Integrand size = 42, antiderivative size = 685 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3 (c i+d i x)^3} \, dx=\frac {B^2 d^4 (a+b x)^2}{4 (b c-a d)^5 g^3 i^3 (c+d x)^2}+\frac {8 A b B d^3 (a+b x)}{(b c-a d)^5 g^3 i^3 (c+d x)}-\frac {8 b B^2 d^3 (a+b x)}{(b c-a d)^5 g^3 i^3 (c+d x)}+\frac {8 b^3 B^2 d (c+d x)}{(b c-a d)^5 g^3 i^3 (a+b x)}-\frac {b^4 B^2 (c+d x)^2}{4 (b c-a d)^5 g^3 i^3 (a+b x)^2}+\frac {8 b B^2 d^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(b c-a d)^5 g^3 i^3 (c+d x)}-\frac {B d^4 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^5 g^3 i^3 (c+d x)^2}+\frac {8 b^3 B d (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^5 g^3 i^3 (a+b x)}-\frac {b^4 B (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^5 g^3 i^3 (a+b x)^2}+\frac {d^4 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 (b c-a d)^5 g^3 i^3 (c+d x)^2}-\frac {4 b d^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^5 g^3 i^3 (c+d x)}+\frac {4 b^3 d (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^5 g^3 i^3 (a+b x)}-\frac {b^4 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 (b c-a d)^5 g^3 i^3 (a+b x)^2}+\frac {2 b^2 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^3}{B (b c-a d)^5 g^3 i^3} \]
1/4*B^2*d^4*(b*x+a)^2/(-a*d+b*c)^5/g^3/i^3/(d*x+c)^2+8*A*b*B*d^3*(b*x+a)/( -a*d+b*c)^5/g^3/i^3/(d*x+c)-8*b*B^2*d^3*(b*x+a)/(-a*d+b*c)^5/g^3/i^3/(d*x+ c)+8*b^3*B^2*d*(d*x+c)/(-a*d+b*c)^5/g^3/i^3/(b*x+a)-1/4*b^4*B^2*(d*x+c)^2/ (-a*d+b*c)^5/g^3/i^3/(b*x+a)^2+8*b*B^2*d^3*(b*x+a)*ln(e*(b*x+a)/(d*x+c))/( -a*d+b*c)^5/g^3/i^3/(d*x+c)-1/2*B*d^4*(b*x+a)^2*(A+B*ln(e*(b*x+a)/(d*x+c)) )/(-a*d+b*c)^5/g^3/i^3/(d*x+c)^2+8*b^3*B*d*(d*x+c)*(A+B*ln(e*(b*x+a)/(d*x+ c)))/(-a*d+b*c)^5/g^3/i^3/(b*x+a)-1/2*b^4*B*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d *x+c)))/(-a*d+b*c)^5/g^3/i^3/(b*x+a)^2+1/2*d^4*(b*x+a)^2*(A+B*ln(e*(b*x+a) /(d*x+c)))^2/(-a*d+b*c)^5/g^3/i^3/(d*x+c)^2-4*b*d^3*(b*x+a)*(A+B*ln(e*(b*x +a)/(d*x+c)))^2/(-a*d+b*c)^5/g^3/i^3/(d*x+c)+4*b^3*d*(d*x+c)*(A+B*ln(e*(b* x+a)/(d*x+c)))^2/(-a*d+b*c)^5/g^3/i^3/(b*x+a)-1/2*b^4*(d*x+c)^2*(A+B*ln(e* (b*x+a)/(d*x+c)))^2/(-a*d+b*c)^5/g^3/i^3/(b*x+a)^2+2*b^2*d^2*(A+B*ln(e*(b* x+a)/(d*x+c)))^3/B/(-a*d+b*c)^5/g^3/i^3
Time = 0.88 (sec) , antiderivative size = 611, normalized size of antiderivative = 0.89 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3 (c i+d i x)^3} \, dx=\frac {\left (2 A^2-2 A B+B^2\right ) d^2 (b c-a d)^2 (a+b x)^2+2 b \left (6 A^2-14 A B+15 B^2\right ) d^2 (b c-a d) (a+b x)^2 (c+d x)-b^2 \left (2 A^2+2 A B+B^2\right ) (b c-a d)^2 (c+d x)^2+2 b^2 \left (6 A^2+14 A B+15 B^2\right ) d (b c-a d) (a+b x) (c+d x)^2+12 b^2 \left (2 A^2+5 B^2\right ) d^2 (a+b x)^2 (c+d x)^2 \log (a+b x)+2 B (b c-a d) \left ((2 A-B) d^2 (b c-a d) (a+b x)^2+2 b (6 A-7 B) d^2 (a+b x)^2 (c+d x)-b^2 (2 A+B) (b c-a d) (c+d x)^2+2 b^2 (6 A+7 B) d (a+b x) (c+d x)^2\right ) \log \left (\frac {e (a+b x)}{c+d x}\right )-2 B \left (-a^4 B d^4+4 a^3 b B d^3 (2 c+d x)-6 a^2 b^2 d^2 \left (2 A (c+d x)^2-B d x (4 c+3 d x)\right )+b^4 \left (-12 A d^2 x^2 (c+d x)^2+B c \left (c^3-4 c^2 d x-18 c d^2 x^2-12 d^3 x^3\right )\right )-4 a b^3 d \left (6 A d x (c+d x)^2+B \left (2 c^3+6 c^2 d x-3 d^3 x^3\right )\right )\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )+8 b^2 B^2 d^2 (a+b x)^2 (c+d x)^2 \log ^3\left (\frac {e (a+b x)}{c+d x}\right )-12 b^2 \left (2 A^2+5 B^2\right ) d^2 (a+b x)^2 (c+d x)^2 \log (c+d x)}{4 (b c-a d)^5 g^3 i^3 (a+b x)^2 (c+d x)^2} \]
((2*A^2 - 2*A*B + B^2)*d^2*(b*c - a*d)^2*(a + b*x)^2 + 2*b*(6*A^2 - 14*A*B + 15*B^2)*d^2*(b*c - a*d)*(a + b*x)^2*(c + d*x) - b^2*(2*A^2 + 2*A*B + B^ 2)*(b*c - a*d)^2*(c + d*x)^2 + 2*b^2*(6*A^2 + 14*A*B + 15*B^2)*d*(b*c - a* d)*(a + b*x)*(c + d*x)^2 + 12*b^2*(2*A^2 + 5*B^2)*d^2*(a + b*x)^2*(c + d*x )^2*Log[a + b*x] + 2*B*(b*c - a*d)*((2*A - B)*d^2*(b*c - a*d)*(a + b*x)^2 + 2*b*(6*A - 7*B)*d^2*(a + b*x)^2*(c + d*x) - b^2*(2*A + B)*(b*c - a*d)*(c + d*x)^2 + 2*b^2*(6*A + 7*B)*d*(a + b*x)*(c + d*x)^2)*Log[(e*(a + b*x))/( c + d*x)] - 2*B*(-(a^4*B*d^4) + 4*a^3*b*B*d^3*(2*c + d*x) - 6*a^2*b^2*d^2* (2*A*(c + d*x)^2 - B*d*x*(4*c + 3*d*x)) + b^4*(-12*A*d^2*x^2*(c + d*x)^2 + B*c*(c^3 - 4*c^2*d*x - 18*c*d^2*x^2 - 12*d^3*x^3)) - 4*a*b^3*d*(6*A*d*x*( c + d*x)^2 + B*(2*c^3 + 6*c^2*d*x - 3*d^3*x^3)))*Log[(e*(a + b*x))/(c + d* x)]^2 + 8*b^2*B^2*d^2*(a + b*x)^2*(c + d*x)^2*Log[(e*(a + b*x))/(c + d*x)] ^3 - 12*b^2*(2*A^2 + 5*B^2)*d^2*(a + b*x)^2*(c + d*x)^2*Log[c + d*x])/(4*( b*c - a*d)^5*g^3*i^3*(a + b*x)^2*(c + d*x)^2)
Time = 0.66 (sec) , antiderivative size = 478, 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.
| \(\displaystyle \int \frac {\left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{(a g+b g x)^3 (c i+d i x)^3} \, dx\) |
| \(\Big \downarrow \) 2962 |
| \(\displaystyle \frac {\int \frac {(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^3}d\frac {a+b x}{c+d x}}{g^3 i^3 (b c-a d)^5}\) |
| \(\Big \downarrow \) 2795 |
| \(\displaystyle \frac {\int \left (\frac {(c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 b^4}{(a+b x)^3}-\frac {4 d (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 b^3}{(a+b x)^2}+\frac {6 d^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 b^2}{a+b x}-4 d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 b+\frac {d^4 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{c+d x}\right )d\frac {a+b x}{c+d x}}{g^3 i^3 (b c-a d)^5}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {b^4 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 (a+b x)^2}-\frac {b^4 B (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (a+b x)^2}+\frac {4 b^3 d (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{a+b x}+\frac {8 b^3 B d (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}+\frac {2 b^2 d^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^3}{B}+\frac {d^4 (a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 (c+d x)^2}-\frac {B d^4 (a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (c+d x)^2}-\frac {4 b d^3 (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{c+d x}+\frac {8 A b B d^3 (a+b x)}{c+d x}-\frac {b^4 B^2 (c+d x)^2}{4 (a+b x)^2}+\frac {8 b^3 B^2 d (c+d x)}{a+b x}+\frac {B^2 d^4 (a+b x)^2}{4 (c+d x)^2}+\frac {8 b B^2 d^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x}-\frac {8 b B^2 d^3 (a+b x)}{c+d x}}{g^3 i^3 (b c-a d)^5}\) |
((B^2*d^4*(a + b*x)^2)/(4*(c + d*x)^2) + (8*A*b*B*d^3*(a + b*x))/(c + d*x) - (8*b*B^2*d^3*(a + b*x))/(c + d*x) + (8*b^3*B^2*d*(c + d*x))/(a + b*x) - (b^4*B^2*(c + d*x)^2)/(4*(a + b*x)^2) + (8*b*B^2*d^3*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)])/(c + d*x) - (B*d^4*(a + b*x)^2*(A + B*Log[(e*(a + b*x) )/(c + d*x)]))/(2*(c + d*x)^2) + (8*b^3*B*d*(c + d*x)*(A + B*Log[(e*(a + b *x))/(c + d*x)]))/(a + b*x) - (b^4*B*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/ (c + d*x)]))/(2*(a + b*x)^2) + (d^4*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/( c + d*x)])^2)/(2*(c + d*x)^2) - (4*b*d^3*(a + b*x)*(A + B*Log[(e*(a + b*x) )/(c + d*x)])^2)/(c + d*x) + (4*b^3*d*(c + d*x)*(A + B*Log[(e*(a + b*x))/( c + d*x)])^2)/(a + b*x) - (b^4*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d *x)])^2)/(2*(a + b*x)^2) + (2*b^2*d^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]) ^3)/B)/((b*c - a*d)^5*g^3*i^3)
3.2.6.3.1 Defintions of rubi rules used
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]))
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]
Leaf count of result is larger than twice the leaf count of optimal. \(1395\) vs. \(2(673)=1346\).
Time = 2.34 (sec) , antiderivative size = 1396, normalized size of antiderivative = 2.04
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1396\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1589\) |
| default | \(\text {Expression too large to display}\) | \(1589\) |
| risch | \(\text {Expression too large to display}\) | \(2582\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(2645\) |
| norman | \(\text {Expression too large to display}\) | \(2758\) |
A^2/g^3/i^3*(-1/2*d^2/(a*d-b*c)^3/(d*x+c)^2+6*d^2/(a*d-b*c)^5*b^2*ln(d*x+c )+3*d^2/(a*d-b*c)^4*b/(d*x+c)+1/2*b^2/(a*d-b*c)^3/(b*x+a)^2-6*d^2/(a*d-b*c )^5*b^2*ln(b*x+a)+3*b^2/(a*d-b*c)^4*d/(b*x+a))-B^2/g^3/i^3*d/(a*d-b*c)^2/e ^2*(d^3/(a*d-b*c)^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)-4/(a*d-b*c)^3*b*d^2*e*((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)+2*d/(a*d-b*c)^3*b^2*e^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3-4/(a*d -b*c)^3*b^3*e^3*(-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/d/(a*d-b*c)^3*b^4*e^4*(-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*B*A/g^3/i^3*d/(a*d-b*c)^2/e^2*(d^3/(a*d-b*c)^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)-4*d^2/(a*d-b*c)^3*b*e*((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)+3*d/(a*d-b*c)^ 3*b^2*e^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2-4/(a*d-b*c)^3*b^3*e^3*(-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...
Leaf count of result is larger than twice the leaf count of optimal. 1517 vs. \(2 (673) = 1346\).
Time = 0.41 (sec) , antiderivative size = 1517, normalized size of antiderivative = 2.21 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3 (c i+d i x)^3} \, dx=\text {Too large to display} \]
-1/4*(60*A*B*a^2*b^2*c^2*d^2 + (2*A^2 + 2*A*B + B^2)*b^4*c^4 - 16*(A^2 + 2 *A*B + 2*B^2)*a*b^3*c^3*d + 16*(A^2 - 2*A*B + 2*B^2)*a^3*b*c*d^3 - (2*A^2 - 2*A*B + B^2)*a^4*d^4 - 12*((2*A^2 + 5*B^2)*b^4*c*d^3 - (2*A^2 + 5*B^2)*a *b^3*d^4)*x^3 - 8*(B^2*b^4*d^4*x^4 + B^2*a^2*b^2*c^2*d^2 + 2*(B^2*b^4*c*d^ 3 + B^2*a*b^3*d^4)*x^3 + (B^2*b^4*c^2*d^2 + 4*B^2*a*b^3*c*d^3 + B^2*a^2*b^ 2*d^4)*x^2 + 2*(B^2*a*b^3*c^2*d^2 + B^2*a^2*b^2*c*d^3)*x)*log((b*e*x + a*e )/(d*x + c))^3 + 6*(8*A*B*a*b^3*c*d^3 - (6*A^2 + 4*A*B + 15*B^2)*b^4*c^2*d ^2 + (6*A^2 - 4*A*B + 15*B^2)*a^2*b^2*d^4)*x^2 - 2*(12*A*B*b^4*d^4*x^4 - B ^2*b^4*c^4 + 8*B^2*a*b^3*c^3*d + 12*A*B*a^2*b^2*c^2*d^2 - 8*B^2*a^3*b*c*d^ 3 + B^2*a^4*d^4 + 12*((2*A*B + B^2)*b^4*c*d^3 + (2*A*B - B^2)*a*b^3*d^4)*x ^3 + 6*(8*A*B*a*b^3*c*d^3 + (2*A*B + 3*B^2)*b^4*c^2*d^2 + (2*A*B - 3*B^2)* a^2*b^2*d^4)*x^2 + 4*(B^2*b^4*c^3*d - B^2*a^3*b*d^4 + 6*(A*B + B^2)*a*b^3* c^2*d^2 + 6*(A*B - B^2)*a^2*b^2*c*d^3)*x)*log((b*e*x + a*e)/(d*x + c))^2 - 4*((2*A^2 + 6*A*B + 7*B^2)*b^4*c^3*d + 6*(2*A^2 - A*B + 4*B^2)*a*b^3*c^2* d^2 - 6*(2*A^2 + A*B + 4*B^2)*a^2*b^2*c*d^3 - (2*A^2 - 6*A*B + 7*B^2)*a^3* b*d^4)*x - 2*(6*(2*A^2 + 5*B^2)*b^4*d^4*x^4 + 12*A^2*a^2*b^2*c^2*d^2 - (2* A*B + B^2)*b^4*c^4 + 16*(A*B + B^2)*a*b^3*c^3*d - 16*(A*B - B^2)*a^3*b*c*d ^3 + (2*A*B - B^2)*a^4*d^4 + 12*((2*A^2 + 2*A*B + 5*B^2)*b^4*c*d^3 + (2*A^ 2 - 2*A*B + 5*B^2)*a*b^3*d^4)*x^3 + 6*((2*A^2 + 6*A*B + 7*B^2)*b^4*c^2*d^2 + 8*(A^2 + 2*B^2)*a*b^3*c*d^3 + (2*A^2 - 6*A*B + 7*B^2)*a^2*b^2*d^4)*x...
Timed out. \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3 (c i+d i x)^3} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 5583 vs. \(2 (673) = 1346\).
Time = 0.55 (sec) , antiderivative size = 5583, normalized size of antiderivative = 8.15 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3 (c i+d i x)^3} \, dx=\text {Too large to display} \]
1/2*B^2*((12*b^3*d^3*x^3 - b^3*c^3 + 7*a*b^2*c^2*d + 7*a^2*b*c*d^2 - a^3*d ^3 + 18*(b^3*c*d^2 + a*b^2*d^3)*x^2 + 4*(b^3*c^2*d + 7*a*b^2*c*d^2 + a^2*b *d^3)*x)/((b^6*c^4*d^2 - 4*a*b^5*c^3*d^3 + 6*a^2*b^4*c^2*d^4 - 4*a^3*b^3*c *d^5 + a^4*b^2*d^6)*g^3*i^3*x^4 + 2*(b^6*c^5*d - 3*a*b^5*c^4*d^2 + 2*a^2*b ^4*c^3*d^3 + 2*a^3*b^3*c^2*d^4 - 3*a^4*b^2*c*d^5 + a^5*b*d^6)*g^3*i^3*x^3 + (b^6*c^6 - 9*a^2*b^4*c^4*d^2 + 16*a^3*b^3*c^3*d^3 - 9*a^4*b^2*c^2*d^4 + a^6*d^6)*g^3*i^3*x^2 + 2*(a*b^5*c^6 - 3*a^2*b^4*c^5*d + 2*a^3*b^3*c^4*d^2 + 2*a^4*b^2*c^3*d^3 - 3*a^5*b*c^2*d^4 + a^6*c*d^5)*g^3*i^3*x + (a^2*b^4*c^ 6 - 4*a^3*b^3*c^5*d + 6*a^4*b^2*c^4*d^2 - 4*a^5*b*c^3*d^3 + a^6*c^2*d^4)*g ^3*i^3) + 12*b^2*d^2*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^3*i^3) - 12*b^2*d ^2*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^3*i^3))*log(b*e*x/(d*x + c) + a*e/( d*x + c))^2 + A*B*((12*b^3*d^3*x^3 - b^3*c^3 + 7*a*b^2*c^2*d + 7*a^2*b*c*d ^2 - a^3*d^3 + 18*(b^3*c*d^2 + a*b^2*d^3)*x^2 + 4*(b^3*c^2*d + 7*a*b^2*c*d ^2 + a^2*b*d^3)*x)/((b^6*c^4*d^2 - 4*a*b^5*c^3*d^3 + 6*a^2*b^4*c^2*d^4 - 4 *a^3*b^3*c*d^5 + a^4*b^2*d^6)*g^3*i^3*x^4 + 2*(b^6*c^5*d - 3*a*b^5*c^4*d^2 + 2*a^2*b^4*c^3*d^3 + 2*a^3*b^3*c^2*d^4 - 3*a^4*b^2*c*d^5 + a^5*b*d^6)*g^ 3*i^3*x^3 + (b^6*c^6 - 9*a^2*b^4*c^4*d^2 + 16*a^3*b^3*c^3*d^3 - 9*a^4*b^2* c^2*d^4 + a^6*d^6)*g^3*i^3*x^2 + 2*(a*b^5*c^6 - 3*a^2*b^4*c^5*d + 2*a^3...
\[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3 (c i+d i x)^3} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}}{{\left (b g x + a g\right )}^{3} {\left (d i x + c i\right )}^{3}} \,d x } \]
Time = 11.14 (sec) , antiderivative size = 2155, normalized size of antiderivative = 3.15 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3 (c i+d i x)^3} \, dx=\text {Too large to display} \]
((2*x*(2*A^2*a^2*b*d^3 + 7*B^2*a^2*b*d^3 + 2*A^2*b^3*c^2*d + 7*B^2*b^3*c^2 *d + 14*A^2*a*b^2*c*d^2 + 31*B^2*a*b^2*c*d^2 - 6*A*B*a^2*b*d^3 + 6*A*B*b^3 *c^2*d))/(a*d - b*c) - (2*A^2*a^3*d^3 + 2*A^2*b^3*c^3 + B^2*a^3*d^3 + B^2* b^3*c^3 - 2*A*B*a^3*d^3 + 2*A*B*b^3*c^3 - 14*A^2*a*b^2*c^2*d - 14*A^2*a^2* b*c*d^2 - 31*B^2*a*b^2*c^2*d - 31*B^2*a^2*b*c*d^2 - 30*A*B*a*b^2*c^2*d + 3 0*A*B*a^2*b*c*d^2)/(2*(a*d - b*c)) + (6*x^3*(2*A^2*b^3*d^3 + 5*B^2*b^3*d^3 ))/(a*d - b*c) + (3*x^2*(6*A^2*a*b^2*d^3 + 15*B^2*a*b^2*d^3 + 6*A^2*b^3*c* d^2 + 15*B^2*b^3*c*d^2 - 4*A*B*a*b^2*d^3 + 4*A*B*b^3*c*d^2))/(a*d - b*c))/ (x^4*(2*a^3*b^2*d^5*g^3*i^3 - 2*b^5*c^3*d^2*g^3*i^3 + 6*a*b^4*c^2*d^3*g^3* i^3 - 6*a^2*b^3*c*d^4*g^3*i^3) - x*(4*a*b^4*c^5*g^3*i^3 - 4*a^5*c*d^4*g^3* i^3 - 8*a^2*b^3*c^4*d*g^3*i^3 + 8*a^4*b*c^2*d^3*g^3*i^3) + x^3*(4*a^4*b*d^ 5*g^3*i^3 - 4*b^5*c^4*d*g^3*i^3 + 8*a*b^4*c^3*d^2*g^3*i^3 - 8*a^3*b^2*c*d^ 4*g^3*i^3) + x^2*(2*a^5*d^5*g^3*i^3 - 2*b^5*c^5*g^3*i^3 - 2*a*b^4*c^4*d*g^ 3*i^3 + 2*a^4*b*c*d^4*g^3*i^3 + 16*a^2*b^3*c^3*d^2*g^3*i^3 - 16*a^3*b^2*c^ 2*d^3*g^3*i^3) - 2*a^2*b^3*c^5*g^3*i^3 + 2*a^5*c^2*d^3*g^3*i^3 + 6*a^3*b^2 *c^4*d*g^3*i^3 - 6*a^4*b*c^3*d^2*g^3*i^3) + log((e*(a + b*x))/(c + d*x))^2 *((x*((3*B^2*(a*d + b*c)^2)/(g^3*i^3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)^2) - B^2/(g^3*i^3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (6*B^2*a*b*c*d)/(g^3*i^3*( a^2*d^2 + b^2*c^2 - 2*a*b*c*d)^2)) - (B^2*(a*d + b*c))/(2*g^3*i^3*(a^2*b*d ^3 + b^3*c^2*d - 2*a*b^2*c*d^2)) + (6*B^2*b^2*d^2*x^3)/(g^3*i^3*(a^2*d^...