Integrand size = 40, antiderivative size = 281 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^7} \, dx=-\frac {B d^2 i^3 (c+d x)^4}{16 (b c-a d)^3 g^7 (a+b x)^4}+\frac {2 b B d i^3 (c+d x)^5}{25 (b c-a d)^3 g^7 (a+b x)^5}-\frac {b^2 B i^3 (c+d x)^6}{36 (b c-a d)^3 g^7 (a+b x)^6}-\frac {d^2 i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 (b c-a d)^3 g^7 (a+b x)^4}+\frac {2 b d i^3 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 (b c-a d)^3 g^7 (a+b x)^5}-\frac {b^2 i^3 (c+d x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 (b c-a d)^3 g^7 (a+b x)^6} \] Output:
-1/16*B*d^2*i^3*(d*x+c)^4/(-a*d+b*c)^3/g^7/(b*x+a)^4+2/25*b*B*d*i^3*(d*x+c )^5/(-a*d+b*c)^3/g^7/(b*x+a)^5-1/36*b^2*B*i^3*(d*x+c)^6/(-a*d+b*c)^3/g^7/( b*x+a)^6-1/4*d^2*i^3*(d*x+c)^4*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^3/g^ 7/(b*x+a)^4+2/5*b*d*i^3*(d*x+c)^5*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^3 /g^7/(b*x+a)^5-1/6*b^2*i^3*(d*x+c)^6*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c )^3/g^7/(b*x+a)^6
Leaf count is larger than twice the leaf count of optimal. \(642\) vs. \(2(281)=562\).
Time = 1.09 (sec) , antiderivative size = 642, normalized size of antiderivative = 2.28 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^7} \, dx=\frac {i^3 \left (-100 B (b c-a d)^6+432 a B d (-b c+a d)^5-432 b B d (b c-a d)^5 x+540 a B d^2 (b c-a d)^4 (a+b x)+120 B d (b c-a d)^5 (a+b x)+540 b B d^2 (b c-a d)^4 x (a+b x)-825 B d^2 (b c-a d)^4 (a+b x)^2+720 a B d^3 (-b c+a d)^3 (a+b x)^2-720 b B d^3 (b c-a d)^3 x (a+b x)^2+1080 a B d^4 (b c-a d)^2 (a+b x)^3+700 B d^3 (b c-a d)^3 (a+b x)^3+1080 b B d^4 (b c-a d)^2 x (a+b x)^3-1050 B d^4 (b c-a d)^2 (a+b x)^4+2160 a B d^5 (-b c+a d) (a+b x)^4-2160 b B d^5 (b c-a d) x (a+b x)^4+2100 B d^5 (b c-a d) (a+b x)^5-2160 a B d^6 (a+b x)^5 \log (a+b x)-2160 b B d^6 x (a+b x)^5 \log (a+b x)+2100 B d^6 (a+b x)^6 \log (a+b x)-600 (b c-a d)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+2160 d (-b c+a d)^5 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-2700 d^2 (b c-a d)^4 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+1200 d^3 (-b c+a d)^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+2160 a B d^6 (a+b x)^5 \log (c+d x)+2160 b B d^6 x (a+b x)^5 \log (c+d x)-2100 B d^6 (a+b x)^6 \log (c+d x)\right )}{3600 b^4 (b c-a d)^3 g^7 (a+b x)^6} \] Input:
Integrate[((c*i + d*i*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b* g*x)^7,x]
Output:
(i^3*(-100*B*(b*c - a*d)^6 + 432*a*B*d*(-(b*c) + a*d)^5 - 432*b*B*d*(b*c - a*d)^5*x + 540*a*B*d^2*(b*c - a*d)^4*(a + b*x) + 120*B*d*(b*c - a*d)^5*(a + b*x) + 540*b*B*d^2*(b*c - a*d)^4*x*(a + b*x) - 825*B*d^2*(b*c - a*d)^4* (a + b*x)^2 + 720*a*B*d^3*(-(b*c) + a*d)^3*(a + b*x)^2 - 720*b*B*d^3*(b*c - a*d)^3*x*(a + b*x)^2 + 1080*a*B*d^4*(b*c - a*d)^2*(a + b*x)^3 + 700*B*d^ 3*(b*c - a*d)^3*(a + b*x)^3 + 1080*b*B*d^4*(b*c - a*d)^2*x*(a + b*x)^3 - 1 050*B*d^4*(b*c - a*d)^2*(a + b*x)^4 + 2160*a*B*d^5*(-(b*c) + a*d)*(a + b*x )^4 - 2160*b*B*d^5*(b*c - a*d)*x*(a + b*x)^4 + 2100*B*d^5*(b*c - a*d)*(a + b*x)^5 - 2160*a*B*d^6*(a + b*x)^5*Log[a + b*x] - 2160*b*B*d^6*x*(a + b*x) ^5*Log[a + b*x] + 2100*B*d^6*(a + b*x)^6*Log[a + b*x] - 600*(b*c - a*d)^6* (A + B*Log[(e*(a + b*x))/(c + d*x)]) + 2160*d*(-(b*c) + a*d)^5*(a + b*x)*( A + B*Log[(e*(a + b*x))/(c + d*x)]) - 2700*d^2*(b*c - a*d)^4*(a + b*x)^2*( A + B*Log[(e*(a + b*x))/(c + d*x)]) + 1200*d^3*(-(b*c) + a*d)^3*(a + b*x)^ 3*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 2160*a*B*d^6*(a + b*x)^5*Log[c + d*x] + 2160*b*B*d^6*x*(a + b*x)^5*Log[c + d*x] - 2100*B*d^6*(a + b*x)^6*Lo g[c + d*x]))/(3600*b^4*(b*c - a*d)^3*g^7*(a + b*x)^6)
Time = 0.43 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.73, 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, 1140, 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.
| \(\displaystyle \int \frac {(c i+d i x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{(a g+b g x)^7} \, dx\) |
| \(\Big \downarrow \) 2962 |
| \(\displaystyle \frac {i^3 \int \frac {(c+d x)^7 \left (b-\frac {d (a+b x)}{c+d x}\right )^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^7}d\frac {a+b x}{c+d x}}{g^7 (b c-a d)^3}\) |
| \(\Big \downarrow \) 2772 |
| \(\displaystyle \frac {i^3 \left (-B \int -\frac {(c+d x)^7 \left (10 b^2-\frac {24 d (a+b x) b}{c+d x}+\frac {15 d^2 (a+b x)^2}{(c+d x)^2}\right )}{60 (a+b x)^7}d\frac {a+b x}{c+d x}-\frac {b^2 (c+d x)^6 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{6 (a+b x)^6}-\frac {d^2 (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 (a+b x)^4}+\frac {2 b d (c+d x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 (a+b x)^5}\right )}{g^7 (b c-a d)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {i^3 \left (\frac {1}{60} B \int \frac {(c+d x)^7 \left (10 b^2-\frac {24 d (a+b x) b}{c+d x}+\frac {15 d^2 (a+b x)^2}{(c+d x)^2}\right )}{(a+b x)^7}d\frac {a+b x}{c+d x}-\frac {b^2 (c+d x)^6 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{6 (a+b x)^6}-\frac {d^2 (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 (a+b x)^4}+\frac {2 b d (c+d x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 (a+b x)^5}\right )}{g^7 (b c-a d)^3}\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \frac {i^3 \left (\frac {1}{60} B \int \left (\frac {10 b^2 (c+d x)^7}{(a+b x)^7}-\frac {24 b d (c+d x)^6}{(a+b x)^6}+\frac {15 d^2 (c+d x)^5}{(a+b x)^5}\right )d\frac {a+b x}{c+d x}-\frac {b^2 (c+d x)^6 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{6 (a+b x)^6}-\frac {d^2 (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 (a+b x)^4}+\frac {2 b d (c+d x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 (a+b x)^5}\right )}{g^7 (b c-a d)^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {i^3 \left (-\frac {b^2 (c+d x)^6 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{6 (a+b x)^6}-\frac {d^2 (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 (a+b x)^4}+\frac {2 b d (c+d x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 (a+b x)^5}+\frac {1}{60} B \left (-\frac {5 b^2 (c+d x)^6}{3 (a+b x)^6}-\frac {15 d^2 (c+d x)^4}{4 (a+b x)^4}+\frac {24 b d (c+d x)^5}{5 (a+b x)^5}\right )\right )}{g^7 (b c-a d)^3}\) |
Input:
Int[((c*i + d*i*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b*g*x)^7 ,x]
Output:
(i^3*((B*((-15*d^2*(c + d*x)^4)/(4*(a + b*x)^4) + (24*b*d*(c + d*x)^5)/(5* (a + b*x)^5) - (5*b^2*(c + d*x)^6)/(3*(a + b*x)^6)))/60 - (d^2*(c + d*x)^4 *(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*(a + b*x)^4) + (2*b*d*(c + d*x)^ 5*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(5*(a + b*x)^5) - (b^2*(c + d*x)^6 *(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(6*(a + b*x)^6)))/((b*c - a*d)^3*g^ 7)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
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 = 2.95 (sec) , antiderivative size = 481, normalized size of antiderivative = 1.71
| method | result | size |
| parts | \(\frac {i^{3} A \left (\frac {3 d^{2} \left (d a -b c \right )}{4 b^{4} \left (b x +a \right )^{4}}-\frac {-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}}{6 b^{4} \left (b x +a \right )^{6}}-\frac {3 d \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}{5 b^{4} \left (b x +a \right )^{5}}-\frac {d^{3}}{3 b^{4} \left (b x +a \right )^{3}}\right )}{g^{7}}-\frac {i^{3} B \left (d a -b c \right )^{4} e^{4} \left (\frac {d^{7} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {1}{16 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}\right )}{\left (d a -b c \right )^{7}}-\frac {2 d^{6} b e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{5 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{5}}-\frac {1}{25 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{5}}\right )}{\left (d a -b c \right )^{7}}+\frac {d^{5} b^{2} e^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{6 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{6}}-\frac {1}{36 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{6}}\right )}{\left (d a -b c \right )^{7}}\right )}{g^{7} d^{5}}\) | \(481\) |
| derivativedivides | \(-\frac {e \left (d a -b c \right ) \left (-\frac {i^{3} d^{2} e^{5} A \,b^{2}}{6 \left (d a -b c \right )^{4} g^{7} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{6}}+\frac {2 i^{3} d^{3} e^{4} A b}{5 \left (d a -b c \right )^{4} g^{7} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{5}}-\frac {i^{3} d^{4} e^{3} A}{4 \left (d a -b c \right )^{4} g^{7} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}+\frac {i^{3} d^{2} e^{5} B \,b^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{6 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{6}}-\frac {1}{36 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{6}}\right )}{\left (d a -b c \right )^{4} g^{7}}-\frac {2 i^{3} d^{3} e^{4} B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{5 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{5}}-\frac {1}{25 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{5}}\right )}{\left (d a -b c \right )^{4} g^{7}}+\frac {i^{3} d^{4} e^{3} B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {1}{16 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}\right )}{\left (d a -b c \right )^{4} g^{7}}\right )}{d^{2}}\) | \(532\) |
| default | \(-\frac {e \left (d a -b c \right ) \left (-\frac {i^{3} d^{2} e^{5} A \,b^{2}}{6 \left (d a -b c \right )^{4} g^{7} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{6}}+\frac {2 i^{3} d^{3} e^{4} A b}{5 \left (d a -b c \right )^{4} g^{7} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{5}}-\frac {i^{3} d^{4} e^{3} A}{4 \left (d a -b c \right )^{4} g^{7} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}+\frac {i^{3} d^{2} e^{5} B \,b^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{6 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{6}}-\frac {1}{36 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{6}}\right )}{\left (d a -b c \right )^{4} g^{7}}-\frac {2 i^{3} d^{3} e^{4} B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{5 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{5}}-\frac {1}{25 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{5}}\right )}{\left (d a -b c \right )^{4} g^{7}}+\frac {i^{3} d^{4} e^{3} B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {1}{16 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}\right )}{\left (d a -b c \right )^{4} g^{7}}\right )}{d^{2}}\) | \(532\) |
| orering | \(-\frac {\left (b x +a \right ) \left (-300 b^{6} d^{6} x^{6}-1500 a \,b^{5} d^{6} x^{5}-300 b^{6} c \,d^{5} x^{5}-2870 a^{2} b^{4} d^{6} x^{4}-1760 a \,b^{5} c \,d^{5} x^{4}+130 b^{6} c^{2} d^{4} x^{4}-11480 a^{2} b^{4} c \,d^{5} x^{3}+7960 a \,b^{5} c^{2} d^{4} x^{3}-2480 b^{6} c^{3} d^{3} x^{3}+777 a^{4} b^{2} d^{6} x^{2}-3108 a^{3} b^{3} c \,d^{5} x^{2}-12558 a^{2} b^{4} c^{2} d^{4} x^{2}+16332 a \,b^{5} c^{3} d^{3} x^{2}-5943 b^{6} c^{4} d^{2} x^{2}+518 a^{5} b \,d^{6} x -1036 a^{4} b^{2} c \,d^{5} x -1036 a^{3} b^{3} c^{2} d^{4} x -7336 a^{2} b^{4} c^{3} d^{3} x +11834 a \,b^{5} c^{4} d^{2} x -4744 b^{6} c^{5} d x +111 d^{6} a^{6}-148 b c \,d^{5} a^{5}-148 c^{2} b^{2} d^{4} a^{4}-148 b^{3} c^{3} d^{3} a^{3}-1723 a^{2} b^{4} c^{4} d^{2}+3056 b^{5} c^{5} d a -1300 b^{6} c^{6}\right ) \left (d i x +c i \right )^{3} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )}{3600 b^{4} \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 (d x +c \right )^{3} \left (b g x +a g \right )^{7}}+\frac {\left (60 b^{5} d^{5} x^{5}+330 a \,b^{4} d^{5} x^{4}-30 b^{5} c \,d^{4} x^{4}+740 a^{2} b^{3} d^{5} x^{3}-160 a \,b^{4} c \,d^{4} x^{3}+20 b^{5} c^{2} d^{3} x^{3}+555 a^{3} b^{2} d^{5} x^{2}+555 a^{2} b^{3} c \,d^{4} x^{2}-795 a \,b^{4} c^{2} d^{3} x^{2}+285 b^{5} c^{3} d^{2} x^{2}+222 a^{4} b \,d^{5} x +222 a^{3} b^{2} c \,d^{4} x +222 a^{2} b^{3} c^{2} d^{3} x -678 a \,b^{4} c^{3} d^{2} x +312 b^{5} c^{4} d x +37 a^{5} d^{5}+37 b c \,d^{4} a^{4}+37 b^{2} c^{2} d^{3} a^{3}+37 b^{3} c^{3} d^{2} a^{2}-188 a \,b^{4} c^{4} d +100 b^{5} c^{5}\right ) \left (b x +a \right )^{2} \left (\frac {3 \left (d i x +c i \right )^{2} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right ) d i}{\left (b g x +a g \right )^{7}}+\frac {\left (d i x +c i \right )^{3} B \left (\frac {e b}{d x +c}-\frac {e \left (b x +a \right ) d}{\left (d x +c \right )^{2}}\right ) \left (d x +c \right )}{e \left (b x +a \right ) \left (b g x +a g \right )^{7}}-\frac {7 \left (d i x +c i \right )^{3} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right ) b g}{\left (b g x +a g \right )^{8}}\right )}{3600 b^{4} \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 (d x +c \right )^{2}}\) | \(914\) |
| risch | \(-\frac {i^{3} B \left (20 b^{3} d^{3} x^{3}+15 a \,b^{2} d^{3} x^{2}+45 b^{3} c \,d^{2} x^{2}+6 a^{2} b \,d^{3} x +18 a \,b^{2} c \,d^{2} x +36 b^{3} c^{2} d x +a^{3} d^{3}+3 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d +10 b^{3} c^{3}\right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{60 \left (b x +a \right )^{6} g^{7} b^{4}}-\frac {\left (-60 B \ln \left (-b x -a \right ) a^{6} d^{6}+900 A \,a^{4} b^{2} d^{6} x^{2}-2700 A \,b^{6} c^{4} d^{2} x^{2}+360 A \,a^{5} b \,d^{6} x -2160 A \,b^{6} c^{5} d x -60 B \ln \left (-b x -a \right ) b^{6} d^{6} x^{6}+60 B \ln \left (d x +c \right ) b^{6} d^{6} x^{6}-3600 A \,a^{2} b^{4} c \,d^{5} x^{3}+3600 A a \,b^{5} c^{2} d^{4} x^{3}-900 B \,a^{2} b^{4} c \,d^{5} x^{3}+180 B a \,b^{5} c^{2} d^{4} x^{3}-5400 A \,a^{2} b^{4} c^{2} d^{4} x^{2}-1350 B \,a^{2} b^{4} c^{2} d^{4} x^{2}+1080 B a \,b^{5} c^{3} d^{3} x^{2}-900 B \,a^{2} b^{4} c^{3} d^{3} x +990 B a \,b^{5} c^{4} d^{2} x +60 B a \,b^{5} d^{6} x^{5}-60 B \,b^{6} c \,d^{5} x^{5}+330 B \,a^{2} b^{4} d^{6} x^{4}+30 B \,b^{6} c^{2} d^{4} x^{4}+1200 A \,a^{3} b^{3} d^{6} x^{3}+740 B \,a^{3} b^{3} d^{6} x^{3}-20 B \,b^{6} c^{3} d^{3} x^{3}+555 B \,a^{4} b^{2} d^{6} x^{2}-285 B \,b^{6} c^{4} d^{2} x^{2}+222 B \,a^{5} b \,d^{6} x -312 B \,b^{6} c^{5} d x -1200 A \,b^{6} c^{3} d^{3} x^{3}+1440 A a \,b^{5} c^{5} d -225 B \,a^{2} b^{4} c^{4} d^{2}+288 B a \,b^{5} c^{5} d -900 A \,a^{2} b^{4} c^{4} d^{2}+60 A \,a^{6} d^{6}-600 A \,b^{6} c^{6}+37 B \,a^{6} d^{6}-100 B \,b^{6} c^{6}-360 B a \,b^{5} c \,d^{5} x^{4}+7200 A a \,b^{5} c^{3} d^{3} x^{2}-3600 A \,a^{2} b^{4} c^{3} d^{3} x +5400 A a \,b^{5} c^{4} d^{2} x -360 B \ln \left (-b x -a \right ) a \,b^{5} d^{6} x^{5}+360 B \ln \left (d x +c \right ) a \,b^{5} d^{6} x^{5}-900 B \ln \left (-b x -a \right ) a^{2} b^{4} d^{6} x^{4}+900 B \ln \left (d x +c \right ) a^{2} b^{4} d^{6} x^{4}-1200 B \ln \left (-b x -a \right ) a^{3} b^{3} d^{6} x^{3}+1200 B \ln \left (d x +c \right ) a^{3} b^{3} d^{6} x^{3}-900 B \ln \left (-b x -a \right ) a^{4} b^{2} d^{6} x^{2}+900 B \ln \left (d x +c \right ) a^{4} b^{2} d^{6} x^{2}-360 B \ln \left (-b x -a \right ) a^{5} b \,d^{6} x +360 B \ln \left (d x +c \right ) a^{5} b \,d^{6} x +60 B \ln \left (d x +c \right ) a^{6} d^{6}\right ) i^{3}}{3600 \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 )^{6} g^{7} b^{4}}\) | \(994\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1190\) |
| norman | \(\text {Expression too large to display}\) | \(1650\) |
Input:
int((d*i*x+c*i)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^7,x,method=_RETU RNVERBOSE)
Output:
i^3*A/g^7*(3/4*d^2*(a*d-b*c)/b^4/(b*x+a)^4-1/6*(-a^3*d^3+3*a^2*b*c*d^2-3*a *b^2*c^2*d+b^3*c^3)/b^4/(b*x+a)^6-3/5*d*(a^2*d^2-2*a*b*c*d+b^2*c^2)/b^4/(b *x+a)^5-1/3*d^3/b^4/(b*x+a)^3)-i^3*B/g^7/d^5*(a*d-b*c)^4*e^4*(d^7/(a*d-b*c )^7*(-1/4/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^4*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))- 1/16/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^4)-2*d^6/(a*d-b*c)^7*b*e*(-1/5/(b*e/d+( a*d-b*c)*e/d/(d*x+c))^5*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/25/(b*e/d+(a*d-b *c)*e/d/(d*x+c))^5)+d^5/(a*d-b*c)^7*b^2*e^2*(-1/6/(b*e/d+(a*d-b*c)*e/d/(d* x+c))^6*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/36/(b*e/d+(a*d-b*c)*e/d/(d*x+c)) ^6))
Leaf count of result is larger than twice the leaf count of optimal. 991 vs. \(2 (269) = 538\).
Time = 0.10 (sec) , antiderivative size = 991, normalized size of antiderivative = 3.53 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^7} \, dx =\text {Too large to display} \] Input:
integrate((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^7,x, algo rithm="fricas")
Output:
-1/3600*(60*(B*b^6*c*d^5 - B*a*b^5*d^6)*i^3*x^5 - 30*(B*b^6*c^2*d^4 - 12*B *a*b^5*c*d^5 + 11*B*a^2*b^4*d^6)*i^3*x^4 + 20*((60*A + B)*b^6*c^3*d^3 - 9* (20*A + B)*a*b^5*c^2*d^4 + 45*(4*A + B)*a^2*b^4*c*d^5 - (60*A + 37*B)*a^3* b^3*d^6)*i^3*x^3 + 15*((180*A + 19*B)*b^6*c^4*d^2 - 24*(20*A + 3*B)*a*b^5* c^3*d^3 + 90*(4*A + B)*a^2*b^4*c^2*d^4 - (60*A + 37*B)*a^4*b^2*d^6)*i^3*x^ 2 + 6*(4*(90*A + 13*B)*b^6*c^5*d - 15*(60*A + 11*B)*a*b^5*c^4*d^2 + 150*(4 *A + B)*a^2*b^4*c^3*d^3 - (60*A + 37*B)*a^5*b*d^6)*i^3*x + (100*(6*A + B)* b^6*c^6 - 288*(5*A + B)*a*b^5*c^5*d + 225*(4*A + B)*a^2*b^4*c^4*d^2 - (60* A + 37*B)*a^6*d^6)*i^3 + 60*(B*b^6*d^6*i^3*x^6 + 6*B*a*b^5*d^6*i^3*x^5 + 1 5*B*a^2*b^4*d^6*i^3*x^4 + 20*(B*b^6*c^3*d^3 - 3*B*a*b^5*c^2*d^4 + 3*B*a^2* b^4*c*d^5)*i^3*x^3 + 15*(3*B*b^6*c^4*d^2 - 8*B*a*b^5*c^3*d^3 + 6*B*a^2*b^4 *c^2*d^4)*i^3*x^2 + 6*(6*B*b^6*c^5*d - 15*B*a*b^5*c^4*d^2 + 10*B*a^2*b^4*c ^3*d^3)*i^3*x + (10*B*b^6*c^6 - 24*B*a*b^5*c^5*d + 15*B*a^2*b^4*c^4*d^2)*i ^3)*log((b*e*x + a*e)/(d*x + c)))/((b^13*c^3 - 3*a*b^12*c^2*d + 3*a^2*b^11 *c*d^2 - a^3*b^10*d^3)*g^7*x^6 + 6*(a*b^12*c^3 - 3*a^2*b^11*c^2*d + 3*a^3* b^10*c*d^2 - a^4*b^9*d^3)*g^7*x^5 + 15*(a^2*b^11*c^3 - 3*a^3*b^10*c^2*d + 3*a^4*b^9*c*d^2 - a^5*b^8*d^3)*g^7*x^4 + 20*(a^3*b^10*c^3 - 3*a^4*b^9*c^2* d + 3*a^5*b^8*c*d^2 - a^6*b^7*d^3)*g^7*x^3 + 15*(a^4*b^9*c^3 - 3*a^5*b^8*c ^2*d + 3*a^6*b^7*c*d^2 - a^7*b^6*d^3)*g^7*x^2 + 6*(a^5*b^8*c^3 - 3*a^6*b^7 *c^2*d + 3*a^7*b^6*c*d^2 - a^8*b^5*d^3)*g^7*x + (a^6*b^7*c^3 - 3*a^7*b^...
Timed out. \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^7} \, dx=\text {Timed out} \] Input:
integrate((d*i*x+c*i)**3*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)**7,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 5524 vs. \(2 (269) = 538\).
Time = 0.35 (sec) , antiderivative size = 5524, normalized size of antiderivative = 19.66 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^7} \, dx=\text {Too large to display} \] Input:
integrate((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^7,x, algo rithm="maxima")
Output:
-1/3600*B*d^3*i^3*(60*(20*b^3*x^3 + 15*a*b^2*x^2 + 6*a^2*b*x + a^3)*log(b* e*x/(d*x + c) + a*e/(d*x + c))/(b^10*g^7*x^6 + 6*a*b^9*g^7*x^5 + 15*a^2*b^ 8*g^7*x^4 + 20*a^3*b^7*g^7*x^3 + 15*a^4*b^6*g^7*x^2 + 6*a^5*b^5*g^7*x + a^ 6*b^4*g^7) + (57*a^3*b^5*c^5 - 405*a^4*b^4*c^4*d + 1470*a^5*b^3*c^3*d^2 - 730*a^6*b^2*c^2*d^3 + 245*a^7*b*c*d^4 - 37*a^8*d^5 + 60*(20*b^8*c^3*d^2 - 15*a*b^7*c^2*d^3 + 6*a^2*b^6*c*d^4 - a^3*b^5*d^5)*x^5 - 30*(20*b^8*c^4*d - 235*a*b^7*c^3*d^2 + 171*a^2*b^6*c^2*d^3 - 67*a^3*b^5*c*d^4 + 11*a^4*b^4*d ^5)*x^4 + 20*(20*b^8*c^5 - 175*a*b^7*c^4*d + 866*a^2*b^6*c^3*d^2 - 604*a^3 *b^5*c^2*d^3 + 230*a^4*b^4*c*d^4 - 37*a^5*b^3*d^5)*x^3 + 15*(35*a*b^7*c^5 - 271*a^2*b^6*c^4*d + 1128*a^3*b^5*c^3*d^2 - 700*a^4*b^4*c^2*d^3 + 245*a^5 *b^3*c*d^4 - 37*a^6*b^2*d^5)*x^2 + 6*(47*a^2*b^6*c^5 - 345*a^3*b^5*c^4*d + 1320*a^4*b^4*c^3*d^2 - 730*a^5*b^3*c^2*d^3 + 245*a^6*b^2*c*d^4 - 37*a^7*b *d^5)*x)/((b^15*c^5 - 5*a*b^14*c^4*d + 10*a^2*b^13*c^3*d^2 - 10*a^3*b^12*c ^2*d^3 + 5*a^4*b^11*c*d^4 - a^5*b^10*d^5)*g^7*x^6 + 6*(a*b^14*c^5 - 5*a^2* b^13*c^4*d + 10*a^3*b^12*c^3*d^2 - 10*a^4*b^11*c^2*d^3 + 5*a^5*b^10*c*d^4 - a^6*b^9*d^5)*g^7*x^5 + 15*(a^2*b^13*c^5 - 5*a^3*b^12*c^4*d + 10*a^4*b^11 *c^3*d^2 - 10*a^5*b^10*c^2*d^3 + 5*a^6*b^9*c*d^4 - a^7*b^8*d^5)*g^7*x^4 + 20*(a^3*b^12*c^5 - 5*a^4*b^11*c^4*d + 10*a^5*b^10*c^3*d^2 - 10*a^6*b^9*c^2 *d^3 + 5*a^7*b^8*c*d^4 - a^8*b^7*d^5)*g^7*x^3 + 15*(a^4*b^11*c^5 - 5*a^5*b ^10*c^4*d + 10*a^6*b^9*c^3*d^2 - 10*a^7*b^8*c^2*d^3 + 5*a^8*b^7*c*d^4 -...
Time = 0.44 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.58 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^7} \, dx=-\frac {1}{3600} \, {\left (\frac {60 \, {\left (10 \, B b^{2} e^{7} i^{3} - \frac {24 \, {\left (b e x + a e\right )} B b d e^{6} i^{3}}{d x + c} + \frac {15 \, {\left (b e x + a e\right )}^{2} B d^{2} e^{5} i^{3}}{{\left (d x + c\right )}^{2}}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{\frac {{\left (b e x + a e\right )}^{6} b^{2} c^{2} g^{7}}{{\left (d x + c\right )}^{6}} - \frac {2 \, {\left (b e x + a e\right )}^{6} a b c d g^{7}}{{\left (d x + c\right )}^{6}} + \frac {{\left (b e x + a e\right )}^{6} a^{2} d^{2} g^{7}}{{\left (d x + c\right )}^{6}}} + \frac {600 \, A b^{2} e^{7} i^{3} + 100 \, B b^{2} e^{7} i^{3} - \frac {1440 \, {\left (b e x + a e\right )} A b d e^{6} i^{3}}{d x + c} - \frac {288 \, {\left (b e x + a e\right )} B b d e^{6} i^{3}}{d x + c} + \frac {900 \, {\left (b e x + a e\right )}^{2} A d^{2} e^{5} i^{3}}{{\left (d x + c\right )}^{2}} + \frac {225 \, {\left (b e x + a e\right )}^{2} B d^{2} e^{5} i^{3}}{{\left (d x + c\right )}^{2}}}{\frac {{\left (b e x + a e\right )}^{6} b^{2} c^{2} g^{7}}{{\left (d x + c\right )}^{6}} - \frac {2 \, {\left (b e x + a e\right )}^{6} a b c d g^{7}}{{\left (d x + c\right )}^{6}} + \frac {{\left (b e x + a e\right )}^{6} a^{2} d^{2} g^{7}}{{\left (d x + c\right )}^{6}}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \] Input:
integrate((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^7,x, algo rithm="giac")
Output:
-1/3600*(60*(10*B*b^2*e^7*i^3 - 24*(b*e*x + a*e)*B*b*d*e^6*i^3/(d*x + c) + 15*(b*e*x + a*e)^2*B*d^2*e^5*i^3/(d*x + c)^2)*log((b*e*x + a*e)/(d*x + c) )/((b*e*x + a*e)^6*b^2*c^2*g^7/(d*x + c)^6 - 2*(b*e*x + a*e)^6*a*b*c*d*g^7 /(d*x + c)^6 + (b*e*x + a*e)^6*a^2*d^2*g^7/(d*x + c)^6) + (600*A*b^2*e^7*i ^3 + 100*B*b^2*e^7*i^3 - 1440*(b*e*x + a*e)*A*b*d*e^6*i^3/(d*x + c) - 288* (b*e*x + a*e)*B*b*d*e^6*i^3/(d*x + c) + 900*(b*e*x + a*e)^2*A*d^2*e^5*i^3/ (d*x + c)^2 + 225*(b*e*x + a*e)^2*B*d^2*e^5*i^3/(d*x + c)^2)/((b*e*x + a*e )^6*b^2*c^2*g^7/(d*x + c)^6 - 2*(b*e*x + a*e)^6*a*b*c*d*g^7/(d*x + c)^6 + (b*e*x + a*e)^6*a^2*d^2*g^7/(d*x + c)^6))*(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 = 31.99 (sec) , antiderivative size = 1396, normalized size of antiderivative = 4.97 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^7} \, dx =\text {Too large to display} \] Input:
int(((c*i + d*i*x)^3*(A + B*log((e*(a + b*x))/(c + d*x))))/(a*g + b*g*x)^7 ,x)
Output:
(B*d^6*i^3*atanh((60*b^7*c^3*g^7 + 60*a^3*b^4*d^3*g^7 - 60*a*b^6*c^2*d*g^7 - 60*a^2*b^5*c*d^2*g^7)/(60*b^4*g^7*(a*d - b*c)^3) + (2*b*d*x*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))/(a*d - b*c)^3))/(30*b^4*g^7*(a*d - b*c)^3) - (log((e *(a + b*x))/(c + d*x))*(x^2*(b*(b*((B*a*d^3*i^3)/(60*b^5*g^7) + (B*c*d^2*i ^3)/(20*b^4*g^7)) + (B*a*d^3*i^3)/(15*b^4*g^7) + (B*c*d^2*i^3)/(5*b^3*g^7) ) + (B*a*d^3*i^3)/(6*b^3*g^7) + (B*c*d^2*i^3)/(2*b^2*g^7)) + x*(b*(a*((B*a *d^3*i^3)/(60*b^5*g^7) + (B*c*d^2*i^3)/(20*b^4*g^7)) + (B*c^2*d*i^3)/(10*b ^3*g^7)) + a*(b*((B*a*d^3*i^3)/(60*b^5*g^7) + (B*c*d^2*i^3)/(20*b^4*g^7)) + (B*a*d^3*i^3)/(15*b^4*g^7) + (B*c*d^2*i^3)/(5*b^3*g^7)) + (B*c^2*d*i^3)/ (2*b^2*g^7)) + a*(a*((B*a*d^3*i^3)/(60*b^5*g^7) + (B*c*d^2*i^3)/(20*b^4*g^ 7)) + (B*c^2*d*i^3)/(10*b^3*g^7)) + (B*c^3*i^3)/(6*b^2*g^7) + (B*d^3*i^3*x ^3)/(3*b^2*g^7)))/(6*a^5*x + a^6/b + b^5*x^6 + 15*a^4*b*x^2 + 6*a*b^4*x^5 + 20*a^3*b^2*x^3 + 15*a^2*b^3*x^4) - ((60*A*a^5*d^5*i^3 + 600*A*b^5*c^5*i^ 3 + 37*B*a^5*d^5*i^3 + 100*B*b^5*c^5*i^3 + 60*A*a^2*b^3*c^3*d^2*i^3 + 60*A *a^3*b^2*c^2*d^3*i^3 + 37*B*a^2*b^3*c^3*d^2*i^3 + 37*B*a^3*b^2*c^2*d^3*i^3 - 840*A*a*b^4*c^4*d*i^3 + 60*A*a^4*b*c*d^4*i^3 - 188*B*a*b^4*c^4*d*i^3 + 37*B*a^4*b*c*d^4*i^3)/(60*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (x^2*(60*A*a^ 3*b^2*d^5*i^3 + 37*B*a^3*b^2*d^5*i^3 + 180*A*b^5*c^3*d^2*i^3 + 19*B*b^5*c^ 3*d^2*i^3 - 300*A*a*b^4*c^2*d^3*i^3 + 60*A*a^2*b^3*c*d^4*i^3 - 53*B*a*b^4* c^2*d^3*i^3 + 37*B*a^2*b^3*c*d^4*i^3))/(4*(a^2*d^2 + b^2*c^2 - 2*a*b*c*...
Time = 0.21 (sec) , antiderivative size = 1696, normalized size of antiderivative = 6.04 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^7} \, dx =\text {Too large to display} \] Input:
int((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^7,x)
Output:
(i*( - 60*log(a + b*x)*a**7*b*d**6 - 360*log(a + b*x)*a**6*b**2*d**6*x - 9 00*log(a + b*x)*a**5*b**3*d**6*x**2 - 1200*log(a + b*x)*a**4*b**4*d**6*x** 3 - 900*log(a + b*x)*a**3*b**5*d**6*x**4 - 360*log(a + b*x)*a**2*b**6*d**6 *x**5 - 60*log(a + b*x)*a*b**7*d**6*x**6 + 60*log(c + d*x)*a**7*b*d**6 + 3 60*log(c + d*x)*a**6*b**2*d**6*x + 900*log(c + d*x)*a**5*b**3*d**6*x**2 + 1200*log(c + d*x)*a**4*b**4*d**6*x**3 + 900*log(c + d*x)*a**3*b**5*d**6*x* *4 + 360*log(c + d*x)*a**2*b**6*d**6*x**5 + 60*log(c + d*x)*a*b**7*d**6*x* *6 + 60*log((a*e + b*e*x)/(c + d*x))*a**7*b*d**6 + 360*log((a*e + b*e*x)/( c + d*x))*a**6*b**2*d**6*x + 900*log((a*e + b*e*x)/(c + d*x))*a**5*b**3*d* *6*x**2 + 1200*log((a*e + b*e*x)/(c + d*x))*a**4*b**4*d**6*x**3 - 900*log( (a*e + b*e*x)/(c + d*x))*a**3*b**5*c**4*d**2 - 3600*log((a*e + b*e*x)/(c + d*x))*a**3*b**5*c**3*d**3*x - 5400*log((a*e + b*e*x)/(c + d*x))*a**3*b**5 *c**2*d**4*x**2 - 3600*log((a*e + b*e*x)/(c + d*x))*a**3*b**5*c*d**5*x**3 + 1440*log((a*e + b*e*x)/(c + d*x))*a**2*b**6*c**5*d + 5400*log((a*e + b*e *x)/(c + d*x))*a**2*b**6*c**4*d**2*x + 7200*log((a*e + b*e*x)/(c + d*x))*a **2*b**6*c**3*d**3*x**2 + 3600*log((a*e + b*e*x)/(c + d*x))*a**2*b**6*c**2 *d**4*x**3 - 600*log((a*e + b*e*x)/(c + d*x))*a*b**7*c**6 - 2160*log((a*e + b*e*x)/(c + d*x))*a*b**7*c**5*d*x - 2700*log((a*e + b*e*x)/(c + d*x))*a* b**7*c**4*d**2*x**2 - 1200*log((a*e + b*e*x)/(c + d*x))*a*b**7*c**3*d**3*x **3 + 60*a**8*d**6 + 360*a**7*b*d**6*x + 27*a**7*b*d**6 + 10*a**6*b**2*...