Integrand size = 34, antiderivative size = 501 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{(a g+b g x)^5} \, dx=\frac {8 B^2 d^3 (c+d x)}{(b c-a d)^4 g^5 (a+b x)}-\frac {3 b B^2 d^2 (c+d x)^2}{(b c-a d)^4 g^5 (a+b x)^2}+\frac {8 b^2 B^2 d (c+d x)^3}{9 (b c-a d)^4 g^5 (a+b x)^3}-\frac {b^3 B^2 (c+d x)^4}{8 (b c-a d)^4 g^5 (a+b x)^4}-\frac {B^2 d^4 \log ^2\left (\frac {c+d x}{a+b x}\right )}{b (b c-a d)^4 g^5}-\frac {4 B d^3 (c+d x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{(b c-a d)^4 g^5 (a+b x)}+\frac {3 b B d^2 (c+d x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{(b c-a d)^4 g^5 (a+b x)^2}-\frac {4 b^2 B d (c+d x)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 (b c-a d)^4 g^5 (a+b x)^3}+\frac {b^3 B (c+d x)^4 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{4 (b c-a d)^4 g^5 (a+b x)^4}+\frac {B d^4 \log \left (\frac {c+d x}{a+b x}\right ) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{b (b c-a d)^4 g^5}-\frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{4 b g^5 (a+b x)^4} \]
8*B^2*d^3*(d*x+c)/(-a*d+b*c)^4/g^5/(b*x+a)-3*b*B^2*d^2*(d*x+c)^2/(-a*d+b*c )^4/g^5/(b*x+a)^2+8/9*b^2*B^2*d*(d*x+c)^3/(-a*d+b*c)^4/g^5/(b*x+a)^3-1/8*b ^3*B^2*(d*x+c)^4/(-a*d+b*c)^4/g^5/(b*x+a)^4-B^2*d^4*ln((d*x+c)/(b*x+a))^2/ b/(-a*d+b*c)^4/g^5-4*B*d^3*(d*x+c)*(A+B*ln(e*(d*x+c)^2/(b*x+a)^2))/(-a*d+b *c)^4/g^5/(b*x+a)+3*b*B*d^2*(d*x+c)^2*(A+B*ln(e*(d*x+c)^2/(b*x+a)^2))/(-a* d+b*c)^4/g^5/(b*x+a)^2-4/3*b^2*B*d*(d*x+c)^3*(A+B*ln(e*(d*x+c)^2/(b*x+a)^2 ))/(-a*d+b*c)^4/g^5/(b*x+a)^3+1/4*b^3*B*(d*x+c)^4*(A+B*ln(e*(d*x+c)^2/(b*x +a)^2))/(-a*d+b*c)^4/g^5/(b*x+a)^4+B*d^4*ln((d*x+c)/(b*x+a))*(A+B*ln(e*(d* x+c)^2/(b*x+a)^2))/b/(-a*d+b*c)^4/g^5-1/4*(A+B*ln(e*(d*x+c)^2/(b*x+a)^2))^ 2/b/g^5/(b*x+a)^4
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.52 (sec) , antiderivative size = 680, normalized size of antiderivative = 1.36 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{(a g+b g x)^5} \, dx=\frac {-18 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2+\frac {B \left (18 A (b c-a d)^4-9 B (b c-a d)^4+28 B d (b c-a d)^3 (a+b x)+24 A d (-b c+a d)^3 (a+b x)+36 A d^2 (b c-a d)^2 (a+b x)^2-78 B d^2 (b c-a d)^2 (a+b x)^2+300 B d^3 (b c-a d) (a+b x)^3+72 A d^3 (-b c+a d) (a+b x)^3-72 A d^4 (a+b x)^4 \log (a+b x)+300 B d^4 (a+b x)^4 \log (a+b x)-72 B d^4 (a+b x)^4 \log ^2(a+b x)+72 A d^4 (a+b x)^4 \log (c+d x)-300 B d^4 (a+b x)^4 \log (c+d x)+144 B d^4 (a+b x)^4 \log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)-72 B d^4 (a+b x)^4 \log ^2(c+d x)+144 B d^4 (a+b x)^4 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )+18 B (b c-a d)^4 \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+24 B d (-b c+a d)^3 (a+b x) \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+36 B d^2 (b c-a d)^2 (a+b x)^2 \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+72 B d^3 (-b c+a d) (a+b x)^3 \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )-72 B d^4 (a+b x)^4 \log (a+b x) \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+72 B d^4 (a+b x)^4 \log (c+d x) \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+144 B d^4 (a+b x)^4 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )+144 B d^4 (a+b x)^4 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )}{(b c-a d)^4}}{72 b g^5 (a+b x)^4} \]
(-18*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2 + (B*(18*A*(b*c - a*d)^4 - 9*B*(b*c - a*d)^4 + 28*B*d*(b*c - a*d)^3*(a + b*x) + 24*A*d*(-(b*c) + a*d )^3*(a + b*x) + 36*A*d^2*(b*c - a*d)^2*(a + b*x)^2 - 78*B*d^2*(b*c - a*d)^ 2*(a + b*x)^2 + 300*B*d^3*(b*c - a*d)*(a + b*x)^3 + 72*A*d^3*(-(b*c) + a*d )*(a + b*x)^3 - 72*A*d^4*(a + b*x)^4*Log[a + b*x] + 300*B*d^4*(a + b*x)^4* Log[a + b*x] - 72*B*d^4*(a + b*x)^4*Log[a + b*x]^2 + 72*A*d^4*(a + b*x)^4* Log[c + d*x] - 300*B*d^4*(a + b*x)^4*Log[c + d*x] + 144*B*d^4*(a + b*x)^4* Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[c + d*x] - 72*B*d^4*(a + b*x)^4*Log[ c + d*x]^2 + 144*B*d^4*(a + b*x)^4*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a *d)] + 18*B*(b*c - a*d)^4*Log[(e*(c + d*x)^2)/(a + b*x)^2] + 24*B*d*(-(b*c ) + a*d)^3*(a + b*x)*Log[(e*(c + d*x)^2)/(a + b*x)^2] + 36*B*d^2*(b*c - a* d)^2*(a + b*x)^2*Log[(e*(c + d*x)^2)/(a + b*x)^2] + 72*B*d^3*(-(b*c) + a*d )*(a + b*x)^3*Log[(e*(c + d*x)^2)/(a + b*x)^2] - 72*B*d^4*(a + b*x)^4*Log[ a + b*x]*Log[(e*(c + d*x)^2)/(a + b*x)^2] + 72*B*d^4*(a + b*x)^4*Log[c + d *x]*Log[(e*(c + d*x)^2)/(a + b*x)^2] + 144*B*d^4*(a + b*x)^4*PolyLog[2, (d *(a + b*x))/(-(b*c) + a*d)] + 144*B*d^4*(a + b*x)^4*PolyLog[2, (b*(c + d*x ))/(b*c - a*d)]))/(b*c - a*d)^4)/(72*b*g^5*(a + b*x)^4)
Time = 0.47 (sec) , antiderivative size = 388, normalized size of antiderivative = 0.77, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2952, 2756, 2772, 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.
| \(\displaystyle \int \frac {\left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )^2}{(a g+b g x)^5} \, dx\) |
| \(\Big \downarrow \) 2952 |
| \(\displaystyle \frac {\int \left (d-\frac {b (c+d x)}{a+b x}\right )^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2d\frac {c+d x}{a+b x}}{g^5 (b c-a d)^4}\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {\frac {B \int \frac {(a+b x) \left (d-\frac {b (c+d x)}{a+b x}\right )^4 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{c+d x}d\frac {c+d x}{a+b x}}{b}-\frac {\left (d-\frac {b (c+d x)}{a+b x}\right )^4 \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )^2}{4 b}}{g^5 (b c-a d)^4}\) |
| \(\Big \downarrow \) 2772 |
| \(\displaystyle \frac {\frac {B \left (-2 B \int \left (\frac {(c+d x)^3 b^4}{4 (a+b x)^3}-\frac {4 d (c+d x)^2 b^3}{3 (a+b x)^2}+\frac {3 d^2 (c+d x) b^2}{a+b x}-4 d^3 b+\frac {d^4 (a+b x) \log \left (\frac {c+d x}{a+b x}\right )}{c+d x}\right )d\frac {c+d x}{a+b x}+\frac {b^4 (c+d x)^4 \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{4 (a+b x)^4}-\frac {4 b^3 d (c+d x)^3 \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{3 (a+b x)^3}+\frac {3 b^2 d^2 (c+d x)^2 \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{(a+b x)^2}+d^4 \log \left (\frac {c+d x}{a+b x}\right ) \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )-\frac {4 b d^3 (c+d x) \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{a+b x}\right )}{b}-\frac {\left (d-\frac {b (c+d x)}{a+b x}\right )^4 \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )^2}{4 b}}{g^5 (b c-a d)^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {B \left (\frac {b^4 (c+d x)^4 \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{4 (a+b x)^4}-\frac {4 b^3 d (c+d x)^3 \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{3 (a+b x)^3}+\frac {3 b^2 d^2 (c+d x)^2 \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{(a+b x)^2}+d^4 \log \left (\frac {c+d x}{a+b x}\right ) \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )-\frac {4 b d^3 (c+d x) \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{a+b x}-2 B \left (\frac {b^4 (c+d x)^4}{16 (a+b x)^4}-\frac {4 b^3 d (c+d x)^3}{9 (a+b x)^3}+\frac {3 b^2 d^2 (c+d x)^2}{2 (a+b x)^2}+\frac {1}{2} d^4 \log ^2\left (\frac {c+d x}{a+b x}\right )-\frac {4 b d^3 (c+d x)}{a+b x}\right )\right )}{b}-\frac {\left (d-\frac {b (c+d x)}{a+b x}\right )^4 \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )^2}{4 b}}{g^5 (b c-a d)^4}\) |
(-1/4*((d - (b*(c + d*x))/(a + b*x))^4*(A + B*Log[(e*(c + d*x)^2)/(a + b*x )^2])^2)/b + (B*(-2*B*((-4*b*d^3*(c + d*x))/(a + b*x) + (3*b^2*d^2*(c + d* x)^2)/(2*(a + b*x)^2) - (4*b^3*d*(c + d*x)^3)/(9*(a + b*x)^3) + (b^4*(c + d*x)^4)/(16*(a + b*x)^4) + (d^4*Log[(c + d*x)/(a + b*x)]^2)/2) - (4*b*d^3* (c + d*x)*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/(a + b*x) + (3*b^2*d^2 *(c + d*x)^2*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/(a + b*x)^2 - (4*b^ 3*d*(c + d*x)^3*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/(3*(a + b*x)^3) + (b^4*(c + d*x)^4*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/(4*(a + b*x)^ 4) + d^4*Log[(c + d*x)/(a + b*x)]*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]) ))/b)/((b*c - a*d)^4*g^5)
3.3.18.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Simp[b*n*(p/(e*(q + 1))) Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (IntegersQ[2*p, 2*q] && !IGtQ[q, 0]) || (EqQ[p, 2] & & NeQ[q, 1]))
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_ .))^(q_.), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^r)^q, x]}, Simp[(a + b*Log[c*x^n]) u, x] - Simp[b*n Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] && !(EqQ[q , 1] && EqQ[m, -1])
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/d)^m Subst[Int[(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] && EqQ[d*f - c*g, 0] && (GtQ[p, 0] || LtQ[m, -1])
Time = 3.06 (sec) , antiderivative size = 886, normalized size of antiderivative = 1.77
| method | result | size |
| derivativedivides | \(-\frac {\frac {A^{2}}{4 g^{5} \left (b x +a \right )^{4}}+\frac {B^{2}}{8 g^{5} \left (b x +a \right )^{4}}-\frac {B^{2} \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )}{4 g^{5} \left (b x +a \right )^{4}}+\frac {B^{2} \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )^{2}}{4 g^{5} \left (b x +a \right )^{4}}+\frac {25 B^{2} d^{3}}{6 g^{5} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (b x +a \right )}+\frac {7 d \,B^{2}}{18 g^{5} \left (a d -c b \right ) \left (b x +a \right )^{3}}+\frac {13 d^{2} B^{2}}{12 g^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (b x +a \right )^{2}}+\frac {25 d^{4} B^{2} \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )}{12 g^{5} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}-\frac {d^{4} B^{2} \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )^{2}}{4 g^{5} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}-\frac {B^{2} d^{3} \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )}{g^{5} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (b x +a \right )}-\frac {d \,B^{2} \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )}{3 g^{5} \left (a d -c b \right ) \left (b x +a \right )^{3}}-\frac {d^{2} B^{2} \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )}{2 g^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (b x +a \right )^{2}}+\frac {2 A B \left (\frac {\ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )}{4 \left (b x +a \right )^{4}}-\left (\frac {a d}{2}-\frac {c b}{2}\right ) \left (\frac {\frac {\left (a d -c b \right ) \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{4 \left (b x +a \right )^{4}}+\frac {d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{3 \left (b x +a \right )^{3}}+\frac {\left (a d -c b \right ) d^{2}}{2 \left (b x +a \right )^{2}}+\frac {d^{3}}{b x +a}}{\left (a d -c b \right )^{4}}+\frac {d^{4} \ln \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )}{\left (a d -c b \right )^{5}}\right )\right )}{g^{5}}}{b}\) | \(886\) |
| default | \(-\frac {\frac {A^{2}}{4 g^{5} \left (b x +a \right )^{4}}+\frac {B^{2}}{8 g^{5} \left (b x +a \right )^{4}}-\frac {B^{2} \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )}{4 g^{5} \left (b x +a \right )^{4}}+\frac {B^{2} \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )^{2}}{4 g^{5} \left (b x +a \right )^{4}}+\frac {25 B^{2} d^{3}}{6 g^{5} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (b x +a \right )}+\frac {7 d \,B^{2}}{18 g^{5} \left (a d -c b \right ) \left (b x +a \right )^{3}}+\frac {13 d^{2} B^{2}}{12 g^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (b x +a \right )^{2}}+\frac {25 d^{4} B^{2} \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )}{12 g^{5} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}-\frac {d^{4} B^{2} \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )^{2}}{4 g^{5} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}-\frac {B^{2} d^{3} \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )}{g^{5} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (b x +a \right )}-\frac {d \,B^{2} \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )}{3 g^{5} \left (a d -c b \right ) \left (b x +a \right )^{3}}-\frac {d^{2} B^{2} \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )}{2 g^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (b x +a \right )^{2}}+\frac {2 A B \left (\frac {\ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )}{4 \left (b x +a \right )^{4}}-\left (\frac {a d}{2}-\frac {c b}{2}\right ) \left (\frac {\frac {\left (a d -c b \right ) \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{4 \left (b x +a \right )^{4}}+\frac {d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{3 \left (b x +a \right )^{3}}+\frac {\left (a d -c b \right ) d^{2}}{2 \left (b x +a \right )^{2}}+\frac {d^{3}}{b x +a}}{\left (a d -c b \right )^{4}}+\frac {d^{4} \ln \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )}{\left (a d -c b \right )^{5}}\right )\right )}{g^{5}}}{b}\) | \(886\) |
| parts | \(\text {Expression too large to display}\) | \(1611\) |
| norman | \(\text {Expression too large to display}\) | \(1816\) |
| risch | \(\text {Expression too large to display}\) | \(1977\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(2110\) |
-1/b*(1/4/g^5*A^2/(b*x+a)^4+1/8/g^5*B^2/(b*x+a)^4-1/4/g^5*B^2/(b*x+a)^4*ln (e*(a*d/(b*x+a)-b*c/(b*x+a)-d)^2/b^2)+1/4/g^5*B^2/(b*x+a)^4*ln(e*(a*d/(b*x +a)-b*c/(b*x+a)-d)^2/b^2)^2+25/6/g^5*B^2*d^3/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^ 2*c^2*d-b^3*c^3)/(b*x+a)+7/18/g^5*d*B^2/(a*d-b*c)/(b*x+a)^3+13/12/g^5*d^2* B^2/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(b*x+a)^2+25/12/g^5*d^4*B^2/(a^4*d^4-4*a^3 *b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)*ln(e*(a*d/(b*x+a)-b*c/(b *x+a)-d)^2/b^2)-1/4/g^5*d^4*B^2/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4 *a*b^3*c^3*d+b^4*c^4)*ln(e*(a*d/(b*x+a)-b*c/(b*x+a)-d)^2/b^2)^2-1/g^5*B^2* d^3/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/(b*x+a)*ln(e*(a*d/(b*x+a )-b*c/(b*x+a)-d)^2/b^2)-1/3/g^5*d*B^2/(a*d-b*c)/(b*x+a)^3*ln(e*(a*d/(b*x+a )-b*c/(b*x+a)-d)^2/b^2)-1/2/g^5*d^2*B^2/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(b*x+a )^2*ln(e*(a*d/(b*x+a)-b*c/(b*x+a)-d)^2/b^2)+2/g^5*A*B*(1/4/(b*x+a)^4*ln(e* (a*d/(b*x+a)-b*c/(b*x+a)-d)^2/b^2)-(1/2*a*d-1/2*c*b)*(1/(a*d-b*c)^4*(1/4*( a*d-b*c)*(a^2*d^2-2*a*b*c*d+b^2*c^2)/(b*x+a)^4+1/3*d*(a^2*d^2-2*a*b*c*d+b^ 2*c^2)/(b*x+a)^3+1/2*(a*d-b*c)*d^2/(b*x+a)^2+d^3/(b*x+a))+d^4/(a*d-b*c)^5* ln(a*d/(b*x+a)-b*c/(b*x+a)-d))))
Leaf count of result is larger than twice the leaf count of optimal. 1088 vs. \(2 (491) = 982\).
Time = 0.29 (sec) , antiderivative size = 1088, normalized size of antiderivative = 2.17 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{(a g+b g x)^5} \, dx=-\frac {9 \, {\left (2 \, A^{2} - 2 \, A B + B^{2}\right )} b^{4} c^{4} - 8 \, {\left (9 \, A^{2} - 12 \, A B + 8 \, B^{2}\right )} a b^{3} c^{3} d + 108 \, {\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} a^{2} b^{2} c^{2} d^{2} - 72 \, {\left (A^{2} - 4 \, A B + 8 \, B^{2}\right )} a^{3} b c d^{3} + {\left (18 \, A^{2} - 150 \, A B + 415 \, B^{2}\right )} a^{4} d^{4} + 12 \, {\left ({\left (6 \, A B - 25 \, B^{2}\right )} b^{4} c d^{3} - {\left (6 \, A B - 25 \, B^{2}\right )} a b^{3} d^{4}\right )} x^{3} - 6 \, {\left ({\left (6 \, A B - 13 \, B^{2}\right )} b^{4} c^{2} d^{2} - 16 \, {\left (3 \, A B - 11 \, B^{2}\right )} a b^{3} c d^{3} + {\left (42 \, A B - 163 \, B^{2}\right )} a^{2} b^{2} d^{4}\right )} x^{2} - 18 \, {\left (B^{2} b^{4} d^{4} x^{4} + 4 \, B^{2} a b^{3} d^{4} x^{3} + 6 \, B^{2} a^{2} b^{2} d^{4} x^{2} + 4 \, B^{2} a^{3} b d^{4} x - B^{2} b^{4} c^{4} + 4 \, B^{2} a b^{3} c^{3} d - 6 \, B^{2} a^{2} b^{2} c^{2} d^{2} + 4 \, B^{2} a^{3} b c d^{3}\right )} \log \left (\frac {d^{2} e x^{2} + 2 \, c d e x + c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )^{2} + 4 \, {\left ({\left (6 \, A B - 7 \, B^{2}\right )} b^{4} c^{3} d - 12 \, {\left (3 \, A B - 5 \, B^{2}\right )} a b^{3} c^{2} d^{2} + 108 \, {\left (A B - 3 \, B^{2}\right )} a^{2} b^{2} c d^{3} - {\left (78 \, A B - 271 \, B^{2}\right )} a^{3} b d^{4}\right )} x - 6 \, {\left ({\left (6 \, A B - 25 \, B^{2}\right )} b^{4} d^{4} x^{4} - 3 \, {\left (2 \, A B - B^{2}\right )} b^{4} c^{4} + 8 \, {\left (3 \, A B - 2 \, B^{2}\right )} a b^{3} c^{3} d - 36 \, {\left (A B - B^{2}\right )} a^{2} b^{2} c^{2} d^{2} + 24 \, {\left (A B - 2 \, B^{2}\right )} a^{3} b c d^{3} - 4 \, {\left (3 \, B^{2} b^{4} c d^{3} - 2 \, {\left (3 \, A B - 11 \, B^{2}\right )} a b^{3} d^{4}\right )} x^{3} + 6 \, {\left (B^{2} b^{4} c^{2} d^{2} - 8 \, B^{2} a b^{3} c d^{3} + 6 \, {\left (A B - 3 \, B^{2}\right )} a^{2} b^{2} d^{4}\right )} x^{2} - 4 \, {\left (B^{2} b^{4} c^{3} d - 6 \, B^{2} a b^{3} c^{2} d^{2} + 18 \, B^{2} a^{2} b^{2} c d^{3} - 6 \, {\left (A B - 2 \, B^{2}\right )} a^{3} b d^{4}\right )} x\right )} \log \left (\frac {d^{2} e x^{2} + 2 \, c d e x + c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )}{72 \, {\left ({\left (b^{9} c^{4} - 4 \, a b^{8} c^{3} d + 6 \, a^{2} b^{7} c^{2} d^{2} - 4 \, a^{3} b^{6} c d^{3} + a^{4} b^{5} d^{4}\right )} g^{5} x^{4} + 4 \, {\left (a b^{8} c^{4} - 4 \, a^{2} b^{7} c^{3} d + 6 \, a^{3} b^{6} c^{2} d^{2} - 4 \, a^{4} b^{5} c d^{3} + a^{5} b^{4} d^{4}\right )} g^{5} x^{3} + 6 \, {\left (a^{2} b^{7} c^{4} - 4 \, a^{3} b^{6} c^{3} d + 6 \, a^{4} b^{5} c^{2} d^{2} - 4 \, a^{5} b^{4} c d^{3} + a^{6} b^{3} d^{4}\right )} g^{5} x^{2} + 4 \, {\left (a^{3} b^{6} c^{4} - 4 \, a^{4} b^{5} c^{3} d + 6 \, a^{5} b^{4} c^{2} d^{2} - 4 \, a^{6} b^{3} c d^{3} + a^{7} b^{2} d^{4}\right )} g^{5} x + {\left (a^{4} b^{5} c^{4} - 4 \, a^{5} b^{4} c^{3} d + 6 \, a^{6} b^{3} c^{2} d^{2} - 4 \, a^{7} b^{2} c d^{3} + a^{8} b d^{4}\right )} g^{5}\right )}} \]
-1/72*(9*(2*A^2 - 2*A*B + B^2)*b^4*c^4 - 8*(9*A^2 - 12*A*B + 8*B^2)*a*b^3* c^3*d + 108*(A^2 - 2*A*B + 2*B^2)*a^2*b^2*c^2*d^2 - 72*(A^2 - 4*A*B + 8*B^ 2)*a^3*b*c*d^3 + (18*A^2 - 150*A*B + 415*B^2)*a^4*d^4 + 12*((6*A*B - 25*B^ 2)*b^4*c*d^3 - (6*A*B - 25*B^2)*a*b^3*d^4)*x^3 - 6*((6*A*B - 13*B^2)*b^4*c ^2*d^2 - 16*(3*A*B - 11*B^2)*a*b^3*c*d^3 + (42*A*B - 163*B^2)*a^2*b^2*d^4) *x^2 - 18*(B^2*b^4*d^4*x^4 + 4*B^2*a*b^3*d^4*x^3 + 6*B^2*a^2*b^2*d^4*x^2 + 4*B^2*a^3*b*d^4*x - B^2*b^4*c^4 + 4*B^2*a*b^3*c^3*d - 6*B^2*a^2*b^2*c^2*d ^2 + 4*B^2*a^3*b*c*d^3)*log((d^2*e*x^2 + 2*c*d*e*x + c^2*e)/(b^2*x^2 + 2*a *b*x + a^2))^2 + 4*((6*A*B - 7*B^2)*b^4*c^3*d - 12*(3*A*B - 5*B^2)*a*b^3*c ^2*d^2 + 108*(A*B - 3*B^2)*a^2*b^2*c*d^3 - (78*A*B - 271*B^2)*a^3*b*d^4)*x - 6*((6*A*B - 25*B^2)*b^4*d^4*x^4 - 3*(2*A*B - B^2)*b^4*c^4 + 8*(3*A*B - 2*B^2)*a*b^3*c^3*d - 36*(A*B - B^2)*a^2*b^2*c^2*d^2 + 24*(A*B - 2*B^2)*a^3 *b*c*d^3 - 4*(3*B^2*b^4*c*d^3 - 2*(3*A*B - 11*B^2)*a*b^3*d^4)*x^3 + 6*(B^2 *b^4*c^2*d^2 - 8*B^2*a*b^3*c*d^3 + 6*(A*B - 3*B^2)*a^2*b^2*d^4)*x^2 - 4*(B ^2*b^4*c^3*d - 6*B^2*a*b^3*c^2*d^2 + 18*B^2*a^2*b^2*c*d^3 - 6*(A*B - 2*B^2 )*a^3*b*d^4)*x)*log((d^2*e*x^2 + 2*c*d*e*x + c^2*e)/(b^2*x^2 + 2*a*b*x + a ^2)))/((b^9*c^4 - 4*a*b^8*c^3*d + 6*a^2*b^7*c^2*d^2 - 4*a^3*b^6*c*d^3 + a^ 4*b^5*d^4)*g^5*x^4 + 4*(a*b^8*c^4 - 4*a^2*b^7*c^3*d + 6*a^3*b^6*c^2*d^2 - 4*a^4*b^5*c*d^3 + a^5*b^4*d^4)*g^5*x^3 + 6*(a^2*b^7*c^4 - 4*a^3*b^6*c^3*d + 6*a^4*b^5*c^2*d^2 - 4*a^5*b^4*c*d^3 + a^6*b^3*d^4)*g^5*x^2 + 4*(a^3*b...
Timed out. \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 2278 vs. \(2 (491) = 982\).
Time = 0.40 (sec) , antiderivative size = 2278, normalized size of antiderivative = 4.55 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \]
-1/72*(6*((12*b^3*d^3*x^3 - 3*b^3*c^3 + 13*a*b^2*c^2*d - 23*a^2*b*c*d^2 + 25*a^3*d^3 - 6*(b^3*c*d^2 - 7*a*b^2*d^3)*x^2 + 4*(b^3*c^2*d - 5*a*b^2*c*d^ 2 + 13*a^2*b*d^3)*x)/((b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a^3*b^5 *d^3)*g^5*x^4 + 4*(a*b^7*c^3 - 3*a^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a^4*b^4 *d^3)*g^5*x^3 + 6*(a^2*b^6*c^3 - 3*a^3*b^5*c^2*d + 3*a^4*b^4*c*d^2 - a^5*b ^3*d^3)*g^5*x^2 + 4*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d^2 - a^6 *b^2*d^3)*g^5*x + (a^4*b^4*c^3 - 3*a^5*b^3*c^2*d + 3*a^6*b^2*c*d^2 - a^7*b *d^3)*g^5) + 12*d^4*log(b*x + a)/((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2 *d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5) - 12*d^4*log(d*x + c)/((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5))*lo g(d^2*e*x^2/(b^2*x^2 + 2*a*b*x + a^2) + 2*c*d*e*x/(b^2*x^2 + 2*a*b*x + a^2 ) + c^2*e/(b^2*x^2 + 2*a*b*x + a^2)) + (9*b^4*c^4 - 64*a*b^3*c^3*d + 216*a ^2*b^2*c^2*d^2 - 576*a^3*b*c*d^3 + 415*a^4*d^4 - 300*(b^4*c*d^3 - a*b^3*d^ 4)*x^3 + 6*(13*b^4*c^2*d^2 - 176*a*b^3*c*d^3 + 163*a^2*b^2*d^4)*x^2 + 72*( b^4*d^4*x^4 + 4*a*b^3*d^4*x^3 + 6*a^2*b^2*d^4*x^2 + 4*a^3*b*d^4*x + a^4*d^ 4)*log(b*x + a)^2 + 72*(b^4*d^4*x^4 + 4*a*b^3*d^4*x^3 + 6*a^2*b^2*d^4*x^2 + 4*a^3*b*d^4*x + a^4*d^4)*log(d*x + c)^2 - 4*(7*b^4*c^3*d - 60*a*b^3*c^2* d^2 + 324*a^2*b^2*c*d^3 - 271*a^3*b*d^4)*x - 300*(b^4*d^4*x^4 + 4*a*b^3*d^ 4*x^3 + 6*a^2*b^2*d^4*x^2 + 4*a^3*b*d^4*x + a^4*d^4)*log(b*x + a) + 12*(25 *b^4*d^4*x^4 + 100*a*b^3*d^4*x^3 + 150*a^2*b^2*d^4*x^2 + 100*a^3*b*d^4*...
Time = 1.10 (sec) , antiderivative size = 883, normalized size of antiderivative = 1.76 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{(a g+b g x)^5} \, dx=\frac {1}{4} \, {\left (\frac {B^{2} d^{4}}{b^{5} c^{4} g^{5} - 4 \, a b^{4} c^{3} d g^{5} + 6 \, a^{2} b^{3} c^{2} d^{2} g^{5} - 4 \, a^{3} b^{2} c d^{3} g^{5} + a^{4} b d^{4} g^{5}} - \frac {B^{2}}{{\left (b g x + a g\right )}^{4} b g}\right )} \log \left (\frac {\frac {b^{2} c^{2} e g^{2}}{{\left (b g x + a g\right )}^{2}} - \frac {2 \, a b c d e g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac {a^{2} d^{2} e g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac {2 \, b c d e g}{b g x + a g} - \frac {2 \, a d^{2} e g}{b g x + a g} + d^{2} e}{b^{2}}\right )^{2} - \frac {1}{12} \, {\left (\frac {12 \, B^{2} d^{3}}{{\left (b^{3} c^{3} g^{3} - 3 \, a b^{2} c^{2} d g^{3} + 3 \, a^{2} b c d^{2} g^{3} - a^{3} d^{3} g^{3}\right )} {\left (b g x + a g\right )} b g} - \frac {6 \, B^{2} d^{2}}{{\left (b^{2} c^{2} g - 2 \, a b c d g + a^{2} d^{2} g\right )} {\left (b g x + a g\right )}^{2} b g^{2}} + \frac {4 \, B^{2} d}{{\left (b g x + a g\right )}^{3} {\left (b c - a d\right )} b g^{2}} + \frac {3 \, {\left (2 \, A B b^{3} g^{3} - B^{2} b^{3} g^{3}\right )}}{{\left (b g x + a g\right )}^{4} b^{4} g^{4}}\right )} \log \left (\frac {\frac {b^{2} c^{2} e g^{2}}{{\left (b g x + a g\right )}^{2}} - \frac {2 \, a b c d e g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac {a^{2} d^{2} e g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac {2 \, b c d e g}{b g x + a g} - \frac {2 \, a d^{2} e g}{b g x + a g} + d^{2} e}{b^{2}}\right ) + \frac {{\left (6 \, A B d^{4} - 25 \, B^{2} d^{4}\right )} \log \left (-\frac {b c g}{b g x + a g} + \frac {a d g}{b g x + a g} - d\right )}{6 \, {\left (b^{5} c^{4} g^{5} - 4 \, a b^{4} c^{3} d g^{5} + 6 \, a^{2} b^{3} c^{2} d^{2} g^{5} - 4 \, a^{3} b^{2} c d^{3} g^{5} + a^{4} b d^{4} g^{5}\right )}} - \frac {6 \, A B d^{3} - 25 \, B^{2} d^{3}}{6 \, {\left (b^{3} c^{3} g^{3} - 3 \, a b^{2} c^{2} d g^{3} + 3 \, a^{2} b c d^{2} g^{3} - a^{3} d^{3} g^{3}\right )} {\left (b g x + a g\right )} b g} + \frac {6 \, A B b d^{2} - 13 \, B^{2} b d^{2}}{12 \, {\left (b^{2} c^{2} g - 2 \, a b c d g + a^{2} d^{2} g\right )} {\left (b g x + a g\right )}^{2} b^{2} g^{2}} - \frac {6 \, A B b^{2} d g - 7 \, B^{2} b^{2} d g}{18 \, {\left (b g x + a g\right )}^{3} {\left (b c - a d\right )} b^{3} g^{3}} - \frac {2 \, A^{2} b^{3} g^{3} - 2 \, A B b^{3} g^{3} + B^{2} b^{3} g^{3}}{8 \, {\left (b g x + a g\right )}^{4} b^{4} g^{4}} \]
1/4*(B^2*d^4/(b^5*c^4*g^5 - 4*a*b^4*c^3*d*g^5 + 6*a^2*b^3*c^2*d^2*g^5 - 4* a^3*b^2*c*d^3*g^5 + a^4*b*d^4*g^5) - B^2/((b*g*x + a*g)^4*b*g))*log((b^2*c ^2*e*g^2/(b*g*x + a*g)^2 - 2*a*b*c*d*e*g^2/(b*g*x + a*g)^2 + a^2*d^2*e*g^2 /(b*g*x + a*g)^2 + 2*b*c*d*e*g/(b*g*x + a*g) - 2*a*d^2*e*g/(b*g*x + a*g) + d^2*e)/b^2)^2 - 1/12*(12*B^2*d^3/((b^3*c^3*g^3 - 3*a*b^2*c^2*d*g^3 + 3*a^ 2*b*c*d^2*g^3 - a^3*d^3*g^3)*(b*g*x + a*g)*b*g) - 6*B^2*d^2/((b^2*c^2*g - 2*a*b*c*d*g + a^2*d^2*g)*(b*g*x + a*g)^2*b*g^2) + 4*B^2*d/((b*g*x + a*g)^3 *(b*c - a*d)*b*g^2) + 3*(2*A*B*b^3*g^3 - B^2*b^3*g^3)/((b*g*x + a*g)^4*b^4 *g^4))*log((b^2*c^2*e*g^2/(b*g*x + a*g)^2 - 2*a*b*c*d*e*g^2/(b*g*x + a*g)^ 2 + a^2*d^2*e*g^2/(b*g*x + a*g)^2 + 2*b*c*d*e*g/(b*g*x + a*g) - 2*a*d^2*e* g/(b*g*x + a*g) + d^2*e)/b^2) + 1/6*(6*A*B*d^4 - 25*B^2*d^4)*log(-b*c*g/(b *g*x + a*g) + a*d*g/(b*g*x + a*g) - d)/(b^5*c^4*g^5 - 4*a*b^4*c^3*d*g^5 + 6*a^2*b^3*c^2*d^2*g^5 - 4*a^3*b^2*c*d^3*g^5 + a^4*b*d^4*g^5) - 1/6*(6*A*B* d^3 - 25*B^2*d^3)/((b^3*c^3*g^3 - 3*a*b^2*c^2*d*g^3 + 3*a^2*b*c*d^2*g^3 - a^3*d^3*g^3)*(b*g*x + a*g)*b*g) + 1/12*(6*A*B*b*d^2 - 13*B^2*b*d^2)/((b^2* c^2*g - 2*a*b*c*d*g + a^2*d^2*g)*(b*g*x + a*g)^2*b^2*g^2) - 1/18*(6*A*B*b^ 2*d*g - 7*B^2*b^2*d*g)/((b*g*x + a*g)^3*(b*c - a*d)*b^3*g^3) - 1/8*(2*A^2* b^3*g^3 - 2*A*B*b^3*g^3 + B^2*b^3*g^3)/((b*g*x + a*g)^4*b^4*g^4)
Time = 8.91 (sec) , antiderivative size = 1882, normalized size of antiderivative = 3.76 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \]
(log((e*(c + d*x)^2)/(a + b*x)^2)*((B^2*d^4*(a*(a*((4*a^2*d^2 + b^2*c^2 - 5*a*b*c*d)/(6*b*d^3) + (a*(a*d - b*c))/(2*b*d^2)) + (6*a^3*d^3 - b^3*c^3 + 5*a*b^2*c^2*d - 10*a^2*b*c*d^2)/(6*b*d^4)) + (4*a^4*d^4 + b^4*c^4 + 10*a^ 2*b^2*c^2*d^2 - 5*a*b^3*c^3*d - 10*a^3*b*c*d^3)/(2*b*d^5)))/(2*b*g^5*(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)) - (A*B )/(2*b^2*d*g^5) + (B^2*d^4*x^2*(b*(b*((4*a^2*d^2 + b^2*c^2 - 5*a*b*c*d)/(6 *b*d^3) + (a*(a*d - b*c))/(2*b*d^2)) + (4*a^2*d^2 + b^2*c^2 - 5*a*b*c*d)/( 3*d^3) + (a*(a*d - b*c))/d^2) - a*((b^2*c - a*b*d)/(2*d^2) - (b*(a*d - b*c ))/d^2) + (b^3*c^2 + 4*a^2*b*d^2 - 5*a*b^2*c*d)/(2*d^3)))/(2*b*g^5*(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 ^4*x^3*(b*((b^2*c - a*b*d)/(2*d^2) - (b*(a*d - b*c))/d^2) + (b^3*c - a*b^2 *d)/(2*d^2)))/(2*b*g^5*(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^4*x*(b*(a*((4*a^2*d^2 + b^2*c^2 - 5*a*b*c*d )/(6*b*d^3) + (a*(a*d - b*c))/(2*b*d^2)) + (6*a^3*d^3 - b^3*c^3 + 5*a*b^2* c^2*d - 10*a^2*b*c*d^2)/(6*b*d^4)) + a*(b*((4*a^2*d^2 + b^2*c^2 - 5*a*b*c* d)/(6*b*d^3) + (a*(a*d - b*c))/(2*b*d^2)) + (4*a^2*d^2 + b^2*c^2 - 5*a*b*c *d)/(3*d^3) + (a*(a*d - b*c))/d^2) + (6*a^3*d^3 - b^3*c^3 + 5*a*b^2*c^2*d - 10*a^2*b*c*d^2)/(2*d^4)))/(2*b*g^5*(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))))/((4*a^3*x)/d + a^4/(b*d) + (b^3*x^4) /d + (6*a^2*b*x^2)/d + (4*a*b^2*x^3)/d) - log((e*(c + d*x)^2)/(a + b*x)...