3*d**3*x**3 + 972*log(a + b*x) 
*a**3*b**3*d**3*x**2 + 1134*log(a + b*x)*a**3*b**3*d**3*x + 216*log(a + b* 
x)*a**2*b**4*c*d**2*x**2 + 396*log(a + b*x)*a**2*b**4*c*d**2*x + 324*log(a 
 + b*x)*a**2*b**4*d**3*x**3 + 1134*log(a + b*x)*a**2*b**4*d**3*x**2 + 72*l 
og(a + b*x)*a*b**5*c*d**2*x**3 + 396*log(a + b*x)*a*b**5*c*d**2*x**2 + 378 
*log(a + b*x)*a*b**5*d**3*x**3 + 132*log(a + b*x)*b**6*c*d**2*x**3 - 108*l 
og(c + d*x)*a**6*d**3 - 324*log(c + d*x)*a**5*b*d**3*x - 324*log(c + d*x)* 
a**5*b*d**3 - 72*log(c + d*x)*a**4*b**2*c*d**2 - 324*log(c + d*x)*a**4*b** 
2*d**3*x**2 - 972*log(c + d*x)*a**4*b**2*d**3*x - 378*log(c + d*x)*a**4*b* 
*2*d**3 - 216*log(c + d*x)*a**3*b**3*c*d**2*x - 132*log(c + d*x)*a**3*b**3 
*c*d**2 - 108*log(c + d*x)*a**3*b**3*d**3*x**3 - 972*log(c + d*x)*a**3*b** 
3*d**3*x**2 - 1134*log(c + d*x)*a**3*b**3*d**3*x - 216*log(c + d*x)*a**2*b 
**4*c*d**2*x**2 - 396*log(c + d*x)*a**2*b**4*c*d**2*x - 324*log(c + d*x)*a 
**2*b**4*d**3*x**3 - 1134*log(c + d*x)*a**2*b**4*d**3*x**2 - 72*log(c + d* 
x)*a*b**5*c*d**2*x**3 - 396*log(c + d*x)*a*b**5*c*d**2*x**2 - 378*log(c + 
d*x)*a*b**5*d**3*x**3 - 132*log(c + d*x)*b**6*c*d**2*x**3 + 36*log((a*e...