Integrand size = 40, antiderivative size = 255 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^3 (c i+d i x)} \, dx=-\frac {B (c+d x)^2 \left (b-\frac {4 d (a+b x)}{c+d x}\right )^2}{4 (b c-a d)^3 g^3 i (a+b x)^2}-\frac {B d^2 \log ^2\left (\frac {a+b x}{c+d x}\right )}{2 (b c-a d)^3 g^3 i}+\frac {2 b d (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 g^3 i (a+b x)}-\frac {b^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^3 g^3 i (a+b x)^2}+\frac {d^2 \log \left (\frac {a+b x}{c+d x}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 g^3 i} \] Output:
-1/4*B*(d*x+c)^2*(b-4*d*(b*x+a)/(d*x+c))^2/(-a*d+b*c)^3/g^3/i/(b*x+a)^2-1/ 2*B*d^2*ln((b*x+a)/(d*x+c))^2/(-a*d+b*c)^3/g^3/i+2*b*d*(d*x+c)*(A+B*ln(e*( b*x+a)/(d*x+c)))/(-a*d+b*c)^3/g^3/i/(b*x+a)-1/2*b^2*(d*x+c)^2*(A+B*ln(e*(b *x+a)/(d*x+c)))/(-a*d+b*c)^3/g^3/i/(b*x+a)^2+d^2*ln((b*x+a)/(d*x+c))*(A+B* ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^3/g^3/i
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.43 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.64 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^3 (c i+d i x)} \, dx=\frac {-2 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+4 d (b c-a d) (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+4 d^2 (a+b x)^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-4 d^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+4 B d (a+b x) (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))-B \left ((b c-a d)^2+2 d (-b c+a d) (a+b x)-2 d^2 (a+b x)^2 \log (a+b x)+2 d^2 (a+b x)^2 \log (c+d x)\right )-2 B d^2 (a+b x)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )+2 B d^2 (a+b x)^2 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )}{4 (b c-a d)^3 g^3 i (a+b x)^2} \] Input:
Integrate[(A + B*Log[(e*(a + b*x))/(c + d*x)])/((a*g + b*g*x)^3*(c*i + d*i *x)),x]
Output:
(-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*((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)]))/(4*(b*c - a*d) ^3*g^3*i*(a + b*x)^2)
Time = 0.47 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.76, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2962, 2772, 27, 2010, 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.
| \(\displaystyle \int \frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{(a g+b g x)^3 (c i+d i x)} \, dx\) |
| \(\Big \downarrow \) 2962 |
| \(\displaystyle \frac {\int \frac {(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^3}d\frac {a+b x}{c+d x}}{g^3 i (b c-a d)^3}\) |
| \(\Big \downarrow \) 2772 |
| \(\displaystyle \frac {-B \int -\frac {(c+d x)^3 \left (b^2-\frac {4 d (a+b x) b}{c+d x}-\frac {2 d^2 (a+b x)^2 \log \left (\frac {a+b x}{c+d x}\right )}{(c+d x)^2}\right )}{2 (a+b x)^3}d\frac {a+b x}{c+d x}-\frac {b^2 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (a+b x)^2}+d^2 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+\frac {2 b d (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}}{g^3 i (b c-a d)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{2} B \int \frac {(c+d x)^3 \left (b^2-\frac {4 d (a+b x) b}{c+d x}-\frac {2 d^2 (a+b x)^2 \log \left (\frac {a+b x}{c+d x}\right )}{(c+d x)^2}\right )}{(a+b x)^3}d\frac {a+b x}{c+d x}-\frac {b^2 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (a+b x)^2}+d^2 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+\frac {2 b d (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}}{g^3 i (b c-a d)^3}\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \frac {\frac {1}{2} B \int \left (\frac {b (c+d x)^3 \left (b-\frac {4 d (a+b x)}{c+d x}\right )}{(a+b x)^3}-\frac {2 d^2 (c+d x) \log \left (\frac {a+b x}{c+d x}\right )}{a+b x}\right )d\frac {a+b x}{c+d x}-\frac {b^2 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (a+b x)^2}+d^2 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+\frac {2 b d (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}}{g^3 i (b c-a d)^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {b^2 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (a+b x)^2}+d^2 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+\frac {2 b d (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}+\frac {1}{2} B \left (-d^2 \log ^2\left (\frac {a+b x}{c+d x}\right )-\frac {(c+d x)^2 \left (b-\frac {4 d (a+b x)}{c+d x}\right )^2}{2 (a+b x)^2}\right )}{g^3 i (b c-a d)^3}\) |
Input:
Int[(A + B*Log[(e*(a + b*x))/(c + d*x)])/((a*g + b*g*x)^3*(c*i + d*i*x)),x ]
Output:
((B*(-1/2*((c + d*x)^2*(b - (4*d*(a + b*x))/(c + d*x))^2)/(a + b*x)^2 - d^ 2*Log[(a + b*x)/(c + d*x)]^2))/2 + (2*b*d*(c + d*x)*(A + B*Log[(e*(a + b*x ))/(c + d*x)]))/(a + b*x) - (b^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(a + b*x)^2) + d^2*Log[(a + b*x)/(c + d*x)]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^3*g^3*i)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
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_.)*((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]
Time = 1.99 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.43
| method | result | size |
| parts | \(\frac {A \left (\frac {d^{2} \ln \left (d x +c \right )}{\left (d a -b c \right )^{3}}+\frac {1}{2 \left (d a -b c \right ) \left (b x +a \right )^{2}}+\frac {d}{\left (d a -b c \right )^{2} \left (b x +a \right )}-\frac {d^{2} \ln \left (b x +a \right )}{\left (d a -b c \right )^{3}}\right )}{g^{3} i}-\frac {B \left (\frac {d^{3} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (d a -b c \right )^{3}}-\frac {2 d^{2} b e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right )}{\left (d a -b c \right )^{3}}+\frac {d \,b^{2} e^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (d a -b c \right )^{3}}\right )}{g^{3} i d}\) | \(365\) |
| risch | \(\frac {A \,d^{2} \ln \left (d x +c \right )}{g^{3} i \left (d a -b c \right )^{3}}+\frac {A}{2 g^{3} i \left (d a -b c \right ) \left (b x +a \right )^{2}}+\frac {A d}{g^{3} i \left (d a -b c \right )^{2} \left (b x +a \right )}-\frac {A \,d^{2} \ln \left (b x +a \right )}{g^{3} i \left (d a -b c \right )^{3}}-\frac {B \,d^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 g^{3} i \left (d a -b c \right )^{3}}-\frac {2 B d b e \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{g^{3} i \left (d a -b c \right )^{3} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )}-\frac {2 B d b e}{g^{3} i \left (d a -b c \right )^{3} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )}+\frac {B \,b^{2} e^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 g^{3} i \left (d a -b c \right )^{3} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )^{2}}+\frac {B \,b^{2} e^{2}}{4 g^{3} i \left (d a -b c \right )^{3} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )^{2}}\) | \(447\) |
| derivativedivides | \(-\frac {e \left (d a -b c \right ) \left (-\frac {d^{2} e A \,b^{2}}{2 i \left (d a -b c \right )^{4} g^{3} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}+\frac {2 d^{3} A b}{i \left (d a -b c \right )^{4} g^{3} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}+\frac {d^{4} A \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{e i \left (d a -b c \right )^{4} g^{3}}+\frac {d^{2} e B \,b^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{i \left (d a -b c \right )^{4} g^{3}}-\frac {2 d^{3} B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right )}{i \left (d a -b c \right )^{4} g^{3}}+\frac {d^{4} B \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 e i \left (d a -b c \right )^{4} g^{3}}\right )}{d^{2}}\) | \(460\) |
| default | \(-\frac {e \left (d a -b c \right ) \left (-\frac {d^{2} e A \,b^{2}}{2 i \left (d a -b c \right )^{4} g^{3} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}+\frac {2 d^{3} A b}{i \left (d a -b c \right )^{4} g^{3} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}+\frac {d^{4} A \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{e i \left (d a -b c \right )^{4} g^{3}}+\frac {d^{2} e B \,b^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{i \left (d a -b c \right )^{4} g^{3}}-\frac {2 d^{3} B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right )}{i \left (d a -b c \right )^{4} g^{3}}+\frac {d^{4} B \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 e i \left (d a -b c \right )^{4} g^{3}}\right )}{d^{2}}\) | \(460\) |
| parallelrisch | \(-\frac {-2 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{4} b^{2} c^{4}+2 A \,x^{2} a^{2} b^{4} c^{4}+B \,x^{2} a^{2} b^{4} c^{4}+2 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{6} c^{2} d^{2}+4 A x \,a^{3} b^{3} c^{4}+4 A \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{6} c^{2} d^{2}+2 B x \,a^{3} b^{3} c^{4}+6 A \,x^{2} a^{4} b^{2} c^{2} d^{2}-8 A \,x^{2} a^{3} b^{3} c^{3} d +7 B \,x^{2} a^{4} b^{2} c^{2} d^{2}-8 B \,x^{2} a^{3} b^{3} c^{3} d +8 A x \,a^{5} b \,c^{2} d^{2}-12 A x \,a^{4} b^{2} c^{3} d +8 B x \,a^{5} b \,c^{2} d^{2}-10 B x \,a^{4} b^{2} c^{3} d +8 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{5} b \,c^{3} d +2 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{4} b^{2} c^{2} d^{2}+4 A \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{4} b^{2} c^{2} d^{2}+6 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{4} b^{2} c^{2} d^{2}+8 A x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{5} b \,c^{2} d^{2}+8 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{5} b \,c^{2} d^{2}+4 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{4} b^{2} c^{3} d +4 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{5} b \,c^{2} d^{2}}{4 i \,g^{3} \left (b x +a \right )^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) a^{4} c^{2}}\) | \(568\) |
| norman | \(\frac {\frac {6 A a \,b^{2} d -2 A \,b^{3} c +7 B a \,b^{2} d -B \,b^{3} c}{4 g i \left (d a -b c \right )^{2} b^{2}}-\frac {\left (2 A \,a^{2} d^{2}+4 B a b c d -B \,b^{2} c^{2}\right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 i g \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {\left (2 A \,b^{2} d +3 B \,b^{2} d \right ) x}{2 i g \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) b}-\frac {B \,a^{2} d^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{2 i g \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {b^{2} \left (2 A \,d^{2}+3 B \,d^{2}\right ) x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 i g \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {b \left (2 A a \,d^{2}+2 B a \,d^{2}+B b c d \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{i g \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {b^{2} B \,d^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{2 i g \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {b B a \,d^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{i g \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}}{\left (b x +a \right )^{2} g^{2}}\) | \(578\) |
Input:
int((A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^3/(d*i*x+c*i),x,method=_RETURN VERBOSE)
Output:
A/g^3/i*(d^2/(a*d-b*c)^3*ln(d*x+c)+1/2/(a*d-b*c)/(b*x+a)^2+d/(a*d-b*c)^2/( b*x+a)-d^2/(a*d-b*c)^3*ln(b*x+a))-B/g^3/i/d*(1/2*d^3/(a*d-b*c)^3*ln(b*e/d+ (a*d-b*c)*e/d/(d*x+c))^2-2*d^2/(a*d-b*c)^3*b*e*(-1/(b*e/d+(a*d-b*c)*e/d/(d *x+c))*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/(b*e/d+(a*d-b*c)*e/d/(d*x+c)))+d/ (a*d-b*c)^3*b^2*e^2*(-1/2/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*ln(b*e/d+(a*d-b* c)*e/d/(d*x+c))-1/4/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2))
Time = 0.09 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.37 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^3 (c i+d i x)} \, dx=-\frac {{\left (2 \, A + B\right )} b^{2} c^{2} - 8 \, {\left (A + B\right )} a b c d + {\left (6 \, A + 7 \, B\right )} a^{2} d^{2} - 2 \, {\left (B b^{2} d^{2} x^{2} + 2 \, B a b d^{2} x + B a^{2} d^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} - 2 \, {\left ({\left (2 \, A + 3 \, B\right )} b^{2} c d - {\left (2 \, A + 3 \, B\right )} a b d^{2}\right )} x - 2 \, {\left ({\left (2 \, A + 3 \, B\right )} b^{2} d^{2} x^{2} - B b^{2} c^{2} + 4 \, B a b c d + 2 \, A a^{2} d^{2} + 2 \, {\left (B b^{2} c d + 2 \, {\left (A + B\right )} a b d^{2}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{4 \, {\left ({\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} g^{3} i x^{2} + 2 \, {\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} g^{3} i x + {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} g^{3} i\right )}} \] Input:
integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^3/(d*i*x+c*i),x, algori thm="fricas")
Output:
-1/4*((2*A + B)*b^2*c^2 - 8*(A + B)*a*b*c*d + (6*A + 7*B)*a^2*d^2 - 2*(B*b ^2*d^2*x^2 + 2*B*a*b*d^2*x + B*a^2*d^2)*log((b*e*x + a*e)/(d*x + c))^2 - 2 *((2*A + 3*B)*b^2*c*d - (2*A + 3*B)*a*b*d^2)*x - 2*((2*A + 3*B)*b^2*d^2*x^ 2 - B*b^2*c^2 + 4*B*a*b*c*d + 2*A*a^2*d^2 + 2*(B*b^2*c*d + 2*(A + B)*a*b*d ^2)*x)*log((b*e*x + a*e)/(d*x + c)))/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3 *c*d^2 - a^3*b^2*d^3)*g^3*i*x^2 + 2*(a*b^4*c^3 - 3*a^2*b^3*c^2*d + 3*a^3*b ^2*c*d^2 - a^4*b*d^3)*g^3*i*x + (a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c *d^2 - a^5*d^3)*g^3*i)
Leaf count of result is larger than twice the leaf count of optimal. 889 vs. \(2 (221) = 442\).
Time = 2.62 (sec) , antiderivative size = 889, normalized size of antiderivative = 3.49 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^3 (c i+d i x)} \, dx =\text {Too large to display} \] Input:
integrate((A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)**3/(d*i*x+c*i),x)
Output:
-B*d**2*log(e*(a + b*x)/(c + d*x))**2/(2*a**3*d**3*g**3*i - 6*a**2*b*c*d** 2*g**3*i + 6*a*b**2*c**2*d*g**3*i - 2*b**3*c**3*g**3*i) + d**2*(2*A + 3*B) *log(x + (2*A*a*d**3 + 2*A*b*c*d**2 + 3*B*a*d**3 + 3*B*b*c*d**2 - a**4*d** 6*(2*A + 3*B)/(a*d - b*c)**3 + 4*a**3*b*c*d**5*(2*A + 3*B)/(a*d - b*c)**3 - 6*a**2*b**2*c**2*d**4*(2*A + 3*B)/(a*d - b*c)**3 + 4*a*b**3*c**3*d**3*(2 *A + 3*B)/(a*d - b*c)**3 - b**4*c**4*d**2*(2*A + 3*B)/(a*d - b*c)**3)/(4*A *b*d**3 + 6*B*b*d**3))/(2*g**3*i*(a*d - b*c)**3) - d**2*(2*A + 3*B)*log(x + (2*A*a*d**3 + 2*A*b*c*d**2 + 3*B*a*d**3 + 3*B*b*c*d**2 + a**4*d**6*(2*A + 3*B)/(a*d - b*c)**3 - 4*a**3*b*c*d**5*(2*A + 3*B)/(a*d - b*c)**3 + 6*a** 2*b**2*c**2*d**4*(2*A + 3*B)/(a*d - b*c)**3 - 4*a*b**3*c**3*d**3*(2*A + 3* B)/(a*d - b*c)**3 + b**4*c**4*d**2*(2*A + 3*B)/(a*d - b*c)**3)/(4*A*b*d**3 + 6*B*b*d**3))/(2*g**3*i*(a*d - b*c)**3) + (3*B*a*d - B*b*c + 2*B*b*d*x)* log(e*(a + b*x)/(c + d*x))/(2*a**4*d**2*g**3*i - 4*a**3*b*c*d*g**3*i + 4*a **3*b*d**2*g**3*i*x + 2*a**2*b**2*c**2*g**3*i - 8*a**2*b**2*c*d*g**3*i*x + 2*a**2*b**2*d**2*g**3*i*x**2 + 4*a*b**3*c**2*g**3*i*x - 4*a*b**3*c*d*g**3 *i*x**2 + 2*b**4*c**2*g**3*i*x**2) + (6*A*a*d - 2*A*b*c + 7*B*a*d - B*b*c + x*(4*A*b*d + 6*B*b*d))/(4*a**4*d**2*g**3*i - 8*a**3*b*c*d*g**3*i + 4*a** 2*b**2*c**2*g**3*i + x**2*(4*a**2*b**2*d**2*g**3*i - 8*a*b**3*c*d*g**3*i + 4*b**4*c**2*g**3*i) + x*(8*a**3*b*d**2*g**3*i - 16*a**2*b**2*c*d*g**3*i + 8*a*b**3*c**2*g**3*i))
Leaf count of result is larger than twice the leaf count of optimal. 885 vs. \(2 (249) = 498\).
Time = 0.09 (sec) , antiderivative size = 885, normalized size of antiderivative = 3.47 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^3 (c i+d i x)} \, dx =\text {Too large to display} \] Input:
integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^3/(d*i*x+c*i),x, algori thm="maxima")
Output:
1/2*B*((2*b*d*x - b*c + 3*a*d)/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*g^3* i*x^2 + 2*(a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2)*g^3*i*x + (a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2)*g^3*i) + 2*d^2*log(b*x + a)/((b^3*c^3 - 3*a*b^2*c^ 2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^3*i) - 2*d^2*log(d*x + c)/((b^3*c^3 - 3*a *b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^3*i))*log(b*e*x/(d*x + c) + a*e/(d *x + c)) + 1/2*A*((2*b*d*x - b*c + 3*a*d)/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^ 2*d^2)*g^3*i*x^2 + 2*(a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2)*g^3*i*x + (a^ 2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2)*g^3*i) + 2*d^2*log(b*x + a)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^3*i) - 2*d^2*log(d*x + c)/((b^ 3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^3*i)) - 1/4*(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))*B/(a^2*b^3*c^3*g^3*i - 3*a^3*b^2*c^2*d *g^3*i + 3*a^4*b*c*d^2*g^3*i - a^5*d^3*g^3*i + (b^5*c^3*g^3*i - 3*a*b^4*c^ 2*d*g^3*i + 3*a^2*b^3*c*d^2*g^3*i - a^3*b^2*d^3*g^3*i)*x^2 + 2*(a*b^4*c^3* g^3*i - 3*a^2*b^3*c^2*d*g^3*i + 3*a^3*b^2*c*d^2*g^3*i - a^4*b*d^3*g^3*i)*x )
Time = 41.45 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.55 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^3 (c i+d i x)} \, dx=-\frac {1}{4} \, {\left (\frac {2 \, {\left (d x + c\right )}^{2} B e^{3} \log \left (\frac {b e x + a e}{d x + c}\right )}{{\left (b e x + a e\right )}^{2} g^{3} i} + \frac {{\left (2 \, A e^{3} + B e^{3}\right )} {\left (d x + c\right )}^{2}}{{\left (b e x + a e\right )}^{2} g^{3} i}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}^{2} \] Input:
integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^3/(d*i*x+c*i),x, algori thm="giac")
Output:
-1/4*(2*(d*x + c)^2*B*e^3*log((b*e*x + a*e)/(d*x + c))/((b*e*x + a*e)^2*g^ 3*i) + (2*A*e^3 + B*e^3)*(d*x + c)^2/((b*e*x + a*e)^2*g^3*i))*(b*c/((b*c*e - a*d*e)*(b*c - a*d)) - a*d/((b*c*e - a*d*e)*(b*c - a*d)))^2
Time = 28.84 (sec) , antiderivative size = 545, normalized size of antiderivative = 2.14 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^3 (c i+d i x)} \, dx=\frac {3\,A\,a\,d}{2\,g^3\,i\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^2}-\frac {B\,d^2\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2}{2\,g^3\,i\,{\left (a\,d-b\,c\right )}^3}-\frac {A\,b\,c}{2\,g^3\,i\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^2}+\frac {7\,B\,a\,d}{4\,g^3\,i\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^2}-\frac {B\,b\,c}{4\,g^3\,i\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^2}+\frac {3\,B\,a^2\,d^2\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{2\,g^3\,i\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^2}+\frac {B\,b^2\,c^2\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{2\,g^3\,i\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^2}+\frac {A\,b\,d\,x}{g^3\,i\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^2}+\frac {3\,B\,b\,d\,x}{2\,g^3\,i\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^2}+\frac {B\,a\,b\,d^2\,x\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^3\,i\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^2}-\frac {B\,b^2\,c\,d\,x\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^3\,i\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^2}-\frac {2\,B\,a\,b\,c\,d\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^3\,i\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^2}+\frac {A\,d^2\,\mathrm {atan}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}\right )\,2{}\mathrm {i}}{g^3\,i\,{\left (a\,d-b\,c\right )}^3}+\frac {B\,d^2\,\mathrm {atan}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}\right )\,3{}\mathrm {i}}{g^3\,i\,{\left (a\,d-b\,c\right )}^3} \] Input:
int((A + B*log((e*(a + b*x))/(c + d*x)))/((a*g + b*g*x)^3*(c*i + d*i*x)),x )
Output:
(A*d^2*atan((a*d*1i + b*c*1i + b*d*x*2i)/(a*d - b*c))*2i)/(g^3*i*(a*d - b* c)^3) + (B*d^2*atan((a*d*1i + b*c*1i + b*d*x*2i)/(a*d - b*c))*3i)/(g^3*i*( a*d - b*c)^3) - (B*d^2*log((e*(a + b*x))/(c + d*x))^2)/(2*g^3*i*(a*d - b*c )^3) + (3*A*a*d)/(2*g^3*i*(a*d - b*c)^2*(a + b*x)^2) - (A*b*c)/(2*g^3*i*(a *d - b*c)^2*(a + b*x)^2) + (7*B*a*d)/(4*g^3*i*(a*d - b*c)^2*(a + b*x)^2) - (B*b*c)/(4*g^3*i*(a*d - b*c)^2*(a + b*x)^2) + (3*B*a^2*d^2*log((e*(a + b* x))/(c + d*x)))/(2*g^3*i*(a*d - b*c)^3*(a + b*x)^2) + (B*b^2*c^2*log((e*(a + b*x))/(c + d*x)))/(2*g^3*i*(a*d - b*c)^3*(a + b*x)^2) + (A*b*d*x)/(g^3* i*(a*d - b*c)^2*(a + b*x)^2) + (3*B*b*d*x)/(2*g^3*i*(a*d - b*c)^2*(a + b*x )^2) + (B*a*b*d^2*x*log((e*(a + b*x))/(c + d*x)))/(g^3*i*(a*d - b*c)^3*(a + b*x)^2) - (B*b^2*c*d*x*log((e*(a + b*x))/(c + d*x)))/(g^3*i*(a*d - b*c)^ 3*(a + b*x)^2) - (2*B*a*b*c*d*log((e*(a + b*x))/(c + d*x)))/(g^3*i*(a*d - b*c)^3*(a + b*x)^2)
Time = 0.20 (sec) , antiderivative size = 679, normalized size of antiderivative = 2.66 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^3 (c i+d i x)} \, dx=\frac {i \left (4 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{2} d^{2} x^{2}+12 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{2} d^{2} x +6 \,\mathrm {log}\left (b x +a \right ) a \,b^{3} d^{2} x^{2}-8 \,\mathrm {log}\left (d x +c \right ) a^{3} b \,d^{2} x -4 \,\mathrm {log}\left (d x +c \right ) a^{2} b^{2} d^{2} x^{2}-12 \,\mathrm {log}\left (d x +c \right ) a^{2} b^{2} d^{2} x -6 \,\mathrm {log}\left (d x +c \right ) a \,b^{3} d^{2} x^{2}+4 \mathrm {log}\left (\frac {b e x +a e}{d x +c}\right )^{2} a^{2} b^{2} d^{2} x +2 \mathrm {log}\left (\frac {b e x +a e}{d x +c}\right )^{2} a \,b^{3} d^{2} x^{2}+8 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a^{2} b^{2} c d -4 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a^{2} b^{2} d^{2} x -2 a \,b^{3} c d \,x^{2}+4 \,\mathrm {log}\left (b x +a \right ) a^{4} d^{2}-4 a^{4} d^{2}+8 \,\mathrm {log}\left (b x +a \right ) a^{3} b \,d^{2} x +4 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a \,b^{3} c d x -3 b^{4} c d \,x^{2}-4 \,\mathrm {log}\left (d x +c \right ) a^{4} d^{2}-4 a^{3} b \,d^{2}-2 a^{2} b^{2} c^{2}-a \,b^{3} c^{2}+6 \,\mathrm {log}\left (b x +a \right ) a^{3} b \,d^{2}-6 \,\mathrm {log}\left (d x +c \right ) a^{3} b \,d^{2}+2 \mathrm {log}\left (\frac {b e x +a e}{d x +c}\right )^{2} a^{3} b \,d^{2}-6 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a^{3} b \,d^{2}-2 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a \,b^{3} c^{2}+6 a^{3} b c d +5 a^{2} b^{2} c d +2 a^{2} b^{2} d^{2} x^{2}+3 a \,b^{3} d^{2} x^{2}\right )}{4 a \,g^{3} \left (a^{3} b^{2} d^{3} x^{2}-3 a^{2} b^{3} c \,d^{2} x^{2}+3 a \,b^{4} c^{2} d \,x^{2}-b^{5} c^{3} x^{2}+2 a^{4} b \,d^{3} x -6 a^{3} b^{2} c \,d^{2} x +6 a^{2} b^{3} c^{2} d x -2 a \,b^{4} c^{3} x +a^{5} d^{3}-3 a^{4} b c \,d^{2}+3 a^{3} b^{2} c^{2} d -a^{2} b^{3} c^{3}\right )} \] Input:
int((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^3/(d*i*x+c*i),x)
Output:
(i*(4*log(a + b*x)*a**4*d**2 + 8*log(a + b*x)*a**3*b*d**2*x + 6*log(a + b* x)*a**3*b*d**2 + 4*log(a + b*x)*a**2*b**2*d**2*x**2 + 12*log(a + b*x)*a**2 *b**2*d**2*x + 6*log(a + b*x)*a*b**3*d**2*x**2 - 4*log(c + d*x)*a**4*d**2 - 8*log(c + d*x)*a**3*b*d**2*x - 6*log(c + d*x)*a**3*b*d**2 - 4*log(c + d* x)*a**2*b**2*d**2*x**2 - 12*log(c + d*x)*a**2*b**2*d**2*x - 6*log(c + d*x) *a*b**3*d**2*x**2 + 2*log((a*e + b*e*x)/(c + d*x))**2*a**3*b*d**2 + 4*log( (a*e + b*e*x)/(c + d*x))**2*a**2*b**2*d**2*x + 2*log((a*e + b*e*x)/(c + d* x))**2*a*b**3*d**2*x**2 - 6*log((a*e + b*e*x)/(c + d*x))*a**3*b*d**2 + 8*l og((a*e + b*e*x)/(c + d*x))*a**2*b**2*c*d - 4*log((a*e + b*e*x)/(c + d*x)) *a**2*b**2*d**2*x - 2*log((a*e + b*e*x)/(c + d*x))*a*b**3*c**2 + 4*log((a* e + b*e*x)/(c + d*x))*a*b**3*c*d*x - 4*a**4*d**2 + 6*a**3*b*c*d - 4*a**3*b *d**2 - 2*a**2*b**2*c**2 + 5*a**2*b**2*c*d + 2*a**2*b**2*d**2*x**2 - a*b** 3*c**2 - 2*a*b**3*c*d*x**2 + 3*a*b**3*d**2*x**2 - 3*b**4*c*d*x**2))/(4*a*g **3*(a**5*d**3 - 3*a**4*b*c*d**2 + 2*a**4*b*d**3*x + 3*a**3*b**2*c**2*d - 6*a**3*b**2*c*d**2*x + a**3*b**2*d**3*x**2 - a**2*b**3*c**3 + 6*a**2*b**3* c**2*d*x - 3*a**2*b**3*c*d**2*x**2 - 2*a*b**4*c**3*x + 3*a*b**4*c**2*d*x** 2 - b**5*c**3*x**2))