Integrand size = 40, antiderivative size = 287 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx=\frac {B^2 d i (c+d x)^2}{4 (b c-a d)^2 g^4 (a+b x)^2}-\frac {2 b B^2 i (c+d x)^3}{27 (b c-a d)^2 g^4 (a+b x)^3}+\frac {B d i (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^2 g^4 (a+b x)^2}-\frac {2 b B i (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9 (b c-a d)^2 g^4 (a+b x)^3}+\frac {d i (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 (b c-a d)^2 g^4 (a+b x)^2}-\frac {b i (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 (b c-a d)^2 g^4 (a+b x)^3} \]
1/4*B^2*d*i*(d*x+c)^2/(-a*d+b*c)^2/g^4/(b*x+a)^2-2/27*b*B^2*i*(d*x+c)^3/(- a*d+b*c)^2/g^4/(b*x+a)^3+1/2*B*d*i*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/( -a*d+b*c)^2/g^4/(b*x+a)^2-2/9*b*B*i*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/ (-a*d+b*c)^2/g^4/(b*x+a)^3+1/2*d*i*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))^2 /(-a*d+b*c)^2/g^4/(b*x+a)^2-1/3*b*i*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))^ 2/(-a*d+b*c)^2/g^4/(b*x+a)^3
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.60 (sec) , antiderivative size = 1032, normalized size of antiderivative = 3.60 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx=-\frac {i \left (36 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+54 d (b c-a d)^2 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+2 B \left (12 A (b c-a d)^3+4 B (b c-a d)^3-18 A d (b c-a d)^2 (a+b x)-15 B d (b c-a d)^2 (a+b x)+36 A d^2 (b c-a d) (a+b x)^2+66 B d^2 (b c-a d) (a+b x)^2+36 A d^3 (a+b x)^3 \log (a+b x)+66 B d^3 (a+b x)^3 \log (a+b x)-18 B d^3 (a+b x)^3 \log ^2(a+b x)+12 B (b c-a d)^3 \log \left (\frac {e (a+b x)}{c+d x}\right )-18 B d (b c-a d)^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+36 B d^2 (b c-a d) (a+b x)^2 \log \left (\frac {e (a+b x)}{c+d x}\right )+36 B d^3 (a+b x)^3 \log (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )-36 A d^3 (a+b x)^3 \log (c+d x)-66 B d^3 (a+b x)^3 \log (c+d x)+36 B d^3 (a+b x)^3 \log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)-36 B d^3 (a+b x)^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)-18 B d^3 (a+b x)^3 \log ^2(c+d x)+36 B d^3 (a+b x)^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )+36 B d^3 (a+b x)^3 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )+36 B d^3 (a+b x)^3 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )+27 B d (a+b x) \left (2 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+4 d (-b c+a d) (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-4 d^2 (a+b x)^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+4 d^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)-4 B d (a+b x) (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))+B \left ((b c-a d)^2+2 d (-b c+a d) (a+b x)-2 d^2 (a+b x)^2 \log (a+b x)+2 d^2 (a+b x)^2 \log (c+d x)\right )+2 B d^2 (a+b x)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )-2 B d^2 (a+b x)^2 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )\right )}{108 b^2 (b c-a d)^2 g^4 (a+b x)^3} \]
-1/108*(i*(36*(b*c - a*d)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + 54*d* (b*c - a*d)^2*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + 2*B*(12*A *(b*c - a*d)^3 + 4*B*(b*c - a*d)^3 - 18*A*d*(b*c - a*d)^2*(a + b*x) - 15*B *d*(b*c - a*d)^2*(a + b*x) + 36*A*d^2*(b*c - a*d)*(a + b*x)^2 + 66*B*d^2*( b*c - a*d)*(a + b*x)^2 + 36*A*d^3*(a + b*x)^3*Log[a + b*x] + 66*B*d^3*(a + b*x)^3*Log[a + b*x] - 18*B*d^3*(a + b*x)^3*Log[a + b*x]^2 + 12*B*(b*c - a *d)^3*Log[(e*(a + b*x))/(c + d*x)] - 18*B*d*(b*c - a*d)^2*(a + b*x)*Log[(e *(a + b*x))/(c + d*x)] + 36*B*d^2*(b*c - a*d)*(a + b*x)^2*Log[(e*(a + b*x) )/(c + d*x)] + 36*B*d^3*(a + b*x)^3*Log[a + b*x]*Log[(e*(a + b*x))/(c + d* x)] - 36*A*d^3*(a + b*x)^3*Log[c + d*x] - 66*B*d^3*(a + b*x)^3*Log[c + d*x ] + 36*B*d^3*(a + b*x)^3*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[c + d*x] - 36*B*d^3*(a + b*x)^3*Log[(e*(a + b*x))/(c + d*x)]*Log[c + d*x] - 18*B*d^3* (a + b*x)^3*Log[c + d*x]^2 + 36*B*d^3*(a + b*x)^3*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)] + 36*B*d^3*(a + b*x)^3*PolyLog[2, (d*(a + b*x))/(-(b*c ) + a*d)] + 36*B*d^3*(a + b*x)^3*PolyLog[2, (b*(c + d*x))/(b*c - a*d)]) + 27*B*d*(a + b*x)*(2*(b*c - a*d)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 4 *d*(-(b*c) + a*d)*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 4*d^2*( a + b*x)^2*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 4*d^2*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x] - 4*B*d*(a + b*x) *(b*c - a*d + d*(a + b*x)*Log[a + b*x] - d*(a + b*x)*Log[c + d*x]) + B*...
Time = 0.42 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.76, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, 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.
| \(\displaystyle \int \frac {(c i+d i x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{(a g+b g x)^4} \, dx\) |
| \(\Big \downarrow \) 2962 |
| \(\displaystyle \frac {i \int \frac {(c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^4}d\frac {a+b x}{c+d x}}{g^4 (b c-a d)^2}\) |
| \(\Big \downarrow \) 2795 |
| \(\displaystyle \frac {i \int \left (\frac {b (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^4}-\frac {d (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^3}\right )d\frac {a+b x}{c+d x}}{g^4 (b c-a d)^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {i \left (-\frac {b (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{3 (a+b x)^3}-\frac {2 b B (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{9 (a+b x)^3}+\frac {d (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 (a+b x)^2}+\frac {B d (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (a+b x)^2}-\frac {2 b B^2 (c+d x)^3}{27 (a+b x)^3}+\frac {B^2 d (c+d x)^2}{4 (a+b x)^2}\right )}{g^4 (b c-a d)^2}\) |
(i*((B^2*d*(c + d*x)^2)/(4*(a + b*x)^2) - (2*b*B^2*(c + d*x)^3)/(27*(a + b *x)^3) + (B*d*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(a + b* x)^2) - (2*b*B*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(9*(a + b *x)^3) + (d*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*(a + b* x)^2) - (b*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(3*(a + b*x )^3)))/((b*c - a*d)^2*g^4)
3.1.62.3.1 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. \(642\) vs. \(2(275)=550\).
Time = 0.99 (sec) , antiderivative size = 643, normalized size of antiderivative = 2.24
| method | result | size |
| parts | \(\frac {i \,A^{2} \left (-\frac {-a d +c b}{3 b^{2} \left (b x +a \right )^{3}}-\frac {d}{2 b^{2} \left (b x +a \right )^{2}}\right )}{g^{4}}-\frac {i \,B^{2} \left (a d -c b \right )^{2} e^{2} \left (\frac {d^{4} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (a d -c b \right )^{4}}-\frac {d^{3} b e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{3 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {2 \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{9 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {2}{27 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (a d -c b \right )^{4}}\right )}{g^{4} d^{3}}-\frac {2 i B A \left (a d -c b \right )^{2} e^{2} \left (\frac {d^{4} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (a d -c b \right )^{4}}-\frac {d^{3} b e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (a d -c b \right )^{4}}\right )}{g^{4} d^{3}}\) | \(643\) |
| derivativedivides | \(-\frac {e \left (a d -c b \right ) \left (\frac {i \,d^{2} e^{2} A^{2} b}{3 \left (a d -c b \right )^{3} g^{4} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {i \,d^{3} e \,A^{2}}{2 \left (a d -c b \right )^{3} g^{4} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {2 i \,d^{2} e^{2} A B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (a d -c b \right )^{3} g^{4}}+\frac {2 i \,d^{3} e A B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (a d -c b \right )^{3} g^{4}}-\frac {i \,d^{2} e^{2} B^{2} b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{3 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {2 \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{9 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {2}{27 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (a d -c b \right )^{3} g^{4}}+\frac {i \,d^{3} e \,B^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (a d -c b \right )^{3} g^{4}}\right )}{d^{2}}\) | \(704\) |
| default | \(-\frac {e \left (a d -c b \right ) \left (\frac {i \,d^{2} e^{2} A^{2} b}{3 \left (a d -c b \right )^{3} g^{4} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {i \,d^{3} e \,A^{2}}{2 \left (a d -c b \right )^{3} g^{4} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {2 i \,d^{2} e^{2} A B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (a d -c b \right )^{3} g^{4}}+\frac {2 i \,d^{3} e A B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (a d -c b \right )^{3} g^{4}}-\frac {i \,d^{2} e^{2} B^{2} b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{3 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {2 \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{9 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {2}{27 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (a d -c b \right )^{3} g^{4}}+\frac {i \,d^{3} e \,B^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (a d -c b \right )^{3} g^{4}}\right )}{d^{2}}\) | \(704\) |
| norman | \(\frac {-\frac {18 A^{2} a^{2} b \,d^{2} i +18 A^{2} a \,b^{2} c d i -36 A^{2} b^{3} c^{2} i +30 A B \,a^{2} b \,d^{2} i +30 A B a \,b^{2} c d i -24 A B \,b^{3} c^{2} i +19 B^{2} a^{2} b \,d^{2} i +19 B^{2} a \,b^{2} c d i -8 B^{2} b^{3} c^{2} i}{108 g \,b^{3} \left (a d -c b \right )}-\frac {\left (18 A^{2} a b \,d^{2} i -18 A^{2} b^{2} c d i +30 A B a b \,d^{2} i -6 A B \,b^{2} c d i +19 B^{2} a b \,d^{2} i +B^{2} b^{2} c d i \right ) x}{36 g \left (a d -c b \right ) b^{2}}-\frac {\left (6 A B b \,d^{2} i +5 B^{2} b \,d^{2} i \right ) x^{2}}{18 b \left (a d -c b \right ) g}+\frac {B^{2} i \,c^{2} \left (3 a d -2 c b \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{6 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) g}+\frac {c^{2} B i \left (18 A a d -12 A b c +9 B a d -4 B b c \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{18 g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {B \,d^{2} i \left (6 A a d +3 B a d +2 B b c \right ) x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{6 g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {B^{2} a i \,d^{3} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{2 g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {b \,d^{3} B^{2} i \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{6 g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {B^{2} i c d \left (2 a d -c b \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) g}+\frac {b \,d^{3} B i \left (6 A +5 B \right ) x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{18 g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {c B i d \left (12 A a d -6 A b c +6 B a d -B b c \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{6 g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (b x +a \right )^{3} g^{3}}\) | \(762\) |
| parallelrisch | \(-\frac {-216 A B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{5} c \,d^{3} i -27 B^{2} a \,b^{5} c^{2} d^{2} i -54 A B a \,b^{5} c^{2} d^{2} i -18 B^{2} x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{6} d^{4} i -30 B^{2} x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{6} d^{4} i +30 B^{2} x^{2} a \,b^{5} d^{4} i -30 B^{2} x^{2} b^{6} c \,d^{3} i +36 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{6} c^{3} d i +54 A^{2} x \,a^{2} b^{4} d^{4} i +54 A^{2} x \,b^{6} c^{2} d^{2} i +57 B^{2} x \,a^{2} b^{4} d^{4} i -3 B^{2} x \,b^{6} c^{2} d^{2} i +24 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{6} c^{3} d i -54 A^{2} a \,b^{5} c^{2} d^{2} i +18 A^{2} a^{3} b^{3} d^{4} i +36 A^{2} b^{6} c^{3} d i +19 B^{2} a^{3} b^{3} d^{4} i +8 B^{2} b^{6} c^{3} d i +24 A B \,b^{6} c^{3} d i +30 A B \,a^{3} b^{3} d^{4} i -36 A B \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{6} d^{4} i -54 B^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a \,b^{5} d^{4} i -54 B^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{5} d^{4} i -36 B^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{6} c \,d^{3} i +54 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{6} c^{2} d^{2} i +36 A B \,x^{2} a \,b^{5} d^{4} i -36 A B \,x^{2} b^{6} c \,d^{3} i +18 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{6} c^{2} d^{2} i -54 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a \,b^{5} c^{2} d^{2} i -108 A^{2} x a \,b^{5} c \,d^{3} i +90 A B x \,a^{2} b^{4} d^{4} i +18 A B x \,b^{6} c^{2} d^{2} i +72 A B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{6} c^{3} d i -54 B^{2} x a \,b^{5} c \,d^{3} i -54 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{5} c^{2} d^{2} i -108 A B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{5} d^{4} i -108 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a \,b^{5} c \,d^{3} i +108 A B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{6} c^{2} d^{2} i -108 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{5} c \,d^{3} i -108 A B x a \,b^{5} c \,d^{3} i -108 A B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{5} c^{2} d^{2} i}{108 g^{4} \left (b x +a \right )^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{5} d}\) | \(952\) |
| risch | \(\text {Expression too large to display}\) | \(2435\) |
i*A^2/g^4*(-1/3*(-a*d+b*c)/b^2/(b*x+a)^3-1/2*d/b^2/(b*x+a)^2)-i*B^2/g^4/d^ 3*(a*d-b*c)^2*e^2*(d^4/(a*d-b*c)^4*(-1/2/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*l n(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^3/(a*d-b *c)^4*b*e*(-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*i*B*A/g^4/d^3*(a*d-b*c)^2*e^2*( d^4/(a*d-b*c)^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))-1/4/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2)-d^3/(a*d-b*c)^4*b*e*(-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))
Leaf count of result is larger than twice the leaf count of optimal. 601 vs. \(2 (275) = 550\).
Time = 0.31 (sec) , antiderivative size = 601, normalized size of antiderivative = 2.09 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx=\frac {6 \, {\left ({\left (6 \, A B + 5 \, B^{2}\right )} b^{3} c d^{2} - {\left (6 \, A B + 5 \, B^{2}\right )} a b^{2} d^{3}\right )} i x^{2} - 3 \, {\left ({\left (18 \, A^{2} + 6 \, A B - B^{2}\right )} b^{3} c^{2} d - 18 \, {\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} a b^{2} c d^{2} + {\left (18 \, A^{2} + 30 \, A B + 19 \, B^{2}\right )} a^{2} b d^{3}\right )} i x + 18 \, {\left (B^{2} b^{3} d^{3} i x^{3} + 3 \, B^{2} a b^{2} d^{3} i x^{2} - 3 \, {\left (B^{2} b^{3} c^{2} d - 2 \, B^{2} a b^{2} c d^{2}\right )} i x - {\left (2 \, B^{2} b^{3} c^{3} - 3 \, B^{2} a b^{2} c^{2} d\right )} i\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} - {\left (4 \, {\left (9 \, A^{2} + 6 \, A B + 2 \, B^{2}\right )} b^{3} c^{3} - 27 \, {\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} a b^{2} c^{2} d + {\left (18 \, A^{2} + 30 \, A B + 19 \, B^{2}\right )} a^{3} d^{3}\right )} i + 6 \, {\left ({\left (6 \, A B + 5 \, B^{2}\right )} b^{3} d^{3} i x^{3} + 3 \, {\left (2 \, B^{2} b^{3} c d^{2} + 3 \, {\left (2 \, A B + B^{2}\right )} a b^{2} d^{3}\right )} i x^{2} - 3 \, {\left ({\left (6 \, A B + B^{2}\right )} b^{3} c^{2} d - 6 \, {\left (2 \, A B + B^{2}\right )} a b^{2} c d^{2}\right )} i x - {\left (4 \, {\left (3 \, A B + B^{2}\right )} b^{3} c^{3} - 9 \, {\left (2 \, A B + B^{2}\right )} a b^{2} c^{2} d\right )} i\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{108 \, {\left ({\left (b^{7} c^{2} - 2 \, a b^{6} c d + a^{2} b^{5} d^{2}\right )} g^{4} x^{3} + 3 \, {\left (a b^{6} c^{2} - 2 \, a^{2} b^{5} c d + a^{3} b^{4} d^{2}\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{5} c^{2} - 2 \, a^{3} b^{4} c d + a^{4} b^{3} d^{2}\right )} g^{4} x + {\left (a^{3} b^{4} c^{2} - 2 \, a^{4} b^{3} c d + a^{5} b^{2} d^{2}\right )} g^{4}\right )}} \]
1/108*(6*((6*A*B + 5*B^2)*b^3*c*d^2 - (6*A*B + 5*B^2)*a*b^2*d^3)*i*x^2 - 3 *((18*A^2 + 6*A*B - B^2)*b^3*c^2*d - 18*(2*A^2 + 2*A*B + B^2)*a*b^2*c*d^2 + (18*A^2 + 30*A*B + 19*B^2)*a^2*b*d^3)*i*x + 18*(B^2*b^3*d^3*i*x^3 + 3*B^ 2*a*b^2*d^3*i*x^2 - 3*(B^2*b^3*c^2*d - 2*B^2*a*b^2*c*d^2)*i*x - (2*B^2*b^3 *c^3 - 3*B^2*a*b^2*c^2*d)*i)*log((b*e*x + a*e)/(d*x + c))^2 - (4*(9*A^2 + 6*A*B + 2*B^2)*b^3*c^3 - 27*(2*A^2 + 2*A*B + B^2)*a*b^2*c^2*d + (18*A^2 + 30*A*B + 19*B^2)*a^3*d^3)*i + 6*((6*A*B + 5*B^2)*b^3*d^3*i*x^3 + 3*(2*B^2* b^3*c*d^2 + 3*(2*A*B + B^2)*a*b^2*d^3)*i*x^2 - 3*((6*A*B + B^2)*b^3*c^2*d - 6*(2*A*B + B^2)*a*b^2*c*d^2)*i*x - (4*(3*A*B + B^2)*b^3*c^3 - 9*(2*A*B + B^2)*a*b^2*c^2*d)*i)*log((b*e*x + a*e)/(d*x + c)))/((b^7*c^2 - 2*a*b^6*c* d + a^2*b^5*d^2)*g^4*x^3 + 3*(a*b^6*c^2 - 2*a^2*b^5*c*d + a^3*b^4*d^2)*g^4 *x^2 + 3*(a^2*b^5*c^2 - 2*a^3*b^4*c*d + a^4*b^3*d^2)*g^4*x + (a^3*b^4*c^2 - 2*a^4*b^3*c*d + a^5*b^2*d^2)*g^4)
Leaf count of result is larger than twice the leaf count of optimal. 1387 vs. \(2 (267) = 534\).
Time = 10.34 (sec) , antiderivative size = 1387, normalized size of antiderivative = 4.83 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx=- \frac {B d^{3} i \left (6 A + 5 B\right ) \log {\left (x + \frac {6 A B a d^{4} i + 6 A B b c d^{3} i + 5 B^{2} a d^{4} i + 5 B^{2} b c d^{3} i - \frac {B a^{3} d^{6} i \left (6 A + 5 B\right )}{\left (a d - b c\right )^{2}} + \frac {3 B a^{2} b c d^{5} i \left (6 A + 5 B\right )}{\left (a d - b c\right )^{2}} - \frac {3 B a b^{2} c^{2} d^{4} i \left (6 A + 5 B\right )}{\left (a d - b c\right )^{2}} + \frac {B b^{3} c^{3} d^{3} i \left (6 A + 5 B\right )}{\left (a d - b c\right )^{2}}}{12 A B b d^{4} i + 10 B^{2} b d^{4} i} \right )}}{18 b^{2} g^{4} \left (a d - b c\right )^{2}} + \frac {B d^{3} i \left (6 A + 5 B\right ) \log {\left (x + \frac {6 A B a d^{4} i + 6 A B b c d^{3} i + 5 B^{2} a d^{4} i + 5 B^{2} b c d^{3} i + \frac {B a^{3} d^{6} i \left (6 A + 5 B\right )}{\left (a d - b c\right )^{2}} - \frac {3 B a^{2} b c d^{5} i \left (6 A + 5 B\right )}{\left (a d - b c\right )^{2}} + \frac {3 B a b^{2} c^{2} d^{4} i \left (6 A + 5 B\right )}{\left (a d - b c\right )^{2}} - \frac {B b^{3} c^{3} d^{3} i \left (6 A + 5 B\right )}{\left (a d - b c\right )^{2}}}{12 A B b d^{4} i + 10 B^{2} b d^{4} i} \right )}}{18 b^{2} g^{4} \left (a d - b c\right )^{2}} + \frac {\left (3 B^{2} a c^{2} d i + 6 B^{2} a c d^{2} i x + 3 B^{2} a d^{3} i x^{2} - 2 B^{2} b c^{3} i - 3 B^{2} b c^{2} d i x + B^{2} b d^{3} i x^{3}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{6 a^{5} d^{2} g^{4} - 12 a^{4} b c d g^{4} + 18 a^{4} b d^{2} g^{4} x + 6 a^{3} b^{2} c^{2} g^{4} - 36 a^{3} b^{2} c d g^{4} x + 18 a^{3} b^{2} d^{2} g^{4} x^{2} + 18 a^{2} b^{3} c^{2} g^{4} x - 36 a^{2} b^{3} c d g^{4} x^{2} + 6 a^{2} b^{3} d^{2} g^{4} x^{3} + 18 a b^{4} c^{2} g^{4} x^{2} - 12 a b^{4} c d g^{4} x^{3} + 6 b^{5} c^{2} g^{4} x^{3}} + \frac {\left (- 6 A B a^{2} d^{2} i - 6 A B a b c d i - 18 A B a b d^{2} i x + 12 A B b^{2} c^{2} i + 18 A B b^{2} c d i x - 5 B^{2} a^{2} d^{2} i - 5 B^{2} a b c d i - 15 B^{2} a b d^{2} i x + 4 B^{2} b^{2} c^{2} i + 3 B^{2} b^{2} c d i x - 6 B^{2} b^{2} d^{2} i x^{2}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{18 a^{4} b^{2} d g^{4} - 18 a^{3} b^{3} c g^{4} + 54 a^{3} b^{3} d g^{4} x - 54 a^{2} b^{4} c g^{4} x + 54 a^{2} b^{4} d g^{4} x^{2} - 54 a b^{5} c g^{4} x^{2} + 18 a b^{5} d g^{4} x^{3} - 18 b^{6} c g^{4} x^{3}} + \frac {- 18 A^{2} a^{2} d^{2} i - 18 A^{2} a b c d i + 36 A^{2} b^{2} c^{2} i - 30 A B a^{2} d^{2} i - 30 A B a b c d i + 24 A B b^{2} c^{2} i - 19 B^{2} a^{2} d^{2} i - 19 B^{2} a b c d i + 8 B^{2} b^{2} c^{2} i + x^{2} \left (- 36 A B b^{2} d^{2} i - 30 B^{2} b^{2} d^{2} i\right ) + x \left (- 54 A^{2} a b d^{2} i + 54 A^{2} b^{2} c d i - 90 A B a b d^{2} i + 18 A B b^{2} c d i - 57 B^{2} a b d^{2} i - 3 B^{2} b^{2} c d i\right )}{108 a^{4} b^{2} d g^{4} - 108 a^{3} b^{3} c g^{4} + x^{3} \cdot \left (108 a b^{5} d g^{4} - 108 b^{6} c g^{4}\right ) + x^{2} \cdot \left (324 a^{2} b^{4} d g^{4} - 324 a b^{5} c g^{4}\right ) + x \left (324 a^{3} b^{3} d g^{4} - 324 a^{2} b^{4} c g^{4}\right )} \]
-B*d**3*i*(6*A + 5*B)*log(x + (6*A*B*a*d**4*i + 6*A*B*b*c*d**3*i + 5*B**2* a*d**4*i + 5*B**2*b*c*d**3*i - B*a**3*d**6*i*(6*A + 5*B)/(a*d - b*c)**2 + 3*B*a**2*b*c*d**5*i*(6*A + 5*B)/(a*d - b*c)**2 - 3*B*a*b**2*c**2*d**4*i*(6 *A + 5*B)/(a*d - b*c)**2 + B*b**3*c**3*d**3*i*(6*A + 5*B)/(a*d - b*c)**2)/ (12*A*B*b*d**4*i + 10*B**2*b*d**4*i))/(18*b**2*g**4*(a*d - b*c)**2) + B*d* *3*i*(6*A + 5*B)*log(x + (6*A*B*a*d**4*i + 6*A*B*b*c*d**3*i + 5*B**2*a*d** 4*i + 5*B**2*b*c*d**3*i + B*a**3*d**6*i*(6*A + 5*B)/(a*d - b*c)**2 - 3*B*a **2*b*c*d**5*i*(6*A + 5*B)/(a*d - b*c)**2 + 3*B*a*b**2*c**2*d**4*i*(6*A + 5*B)/(a*d - b*c)**2 - B*b**3*c**3*d**3*i*(6*A + 5*B)/(a*d - b*c)**2)/(12*A *B*b*d**4*i + 10*B**2*b*d**4*i))/(18*b**2*g**4*(a*d - b*c)**2) + (3*B**2*a *c**2*d*i + 6*B**2*a*c*d**2*i*x + 3*B**2*a*d**3*i*x**2 - 2*B**2*b*c**3*i - 3*B**2*b*c**2*d*i*x + B**2*b*d**3*i*x**3)*log(e*(a + b*x)/(c + d*x))**2/( 6*a**5*d**2*g**4 - 12*a**4*b*c*d*g**4 + 18*a**4*b*d**2*g**4*x + 6*a**3*b** 2*c**2*g**4 - 36*a**3*b**2*c*d*g**4*x + 18*a**3*b**2*d**2*g**4*x**2 + 18*a **2*b**3*c**2*g**4*x - 36*a**2*b**3*c*d*g**4*x**2 + 6*a**2*b**3*d**2*g**4* x**3 + 18*a*b**4*c**2*g**4*x**2 - 12*a*b**4*c*d*g**4*x**3 + 6*b**5*c**2*g* *4*x**3) + (-6*A*B*a**2*d**2*i - 6*A*B*a*b*c*d*i - 18*A*B*a*b*d**2*i*x + 1 2*A*B*b**2*c**2*i + 18*A*B*b**2*c*d*i*x - 5*B**2*a**2*d**2*i - 5*B**2*a*b* c*d*i - 15*B**2*a*b*d**2*i*x + 4*B**2*b**2*c**2*i + 3*B**2*b**2*c*d*i*x - 6*B**2*b**2*d**2*i*x**2)*log(e*(a + b*x)/(c + d*x))/(18*a**4*b**2*d*g**...
Leaf count of result is larger than twice the leaf count of optimal. 3282 vs. \(2 (275) = 550\).
Time = 0.42 (sec) , antiderivative size = 3282, normalized size of antiderivative = 11.44 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx=\text {Too large to display} \]
-1/6*(3*b*x + a)*B^2*d*i*log(b*e*x/(d*x + c) + a*e/(d*x + c))^2/(b^5*g^4*x ^3 + 3*a*b^4*g^4*x^2 + 3*a^2*b^3*g^4*x + a^3*b^2*g^4) - 1/54*(6*((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^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*g^4*x^3 + 3*(a*b^5*c^2 - 2*a^2*b^4*c* d + a^3*b^3*d^2)*g^4*x^2 + 3*(a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)*g ^4*x + (a^3*b^3*c^2 - 2*a^4*b^2*c*d + a^5*b*d^2)*g^4) + 6*d^3*log(b*x + a) /((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*g^4) - 6*d^3*log (d*x + c)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*g^4))*l og(b*e*x/(d*x + c) + a*e/(d*x + c)) + (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^3*d^3)*log(d*x + c)^2 - 3*(5*b^3 *c^2*d - 54*a*b^2*c*d^2 + 49*a^2*b*d^3)*x + 66*(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) - 6*(11*b^3*d^3*x^3 + 33*a*b^2 *d^3*x^2 + 33*a^2*b*d^3*x + 11*a^3*d^3 - 6*(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))*log(d*x + c))/(a^3*b^4*c^3*g^4 - 3*a^4*b^3*c^2*d*g^4 + 3*a^5*b^2*c*d^2*g^4 - a^6*b*d^3*g^4 + (b^7*c^3*g^4 - 3*a*b^6*c^2*d*g^4 + 3*a^2*b^5*c*d^2*g^4 - a^3*b^4*d^3*g^4)*x^3 + 3*(a*b^6 *c^3*g^4 - 3*a^2*b^5*c^2*d*g^4 + 3*a^3*b^4*c*d^2*g^4 - a^4*b^3*d^3*g^4)*x^ 2 + 3*(a^2*b^5*c^3*g^4 - 3*a^3*b^4*c^2*d*g^4 + 3*a^4*b^3*c*d^2*g^4 - a^...
Time = 0.51 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.59 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx=-\frac {1}{108} \, {\left (\frac {18 \, {\left (2 \, B^{2} b e^{4} i - \frac {3 \, {\left (b e x + a e\right )} B^{2} d e^{3} i}{d x + c}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2}}{\frac {{\left (b e x + a e\right )}^{3} b c g^{4}}{{\left (d x + c\right )}^{3}} - \frac {{\left (b e x + a e\right )}^{3} a d g^{4}}{{\left (d x + c\right )}^{3}}} + \frac {6 \, {\left (12 \, A B b e^{4} i + 4 \, B^{2} b e^{4} i - \frac {18 \, {\left (b e x + a e\right )} A B d e^{3} i}{d x + c} - \frac {9 \, {\left (b e x + a e\right )} B^{2} d e^{3} i}{d x + c}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{\frac {{\left (b e x + a e\right )}^{3} b c g^{4}}{{\left (d x + c\right )}^{3}} - \frac {{\left (b e x + a e\right )}^{3} a d g^{4}}{{\left (d x + c\right )}^{3}}} + \frac {36 \, A^{2} b e^{4} i + 24 \, A B b e^{4} i + 8 \, B^{2} b e^{4} i - \frac {54 \, {\left (b e x + a e\right )} A^{2} d e^{3} i}{d x + c} - \frac {54 \, {\left (b e x + a e\right )} A B d e^{3} i}{d x + c} - \frac {27 \, {\left (b e x + a e\right )} B^{2} d e^{3} i}{d x + c}}{\frac {{\left (b e x + a e\right )}^{3} b c g^{4}}{{\left (d x + c\right )}^{3}} - \frac {{\left (b e x + a e\right )}^{3} a d g^{4}}{{\left (d x + c\right )}^{3}}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \]
-1/108*(18*(2*B^2*b*e^4*i - 3*(b*e*x + a*e)*B^2*d*e^3*i/(d*x + c))*log((b* e*x + a*e)/(d*x + c))^2/((b*e*x + a*e)^3*b*c*g^4/(d*x + c)^3 - (b*e*x + a* e)^3*a*d*g^4/(d*x + c)^3) + 6*(12*A*B*b*e^4*i + 4*B^2*b*e^4*i - 18*(b*e*x + a*e)*A*B*d*e^3*i/(d*x + c) - 9*(b*e*x + a*e)*B^2*d*e^3*i/(d*x + c))*log( (b*e*x + a*e)/(d*x + c))/((b*e*x + a*e)^3*b*c*g^4/(d*x + c)^3 - (b*e*x + a *e)^3*a*d*g^4/(d*x + c)^3) + (36*A^2*b*e^4*i + 24*A*B*b*e^4*i + 8*B^2*b*e^ 4*i - 54*(b*e*x + a*e)*A^2*d*e^3*i/(d*x + c) - 54*(b*e*x + a*e)*A*B*d*e^3* i/(d*x + c) - 27*(b*e*x + a*e)*B^2*d*e^3*i/(d*x + c))/((b*e*x + a*e)^3*b*c *g^4/(d*x + c)^3 - (b*e*x + a*e)^3*a*d*g^4/(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)))
Time = 4.17 (sec) , antiderivative size = 955, normalized size of antiderivative = 3.33 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx=-{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2\,\left (\frac {\frac {B^2\,c\,i}{3\,b^2\,g^4}+\frac {B^2\,a\,d\,i}{6\,b^3\,g^4}+\frac {B^2\,d\,i\,x}{2\,b^2\,g^4}}{3\,a^2\,x+\frac {a^3}{b}+b^2\,x^3+3\,a\,b\,x^2}-\frac {B^2\,d^3\,i}{6\,b^2\,g^4\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )-\frac {\frac {18\,i\,A^2\,a^2\,d^2+18\,i\,A^2\,a\,b\,c\,d-36\,i\,A^2\,b^2\,c^2+30\,i\,A\,B\,a^2\,d^2+30\,i\,A\,B\,a\,b\,c\,d-24\,i\,A\,B\,b^2\,c^2+19\,i\,B^2\,a^2\,d^2+19\,i\,B^2\,a\,b\,c\,d-8\,i\,B^2\,b^2\,c^2}{6\,\left (a\,d-b\,c\right )}+\frac {x^2\,\left (5\,i\,B^2\,b^2\,d^2+6\,A\,i\,B\,b^2\,d^2\right )}{a\,d-b\,c}+\frac {x\,\left (-18\,c\,i\,A^2\,b^2\,d+18\,a\,i\,A^2\,b\,d^2-6\,c\,i\,A\,B\,b^2\,d+30\,a\,i\,A\,B\,b\,d^2+c\,i\,B^2\,b^2\,d+19\,a\,i\,B^2\,b\,d^2\right )}{2\,\left (a\,d-b\,c\right )}}{18\,a^3\,b^2\,g^4+54\,a^2\,b^3\,g^4\,x+54\,a\,b^4\,g^4\,x^2+18\,b^5\,g^4\,x^3}-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (x\,\left (\frac {A\,B\,i}{b^2\,g^4}+\frac {B^2\,d^3\,i\,\left (b\,\left (\frac {3\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2}{6\,b\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{3\,b\,d^2}\right )+\frac {3\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2}{3\,d^3}+\frac {2\,a\,\left (a\,d-b\,c\right )}{3\,d^2}\right )}{3\,b^2\,g^4\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )+\frac {A\,B\,a\,i}{3\,b^3\,g^4}+\frac {B\,i\,\left (2\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{3\,b^3\,d\,g^4}+\frac {B^2\,d^3\,i\,\left (a\,\left (\frac {3\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2}{6\,b\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{3\,b\,d^2}\right )+\frac {3\,a^3\,d^3-6\,a^2\,b\,c\,d^2+4\,a\,b^2\,c^2\,d-b^3\,c^3}{3\,b\,d^4}\right )}{3\,b^2\,g^4\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {B^2\,d^3\,i\,x^2\,\left (\frac {b^2\,c-a\,b\,d}{3\,d^2}-\frac {2\,b\,\left (a\,d-b\,c\right )}{3\,d^2}\right )}{3\,b^2\,g^4\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )}{\frac {3\,a^2\,x}{d}+\frac {a^3}{b\,d}+\frac {b^2\,x^3}{d}+\frac {3\,a\,b\,x^2}{d}}-\frac {B\,d^3\,i\,\mathrm {atan}\left (\frac {\left (2\,b\,d\,x-\frac {18\,b^4\,c^2\,g^4-18\,a^2\,b^2\,d^2\,g^4}{18\,b^2\,g^4\,\left (a\,d-b\,c\right )}\right )\,1{}\mathrm {i}}{a\,d-b\,c}\right )\,\left (6\,A+5\,B\right )\,1{}\mathrm {i}}{9\,b^2\,g^4\,{\left (a\,d-b\,c\right )}^2} \]
- log((e*(a + b*x))/(c + d*x))^2*(((B^2*c*i)/(3*b^2*g^4) + (B^2*a*d*i)/(6* b^3*g^4) + (B^2*d*i*x)/(2*b^2*g^4))/(3*a^2*x + a^3/b + b^2*x^3 + 3*a*b*x^2 ) - (B^2*d^3*i)/(6*b^2*g^4*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))) - ((18*A^2*a^ 2*d^2*i - 36*A^2*b^2*c^2*i + 19*B^2*a^2*d^2*i - 8*B^2*b^2*c^2*i + 30*A*B*a ^2*d^2*i - 24*A*B*b^2*c^2*i + 18*A^2*a*b*c*d*i + 19*B^2*a*b*c*d*i + 30*A*B *a*b*c*d*i)/(6*(a*d - b*c)) + (x^2*(5*B^2*b^2*d^2*i + 6*A*B*b^2*d^2*i))/(a *d - b*c) + (x*(18*A^2*a*b*d^2*i + 19*B^2*a*b*d^2*i - 18*A^2*b^2*c*d*i + B ^2*b^2*c*d*i + 30*A*B*a*b*d^2*i - 6*A*B*b^2*c*d*i))/(2*(a*d - b*c)))/(18*a ^3*b^2*g^4 + 18*b^5*g^4*x^3 + 54*a^2*b^3*g^4*x + 54*a*b^4*g^4*x^2) - (log( (e*(a + b*x))/(c + d*x))*(x*((A*B*i)/(b^2*g^4) + (B^2*d^3*i*(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)))/(3*b^2*g^4* (a^2*d^2 + b^2*c^2 - 2*a*b*c*d))) + (A*B*a*i)/(3*b^3*g^4) + (B*i*(2*A*b*c - B*a*d + B*b*c))/(3*b^3*d*g^4) + (B^2*d^3*i*(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)))/(3*b^2*g^4*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) - (B^2*d^3*i*x^2*((b^2*c - a*b*d)/(3*d^2) - (2*b*(a*d - b*c))/ (3*d^2)))/(3*b^2*g^4*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))))/((3*a^2*x)/d + a^3 /(b*d) + (b^2*x^3)/d + (3*a*b*x^2)/d) - (B*d^3*i*atan(((2*b*d*x - (18*b^4* c^2*g^4 - 18*a^2*b^2*d^2*g^4)/(18*b^2*g^4*(a*d - b*c)))*1i)/(a*d - b*c)...