Integrand size = 34, antiderivative size = 272 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^3} \, dx=\frac {8 B^2 d (c+d x)}{(b c-a d)^2 g^3 (a+b x)}-\frac {b B^2 (c+d x)^2}{(b c-a d)^2 g^3 (a+b x)^2}+\frac {4 B d (c+d x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d)^2 g^3 (a+b x)}-\frac {b B (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d)^2 g^3 (a+b x)^2}+\frac {d (c+d x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(b c-a d)^2 g^3 (a+b x)}-\frac {b (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{2 (b c-a d)^2 g^3 (a+b x)^2} \]
8*B^2*d*(d*x+c)/(-a*d+b*c)^2/g^3/(b*x+a)-b*B^2*(d*x+c)^2/(-a*d+b*c)^2/g^3/ (b*x+a)^2+4*B*d*(d*x+c)*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))/(-a*d+b*c)^2/g^3/( b*x+a)-b*B*(d*x+c)^2*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))/(-a*d+b*c)^2/g^3/(b*x +a)^2+d*(d*x+c)*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))^2/(-a*d+b*c)^2/g^3/(b*x+a) -1/2*b*(d*x+c)^2*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))^2/(-a*d+b*c)^2/g^3/(b*x+a )^2
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.29 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.66 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^3} \, dx=-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2+\frac {2 B \left ((b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )+2 d (-b c+a d) (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )-2 d^2 (a+b x)^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )+2 d^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)-4 B d (a+b x) (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))+B \left ((b c-a d)^2+2 d (-b c+a d) (a+b x)-2 d^2 (a+b x)^2 \log (a+b x)+2 d^2 (a+b x)^2 \log (c+d x)\right )+2 B d^2 (a+b x)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )-2 B d^2 (a+b x)^2 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{(b c-a d)^2}}{2 b g^3 (a+b x)^2} \]
-1/2*((A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2 + (2*B*((b*c - a*d)^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]) + 2*d*(-(b*c) + a*d)*(a + b*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]) - 2*d^2*(a + b*x)^2*Log[a + b*x]*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]) + 2*d^2*(a + b*x)^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])*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*((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]) + 2*B*d^2*(a + b*x)^2*(Log[a + b*x]*(Log[a + b*x] - 2*Log[( b*(c + d*x))/(b*c - a*d)]) - 2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)]) - 2*B*d^2*(a + b*x)^2*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x]) *Log[c + d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])))/(b*c - a*d)^2)/ (b*g^3*(a + b*x)^2)
Time = 0.39 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.76, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {2950, 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.
| \(\displaystyle \int \frac {\left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{(a g+b g x)^3} \, dx\) |
| \(\Big \downarrow \) 2950 |
| \(\displaystyle \frac {\int \frac {(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right ) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a+b x)^3}d\frac {a+b x}{c+d x}}{g^3 (b c-a d)^2}\) |
| \(\Big \downarrow \) 2795 |
| \(\displaystyle \frac {\int \left (\frac {b (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a+b x)^3}-\frac {d (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a+b x)^2}\right )d\frac {a+b x}{c+d x}}{g^3 (b c-a d)^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {b B (c+d x)^2 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{(a+b x)^2}+\frac {4 B d (c+d x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{a+b x}-\frac {b (c+d x)^2 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{2 (a+b x)^2}+\frac {d (c+d x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{a+b x}-\frac {b B^2 (c+d x)^2}{(a+b x)^2}+\frac {8 B^2 d (c+d x)}{a+b x}}{g^3 (b c-a d)^2}\) |
((8*B^2*d*(c + d*x))/(a + b*x) - (b*B^2*(c + d*x)^2)/(a + b*x)^2 + (4*B*d* (c + d*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(a + b*x) - (b*B*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(a + b*x)^2 + (d*(c + d*x )*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2)/(a + b*x) - (b*(c + d*x)^2*( A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2)/(2*(a + b*x)^2))/((b*c - a*d)^2 *g^3)
3.2.34.3.1 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_.), x_Symbol] :> Simp[(b*c - a*d)^( m + 1)*(g/b)^m Subst[Int[x^m*((A + B*Log[e*x^n])^p/(b - d*x)^(m + 2)), x] , x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] & & EqQ[n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && IntegersQ[m, p] && E qQ[b*f - a*g, 0] && (GtQ[p, 0] || LtQ[m, -1])
Time = 1.51 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.80
| method | result | size |
| norman | \(\frac {\frac {\left (A^{2} a d -A^{2} b c +4 A B a d -2 A B b c +8 B^{2} a d -2 B^{2} b c \right ) x}{a g \left (a d -c b \right )}+\frac {B c \left (2 A a d -A b c +4 B a d -B b c \right ) \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {B^{2} a \,d^{2} x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )^{2}}{g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {b B \,d^{2} \left (A +3 B \right ) x^{2} \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {B^{2} c \left (2 a d -c b \right ) \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )^{2}}{2 g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (A^{2} a d -A^{2} b c +6 A B a d -2 A B b c +14 B^{2} a d -2 B^{2} b c \right ) b \,x^{2}}{2 a^{2} g \left (a d -c b \right )}+\frac {2 B d \left (A a d +2 B a d +B b c \right ) x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {b \,d^{2} B^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )^{2}}{2 g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{g^{2} \left (b x +a \right )^{2}}\) | \(490\) |
| parallelrisch | \(-\frac {-2 A B \,x^{2} \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) b^{5} d^{3}-2 B^{2} x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )^{2} a \,b^{4} d^{3}-8 B^{2} x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) a \,b^{4} d^{3}-4 B^{2} x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) b^{5} c \,d^{2}-2 B^{2} \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )^{2} a \,b^{4} c \,d^{2}+2 A B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) b^{5} c^{2} d -8 B^{2} \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) a \,b^{4} c \,d^{2}-2 A^{2} a \,b^{4} c \,d^{2}+6 A B \,a^{2} b^{3} d^{3}+2 A B \,b^{5} c^{2} d -16 B^{2} a \,b^{4} c \,d^{2}-8 A B a \,b^{4} c \,d^{2}+4 A B x a \,b^{4} d^{3}-4 A B x \,b^{5} c \,d^{2}+12 B^{2} x a \,b^{4} d^{3}-12 B^{2} x \,b^{5} c \,d^{2}-B^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )^{2} b^{5} d^{3}-6 B^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) b^{5} d^{3}+B^{2} \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )^{2} b^{5} c^{2} d +2 B^{2} \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) b^{5} c^{2} d +A^{2} a^{2} b^{3} d^{3}+A^{2} b^{5} c^{2} d +14 B^{2} a^{2} b^{3} d^{3}+2 B^{2} b^{5} c^{2} d -4 A B x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) a \,b^{4} d^{3}-4 A B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) a \,b^{4} c \,d^{2}}{2 g^{3} \left (b x +a \right )^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{4} d}\) | \(598\) |
| derivativedivides | \(-\frac {\frac {d^{3} A^{2} \left (-\frac {1}{\left (a d -c b \right )^{2} \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}+\frac {b}{2 \left (a d -c b \right )^{2} \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}\right )}{g^{3}}+\frac {\frac {7 B^{2} d^{3}}{b g \left (d x +c \right )^{2}}-\frac {3 b \,B^{2} d^{3} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {6 B^{2} d^{3}}{g \left (a d -c b \right ) \left (d x +c \right )}-\frac {4 B^{2} d^{3} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{g \left (a d -c b \right ) \left (d x +c \right )}-\frac {B^{2} b \,d^{3} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )^{2}}{2 g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {B^{2} d^{3} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )^{2}}{g \left (a d -c b \right ) \left (d x +c \right )}}{\left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2} g^{2}}+\frac {\frac {3 A B \,d^{3}}{b g \left (d x +c \right )^{2}}-\frac {b A B \,d^{3} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {2 A B \,d^{3}}{g \left (a d -c b \right ) \left (d x +c \right )}-\frac {2 A B \,d^{3} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{g \left (a d -c b \right ) \left (d x +c \right )}}{\left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2} g^{2}}}{d}\) | \(626\) |
| default | \(-\frac {\frac {d^{3} A^{2} \left (-\frac {1}{\left (a d -c b \right )^{2} \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}+\frac {b}{2 \left (a d -c b \right )^{2} \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}\right )}{g^{3}}+\frac {\frac {7 B^{2} d^{3}}{b g \left (d x +c \right )^{2}}-\frac {3 b \,B^{2} d^{3} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {6 B^{2} d^{3}}{g \left (a d -c b \right ) \left (d x +c \right )}-\frac {4 B^{2} d^{3} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{g \left (a d -c b \right ) \left (d x +c \right )}-\frac {B^{2} b \,d^{3} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )^{2}}{2 g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {B^{2} d^{3} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )^{2}}{g \left (a d -c b \right ) \left (d x +c \right )}}{\left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2} g^{2}}+\frac {\frac {3 A B \,d^{3}}{b g \left (d x +c \right )^{2}}-\frac {b A B \,d^{3} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {2 A B \,d^{3}}{g \left (a d -c b \right ) \left (d x +c \right )}-\frac {2 A B \,d^{3} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{g \left (a d -c b \right ) \left (d x +c \right )}}{\left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2} g^{2}}}{d}\) | \(626\) |
| risch | \(-\frac {A^{2}}{2 g^{3} \left (b x +a \right )^{2} b}+\frac {\frac {b \left (7 B^{2} a d -B^{2} b c \right ) x^{2}}{a^{2} g \left (a d -c b \right )}+\frac {\left (4 a d -c b \right ) B^{2} c \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{g \left (a d -c b \right )^{2}}+\frac {B^{2} a \,d^{2} x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )^{2}}{g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {2 \left (4 B^{2} a d -B^{2} b c \right ) x}{a g \left (a d -c b \right )}+\frac {B^{2} c \left (2 a d -c b \right ) \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )^{2}}{2 g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {3 b \,d^{2} B^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {b \,d^{2} B^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )^{2}}{2 g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {2 \left (2 a d +c b \right ) B^{2} d x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{g^{2} \left (b x +a \right )^{2}}+\frac {\frac {\left (3 A B a d -A B b c \right ) b \,x^{2}}{g \,a^{2} \left (a d -c b \right )}+\frac {A B c \left (2 a d -c b \right ) \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {A B \,d^{2} b \,x^{2} \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {2 \left (2 A B a d -A B b c \right ) x}{a g \left (a d -c b \right )}+\frac {2 A B a \,d^{2} x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{g^{2} \left (b x +a \right )^{2}}\) | \(669\) |
| parts | \(-\frac {A^{2}}{2 g^{3} \left (b x +a \right )^{2} b}+\frac {\frac {b \left (7 B^{2} a d -B^{2} b c \right ) x^{2}}{a^{2} g \left (a d -c b \right )}+\frac {\left (4 a d -c b \right ) B^{2} c \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{g \left (a d -c b \right )^{2}}+\frac {B^{2} a \,d^{2} x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )^{2}}{g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {2 \left (4 B^{2} a d -B^{2} b c \right ) x}{a g \left (a d -c b \right )}+\frac {B^{2} c \left (2 a d -c b \right ) \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )^{2}}{2 g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {3 b \,d^{2} B^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {b \,d^{2} B^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )^{2}}{2 g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {2 \left (2 a d +c b \right ) B^{2} d x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{g^{2} \left (b x +a \right )^{2}}+\frac {\frac {\left (3 A B a d -A B b c \right ) b \,x^{2}}{g \,a^{2} \left (a d -c b \right )}+\frac {A B c \left (2 a d -c b \right ) \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {A B \,d^{2} b \,x^{2} \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {2 \left (2 A B a d -A B b c \right ) x}{a g \left (a d -c b \right )}+\frac {2 A B a \,d^{2} x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{g^{2} \left (b x +a \right )^{2}}\) | \(669\) |
((A^2*a*d-A^2*b*c+4*A*B*a*d-2*A*B*b*c+8*B^2*a*d-2*B^2*b*c)/a/g/(a*d-b*c)*x +B*c*(2*A*a*d-A*b*c+4*B*a*d-B*b*c)/g/(a^2*d^2-2*a*b*c*d+b^2*c^2)*ln(e*(b*x +a)^2/(d*x+c)^2)+B^2*a*d^2/g/(a^2*d^2-2*a*b*c*d+b^2*c^2)*x*ln(e*(b*x+a)^2/ (d*x+c)^2)^2+b*B/g*d^2*(A+3*B)/(a^2*d^2-2*a*b*c*d+b^2*c^2)*x^2*ln(e*(b*x+a )^2/(d*x+c)^2)+1/2*B^2*c*(2*a*d-b*c)/g/(a^2*d^2-2*a*b*c*d+b^2*c^2)*ln(e*(b *x+a)^2/(d*x+c)^2)^2+1/2*(A^2*a*d-A^2*b*c+6*A*B*a*d-2*A*B*b*c+14*B^2*a*d-2 *B^2*b*c)/a^2/g*b/(a*d-b*c)*x^2+2*B/g*d*(A*a*d+2*B*a*d+B*b*c)/(a^2*d^2-2*a *b*c*d+b^2*c^2)*x*ln(e*(b*x+a)^2/(d*x+c)^2)+1/2*b*d^2*B^2/g/(a^2*d^2-2*a*b *c*d+b^2*c^2)*x^2*ln(e*(b*x+a)^2/(d*x+c)^2)^2)/g^2/(b*x+a)^2
Time = 0.27 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.51 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^3} \, dx=-\frac {{\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} b^{2} c^{2} - 2 \, {\left (A^{2} + 4 \, A B + 8 \, B^{2}\right )} a b c d + {\left (A^{2} + 6 \, A B + 14 \, B^{2}\right )} a^{2} d^{2} - {\left (B^{2} b^{2} d^{2} x^{2} + 2 \, B^{2} a b d^{2} x - B^{2} b^{2} c^{2} + 2 \, B^{2} a b c d\right )} \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )^{2} - 4 \, {\left ({\left (A B + 3 \, B^{2}\right )} b^{2} c d - {\left (A B + 3 \, B^{2}\right )} a b d^{2}\right )} x - 2 \, {\left ({\left (A B + 3 \, B^{2}\right )} b^{2} d^{2} x^{2} - {\left (A B + B^{2}\right )} b^{2} c^{2} + 2 \, {\left (A B + 2 \, B^{2}\right )} a b c d + 2 \, {\left (B^{2} b^{2} c d + {\left (A B + 2 \, B^{2}\right )} a b d^{2}\right )} x\right )} \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{2 \, {\left ({\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} g^{3} x^{2} + 2 \, {\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} g^{3} x + {\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} g^{3}\right )}} \]
-1/2*((A^2 + 2*A*B + 2*B^2)*b^2*c^2 - 2*(A^2 + 4*A*B + 8*B^2)*a*b*c*d + (A ^2 + 6*A*B + 14*B^2)*a^2*d^2 - (B^2*b^2*d^2*x^2 + 2*B^2*a*b*d^2*x - B^2*b^ 2*c^2 + 2*B^2*a*b*c*d)*log((b^2*e*x^2 + 2*a*b*e*x + a^2*e)/(d^2*x^2 + 2*c* d*x + c^2))^2 - 4*((A*B + 3*B^2)*b^2*c*d - (A*B + 3*B^2)*a*b*d^2)*x - 2*(( A*B + 3*B^2)*b^2*d^2*x^2 - (A*B + B^2)*b^2*c^2 + 2*(A*B + 2*B^2)*a*b*c*d + 2*(B^2*b^2*c*d + (A*B + 2*B^2)*a*b*d^2)*x)*log((b^2*e*x^2 + 2*a*b*e*x + a ^2*e)/(d^2*x^2 + 2*c*d*x + c^2)))/((b^5*c^2 - 2*a*b^4*c*d + a^2*b^3*d^2)*g ^3*x^2 + 2*(a*b^4*c^2 - 2*a^2*b^3*c*d + a^3*b^2*d^2)*g^3*x + (a^2*b^3*c^2 - 2*a^3*b^2*c*d + a^4*b*d^2)*g^3)
Leaf count of result is larger than twice the leaf count of optimal. 879 vs. \(2 (252) = 504\).
Time = 2.08 (sec) , antiderivative size = 879, normalized size of antiderivative = 3.23 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^3} \, dx=- \frac {2 B d^{2} \left (A + 3 B\right ) \log {\left (x + \frac {2 A B a d^{3} + 2 A B b c d^{2} + 6 B^{2} a d^{3} + 6 B^{2} b c d^{2} - \frac {2 B a^{3} d^{5} \left (A + 3 B\right )}{\left (a d - b c\right )^{2}} + \frac {6 B a^{2} b c d^{4} \left (A + 3 B\right )}{\left (a d - b c\right )^{2}} - \frac {6 B a b^{2} c^{2} d^{3} \left (A + 3 B\right )}{\left (a d - b c\right )^{2}} + \frac {2 B b^{3} c^{3} d^{2} \left (A + 3 B\right )}{\left (a d - b c\right )^{2}}}{4 A B b d^{3} + 12 B^{2} b d^{3}} \right )}}{b g^{3} \left (a d - b c\right )^{2}} + \frac {2 B d^{2} \left (A + 3 B\right ) \log {\left (x + \frac {2 A B a d^{3} + 2 A B b c d^{2} + 6 B^{2} a d^{3} + 6 B^{2} b c d^{2} + \frac {2 B a^{3} d^{5} \left (A + 3 B\right )}{\left (a d - b c\right )^{2}} - \frac {6 B a^{2} b c d^{4} \left (A + 3 B\right )}{\left (a d - b c\right )^{2}} + \frac {6 B a b^{2} c^{2} d^{3} \left (A + 3 B\right )}{\left (a d - b c\right )^{2}} - \frac {2 B b^{3} c^{3} d^{2} \left (A + 3 B\right )}{\left (a d - b c\right )^{2}}}{4 A B b d^{3} + 12 B^{2} b d^{3}} \right )}}{b g^{3} \left (a d - b c\right )^{2}} + \frac {\left (2 B^{2} a c d + 2 B^{2} a d^{2} x - B^{2} b c^{2} + B^{2} b d^{2} x^{2}\right ) \log {\left (\frac {e \left (a + b x\right )^{2}}{\left (c + d x\right )^{2}} \right )}^{2}}{2 a^{4} d^{2} g^{3} - 4 a^{3} b c d g^{3} + 4 a^{3} b d^{2} g^{3} x + 2 a^{2} b^{2} c^{2} g^{3} - 8 a^{2} b^{2} c d g^{3} x + 2 a^{2} b^{2} d^{2} g^{3} x^{2} + 4 a b^{3} c^{2} g^{3} x - 4 a b^{3} c d g^{3} x^{2} + 2 b^{4} c^{2} g^{3} x^{2}} + \frac {\left (- A B a d + A B b c - 3 B^{2} a d + B^{2} b c - 2 B^{2} b d x\right ) \log {\left (\frac {e \left (a + b x\right )^{2}}{\left (c + d x\right )^{2}} \right )}}{a^{3} b d g^{3} - a^{2} b^{2} c g^{3} + 2 a^{2} b^{2} d g^{3} x - 2 a b^{3} c g^{3} x + a b^{3} d g^{3} x^{2} - b^{4} c g^{3} x^{2}} + \frac {- A^{2} a d + A^{2} b c - 6 A B a d + 2 A B b c - 14 B^{2} a d + 2 B^{2} b c + x \left (- 4 A B b d - 12 B^{2} b d\right )}{2 a^{3} b d g^{3} - 2 a^{2} b^{2} c g^{3} + x^{2} \cdot \left (2 a b^{3} d g^{3} - 2 b^{4} c g^{3}\right ) + x \left (4 a^{2} b^{2} d g^{3} - 4 a b^{3} c g^{3}\right )} \]
-2*B*d**2*(A + 3*B)*log(x + (2*A*B*a*d**3 + 2*A*B*b*c*d**2 + 6*B**2*a*d**3 + 6*B**2*b*c*d**2 - 2*B*a**3*d**5*(A + 3*B)/(a*d - b*c)**2 + 6*B*a**2*b*c *d**4*(A + 3*B)/(a*d - b*c)**2 - 6*B*a*b**2*c**2*d**3*(A + 3*B)/(a*d - b*c )**2 + 2*B*b**3*c**3*d**2*(A + 3*B)/(a*d - b*c)**2)/(4*A*B*b*d**3 + 12*B** 2*b*d**3))/(b*g**3*(a*d - b*c)**2) + 2*B*d**2*(A + 3*B)*log(x + (2*A*B*a*d **3 + 2*A*B*b*c*d**2 + 6*B**2*a*d**3 + 6*B**2*b*c*d**2 + 2*B*a**3*d**5*(A + 3*B)/(a*d - b*c)**2 - 6*B*a**2*b*c*d**4*(A + 3*B)/(a*d - b*c)**2 + 6*B*a *b**2*c**2*d**3*(A + 3*B)/(a*d - b*c)**2 - 2*B*b**3*c**3*d**2*(A + 3*B)/(a *d - b*c)**2)/(4*A*B*b*d**3 + 12*B**2*b*d**3))/(b*g**3*(a*d - b*c)**2) + ( 2*B**2*a*c*d + 2*B**2*a*d**2*x - B**2*b*c**2 + B**2*b*d**2*x**2)*log(e*(a + b*x)**2/(c + d*x)**2)**2/(2*a**4*d**2*g**3 - 4*a**3*b*c*d*g**3 + 4*a**3* b*d**2*g**3*x + 2*a**2*b**2*c**2*g**3 - 8*a**2*b**2*c*d*g**3*x + 2*a**2*b* *2*d**2*g**3*x**2 + 4*a*b**3*c**2*g**3*x - 4*a*b**3*c*d*g**3*x**2 + 2*b**4 *c**2*g**3*x**2) + (-A*B*a*d + A*B*b*c - 3*B**2*a*d + B**2*b*c - 2*B**2*b* d*x)*log(e*(a + b*x)**2/(c + d*x)**2)/(a**3*b*d*g**3 - a**2*b**2*c*g**3 + 2*a**2*b**2*d*g**3*x - 2*a*b**3*c*g**3*x + a*b**3*d*g**3*x**2 - b**4*c*g** 3*x**2) + (-A**2*a*d + A**2*b*c - 6*A*B*a*d + 2*A*B*b*c - 14*B**2*a*d + 2* B**2*b*c + x*(-4*A*B*b*d - 12*B**2*b*d))/(2*a**3*b*d*g**3 - 2*a**2*b**2*c* g**3 + x**2*(2*a*b**3*d*g**3 - 2*b**4*c*g**3) + x*(4*a**2*b**2*d*g**3 - 4* a*b**3*c*g**3))
Leaf count of result is larger than twice the leaf count of optimal. 1001 vs. \(2 (270) = 540\).
Time = 0.26 (sec) , antiderivative size = 1001, normalized size of antiderivative = 3.68 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^3} \, dx={\left ({\left (\frac {2 \, b d x - b c + 3 \, a d}{{\left (b^{4} c - a b^{3} d\right )} g^{3} x^{2} + 2 \, {\left (a b^{3} c - a^{2} b^{2} d\right )} g^{3} x + {\left (a^{2} b^{2} c - a^{3} b d\right )} g^{3}} + \frac {2 \, d^{2} \log \left (b x + a\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}} - \frac {2 \, d^{2} \log \left (d x + c\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}}\right )} \log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - \frac {b^{2} c^{2} - 8 \, a b c d + 7 \, a^{2} d^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (b x + a\right )^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (d x + c\right )^{2} - 6 \, {\left (b^{2} c d - a b d^{2}\right )} x - 6 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (b x + a\right ) + 2 \, {\left (3 \, b^{2} d^{2} x^{2} + 6 \, a b d^{2} x + 3 \, a^{2} d^{2} - 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{a^{2} b^{3} c^{2} g^{3} - 2 \, a^{3} b^{2} c d g^{3} + a^{4} b d^{2} g^{3} + {\left (b^{5} c^{2} g^{3} - 2 \, a b^{4} c d g^{3} + a^{2} b^{3} d^{2} g^{3}\right )} x^{2} + 2 \, {\left (a b^{4} c^{2} g^{3} - 2 \, a^{2} b^{3} c d g^{3} + a^{3} b^{2} d^{2} g^{3}\right )} x}\right )} B^{2} + A B {\left (\frac {2 \, b d x - b c + 3 \, a d}{{\left (b^{4} c - a b^{3} d\right )} g^{3} x^{2} + 2 \, {\left (a b^{3} c - a^{2} b^{2} d\right )} g^{3} x + {\left (a^{2} b^{2} c - a^{3} b d\right )} g^{3}} - \frac {\log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}} + \frac {2 \, d^{2} \log \left (b x + a\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}} - \frac {2 \, d^{2} \log \left (d x + c\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}}\right )} - \frac {B^{2} \log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )^{2}}{2 \, {\left (b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}\right )}} - \frac {A^{2}}{2 \, {\left (b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}\right )}} \]
(((2*b*d*x - b*c + 3*a*d)/((b^4*c - a*b^3*d)*g^3*x^2 + 2*(a*b^3*c - a^2*b^ 2*d)*g^3*x + (a^2*b^2*c - a^3*b*d)*g^3) + 2*d^2*log(b*x + a)/((b^3*c^2 - 2 *a*b^2*c*d + a^2*b*d^2)*g^3) - 2*d^2*log(d*x + c)/((b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*g^3))*log(b^2*e*x^2/(d^2*x^2 + 2*c*d*x + c^2) + 2*a*b*e*x/(d^ 2*x^2 + 2*c*d*x + c^2) + a^2*e/(d^2*x^2 + 2*c*d*x + c^2)) - (b^2*c^2 - 8*a *b*c*d + 7*a^2*d^2 + 2*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(b*x + a)^ 2 + 2*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(d*x + c)^2 - 6*(b^2*c*d - a*b*d^2)*x - 6*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(b*x + a) + 2*(3*b ^2*d^2*x^2 + 6*a*b*d^2*x + 3*a^2*d^2 - 2*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2* d^2)*log(b*x + a))*log(d*x + c))/(a^2*b^3*c^2*g^3 - 2*a^3*b^2*c*d*g^3 + a^ 4*b*d^2*g^3 + (b^5*c^2*g^3 - 2*a*b^4*c*d*g^3 + a^2*b^3*d^2*g^3)*x^2 + 2*(a *b^4*c^2*g^3 - 2*a^2*b^3*c*d*g^3 + a^3*b^2*d^2*g^3)*x))*B^2 + A*B*((2*b*d* x - b*c + 3*a*d)/((b^4*c - a*b^3*d)*g^3*x^2 + 2*(a*b^3*c - a^2*b^2*d)*g^3* x + (a^2*b^2*c - a^3*b*d)*g^3) - log(b^2*e*x^2/(d^2*x^2 + 2*c*d*x + c^2) + 2*a*b*e*x/(d^2*x^2 + 2*c*d*x + c^2) + a^2*e/(d^2*x^2 + 2*c*d*x + c^2))/(b ^3*g^3*x^2 + 2*a*b^2*g^3*x + a^2*b*g^3) + 2*d^2*log(b*x + a)/((b^3*c^2 - 2 *a*b^2*c*d + a^2*b*d^2)*g^3) - 2*d^2*log(d*x + c)/((b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*g^3)) - 1/2*B^2*log(b^2*e*x^2/(d^2*x^2 + 2*c*d*x + c^2) + 2*a *b*e*x/(d^2*x^2 + 2*c*d*x + c^2) + a^2*e/(d^2*x^2 + 2*c*d*x + c^2))^2/(b^3 *g^3*x^2 + 2*a*b^2*g^3*x + a^2*b*g^3) - 1/2*A^2/(b^3*g^3*x^2 + 2*a*b^2*...
\[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^3} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{2} e}{{\left (d x + c\right )}^{2}}\right ) + A\right )}^{2}}{{\left (b g x + a g\right )}^{3}} \,d x } \]
Time = 2.68 (sec) , antiderivative size = 503, normalized size of antiderivative = 1.85 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^3} \, dx=-\frac {\frac {A^2\,a\,d-A^2\,b\,c+14\,B^2\,a\,d-2\,B^2\,b\,c+6\,A\,B\,a\,d-2\,A\,B\,b\,c}{2\,\left (a\,d-b\,c\right )}+\frac {2\,x\,\left (3\,b\,d\,B^2+A\,b\,d\,B\right )}{a\,d-b\,c}}{a^2\,b\,g^3+2\,a\,b^2\,g^3\,x+b^3\,g^3\,x^2}-{\ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )}^2\,\left (\frac {B^2}{2\,b^2\,g^3\,\left (2\,a\,x+b\,x^2+\frac {a^2}{b}\right )}-\frac {B^2\,d^2}{2\,b\,g^3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )-\frac {\ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )\,\left (\frac {A\,B}{b^2\,d\,g^3}+\frac {2\,B^2\,x\,\left (a\,d-b\,c\right )}{b\,g^3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {B^2\,d^2\,\left (\frac {2\,a^2\,d^2-3\,a\,b\,c\,d+b^2\,c^2}{b\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{b\,d^2}\right )}{b\,g^3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )}{\frac {b\,x^2}{d}+\frac {a^2}{b\,d}+\frac {2\,a\,x}{d}}-\frac {B\,d^2\,\mathrm {atan}\left (\frac {B\,d^2\,\left (2\,b\,d\,x-\frac {b^3\,c^2\,g^3-a^2\,b\,d^2\,g^3}{b\,g^3\,\left (a\,d-b\,c\right )}\right )\,\left (A+3\,B\right )\,2{}\mathrm {i}}{\left (a\,d-b\,c\right )\,\left (6\,B^2\,d^2+2\,A\,B\,d^2\right )}\right )\,\left (A+3\,B\right )\,4{}\mathrm {i}}{b\,g^3\,{\left (a\,d-b\,c\right )}^2} \]
- ((A^2*a*d - A^2*b*c + 14*B^2*a*d - 2*B^2*b*c + 6*A*B*a*d - 2*A*B*b*c)/(2 *(a*d - b*c)) + (2*x*(3*B^2*b*d + A*B*b*d))/(a*d - b*c))/(a^2*b*g^3 + b^3* g^3*x^2 + 2*a*b^2*g^3*x) - log((e*(a + b*x)^2)/(c + d*x)^2)^2*(B^2/(2*b^2* g^3*(2*a*x + b*x^2 + a^2/b)) - (B^2*d^2)/(2*b*g^3*(a^2*d^2 + b^2*c^2 - 2*a *b*c*d))) - (log((e*(a + b*x)^2)/(c + d*x)^2)*((A*B)/(b^2*d*g^3) + (2*B^2* x*(a*d - b*c))/(b*g^3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (B^2*d^2*((2*a^2* d^2 + b^2*c^2 - 3*a*b*c*d)/(b*d^3) + (a*(a*d - b*c))/(b*d^2)))/(b*g^3*(a^2 *d^2 + b^2*c^2 - 2*a*b*c*d))))/((b*x^2)/d + a^2/(b*d) + (2*a*x)/d) - (B*d^ 2*atan((B*d^2*(2*b*d*x - (b^3*c^2*g^3 - a^2*b*d^2*g^3)/(b*g^3*(a*d - b*c)) )*(A + 3*B)*2i)/((a*d - b*c)*(6*B^2*d^2 + 2*A*B*d^2)))*(A + 3*B)*4i)/(b*g^ 3*(a*d - b*c)^2)