Integrand size = 20, antiderivative size = 277 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^2} \, dx=-\frac {3 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e) x}{e^7}+\frac {(b d-a e)^6 (B d-A e)}{e^8 (d+e x)}+\frac {5 b^2 (b d-a e)^3 (7 b B d-4 A b e-3 a B e) (d+e x)^2}{2 e^8}-\frac {5 b^3 (b d-a e)^2 (7 b B d-3 A b e-4 a B e) (d+e x)^3}{3 e^8}+\frac {3 b^4 (b d-a e) (7 b B d-2 A b e-5 a B e) (d+e x)^4}{4 e^8}-\frac {b^5 (7 b B d-A b e-6 a B e) (d+e x)^5}{5 e^8}+\frac {b^6 B (d+e x)^6}{6 e^8}+\frac {(b d-a e)^5 (7 b B d-6 A b e-a B e) \log (d+e x)}{e^8} \]
-3*b*(-a*e+b*d)^4*(-5*A*b*e-2*B*a*e+7*B*b*d)*x/e^7+(-a*e+b*d)^6*(-A*e+B*d) /e^8/(e*x+d)+5/2*b^2*(-a*e+b*d)^3*(-4*A*b*e-3*B*a*e+7*B*b*d)*(e*x+d)^2/e^8 -5/3*b^3*(-a*e+b*d)^2*(-3*A*b*e-4*B*a*e+7*B*b*d)*(e*x+d)^3/e^8+3/4*b^4*(-a *e+b*d)*(-2*A*b*e-5*B*a*e+7*B*b*d)*(e*x+d)^4/e^8-1/5*b^5*(-A*b*e-6*B*a*e+7 *B*b*d)*(e*x+d)^5/e^8+1/6*b^6*B*(e*x+d)^6/e^8+(-a*e+b*d)^5*(-6*A*b*e-B*a*e +7*B*b*d)*ln(e*x+d)/e^8
Leaf count is larger than twice the leaf count of optimal. \(643\) vs. \(2(277)=554\).
Time = 0.18 (sec) , antiderivative size = 643, normalized size of antiderivative = 2.32 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^2} \, dx=\frac {60 a^6 e^6 (B d-A e)+360 a^5 b e^5 \left (A d e+B \left (-d^2+d e x+e^2 x^2\right )\right )+450 a^4 b^2 e^4 \left (2 A e \left (-d^2+d e x+e^2 x^2\right )+B \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )\right )+200 a^3 b^3 e^3 \left (3 A e \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )+2 B \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )\right )+75 a^2 b^4 e^2 \left (4 A e \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )+B \left (12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right )\right )+6 a b^5 e \left (5 A e \left (12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right )-6 B \left (10 d^6-50 d^5 e x-30 d^4 e^2 x^2+10 d^3 e^3 x^3-5 d^2 e^4 x^4+3 d e^5 x^5-2 e^6 x^6\right )\right )+b^6 \left (6 A e \left (-10 d^6+50 d^5 e x+30 d^4 e^2 x^2-10 d^3 e^3 x^3+5 d^2 e^4 x^4-3 d e^5 x^5+2 e^6 x^6\right )+B \left (60 d^7-360 d^6 e x-210 d^5 e^2 x^2+70 d^4 e^3 x^3-35 d^3 e^4 x^4+21 d^2 e^5 x^5-14 d e^6 x^6+10 e^7 x^7\right )\right )+60 (b d-a e)^5 (7 b B d-6 A b e-a B e) (d+e x) \log (d+e x)}{60 e^8 (d+e x)} \]
(60*a^6*e^6*(B*d - A*e) + 360*a^5*b*e^5*(A*d*e + B*(-d^2 + d*e*x + e^2*x^2 )) + 450*a^4*b^2*e^4*(2*A*e*(-d^2 + d*e*x + e^2*x^2) + B*(2*d^3 - 4*d^2*e* x - 3*d*e^2*x^2 + e^3*x^3)) + 200*a^3*b^3*e^3*(3*A*e*(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3) + 2*B*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3 *x^3 + e^4*x^4)) + 75*a^2*b^4*e^2*(4*A*e*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x ^2 - 2*d*e^3*x^3 + e^4*x^4) + B*(12*d^5 - 48*d^4*e*x - 30*d^3*e^2*x^2 + 10 *d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^5)) + 6*a*b^5*e*(5*A*e*(12*d^5 - 48*d ^4*e*x - 30*d^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^5) - 6*B* (10*d^6 - 50*d^5*e*x - 30*d^4*e^2*x^2 + 10*d^3*e^3*x^3 - 5*d^2*e^4*x^4 + 3 *d*e^5*x^5 - 2*e^6*x^6)) + b^6*(6*A*e*(-10*d^6 + 50*d^5*e*x + 30*d^4*e^2*x ^2 - 10*d^3*e^3*x^3 + 5*d^2*e^4*x^4 - 3*d*e^5*x^5 + 2*e^6*x^6) + B*(60*d^7 - 360*d^6*e*x - 210*d^5*e^2*x^2 + 70*d^4*e^3*x^3 - 35*d^3*e^4*x^4 + 21*d^ 2*e^5*x^5 - 14*d*e^6*x^6 + 10*e^7*x^7)) + 60*(b*d - a*e)^5*(7*b*B*d - 6*A* b*e - a*B*e)*(d + e*x)*Log[d + e*x])/(60*e^8*(d + e*x))
Time = 0.75 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {86, 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.
| \(\displaystyle \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {b^5 (d+e x)^4 (6 a B e+A b e-7 b B d)}{e^7}-\frac {3 b^4 (d+e x)^3 (b d-a e) (5 a B e+2 A b e-7 b B d)}{e^7}+\frac {5 b^3 (d+e x)^2 (b d-a e)^2 (4 a B e+3 A b e-7 b B d)}{e^7}-\frac {5 b^2 (d+e x) (b d-a e)^3 (3 a B e+4 A b e-7 b B d)}{e^7}+\frac {(a e-b d)^5 (a B e+6 A b e-7 b B d)}{e^7 (d+e x)}+\frac {(a e-b d)^6 (A e-B d)}{e^7 (d+e x)^2}+\frac {3 b (b d-a e)^4 (2 a B e+5 A b e-7 b B d)}{e^7}+\frac {b^6 B (d+e x)^5}{e^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {b^5 (d+e x)^5 (-6 a B e-A b e+7 b B d)}{5 e^8}+\frac {3 b^4 (d+e x)^4 (b d-a e) (-5 a B e-2 A b e+7 b B d)}{4 e^8}-\frac {5 b^3 (d+e x)^3 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{3 e^8}+\frac {5 b^2 (d+e x)^2 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{2 e^8}+\frac {(b d-a e)^6 (B d-A e)}{e^8 (d+e x)}+\frac {(b d-a e)^5 \log (d+e x) (-a B e-6 A b e+7 b B d)}{e^8}-\frac {3 b x (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{e^7}+\frac {b^6 B (d+e x)^6}{6 e^8}\) |
(-3*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e)*x)/e^7 + ((b*d - a*e)^6* (B*d - A*e))/(e^8*(d + e*x)) + (5*b^2*(b*d - a*e)^3*(7*b*B*d - 4*A*b*e - 3 *a*B*e)*(d + e*x)^2)/(2*e^8) - (5*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4 *a*B*e)*(d + e*x)^3)/(3*e^8) + (3*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a *B*e)*(d + e*x)^4)/(4*e^8) - (b^5*(7*b*B*d - A*b*e - 6*a*B*e)*(d + e*x)^5) /(5*e^8) + (b^6*B*(d + e*x)^6)/(6*e^8) + ((b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e)*Log[d + e*x])/e^8
3.11.61.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Leaf count of result is larger than twice the leaf count of optimal. \(794\) vs. \(2(267)=534\).
Time = 0.70 (sec) , antiderivative size = 795, normalized size of antiderivative = 2.87
| method | result | size |
| norman | \(\frac {\frac {\left (A \,a^{6} e^{7}-6 A \,a^{5} b d \,e^{6}+30 A \,a^{4} b^{2} d^{2} e^{5}-60 A \,a^{3} b^{3} d^{3} e^{4}+60 A \,a^{2} b^{4} d^{4} e^{3}-30 A a \,b^{5} d^{5} e^{2}+6 A \,b^{6} d^{6} e -B \,a^{6} d \,e^{6}+12 B \,a^{5} b \,d^{2} e^{5}-45 B \,a^{4} b^{2} d^{3} e^{4}+80 B \,a^{3} b^{3} d^{4} e^{3}-75 B \,a^{2} b^{4} d^{5} e^{2}+36 B a \,b^{5} d^{6} e -7 b^{6} B \,d^{7}\right ) x}{e^{7} d}+\frac {b \left (30 A \,a^{4} b \,e^{5}-60 A \,a^{3} b^{2} d \,e^{4}+60 A \,a^{2} b^{3} d^{2} e^{3}-30 A a \,b^{4} d^{3} e^{2}+6 A \,b^{5} d^{4} e +12 B \,a^{5} e^{5}-45 B \,a^{4} b d \,e^{4}+80 B \,a^{3} b^{2} d^{2} e^{3}-75 B \,a^{2} b^{3} d^{3} e^{2}+36 B a \,b^{4} d^{4} e -7 B \,b^{5} d^{5}\right ) x^{2}}{2 e^{6}}+\frac {b^{2} \left (60 A \,a^{3} b \,e^{4}-60 A \,a^{2} b^{2} d \,e^{3}+30 A a \,b^{3} d^{2} e^{2}-6 A \,b^{4} d^{3} e +45 B \,a^{4} e^{4}-80 B \,a^{3} b d \,e^{3}+75 B \,a^{2} b^{2} d^{2} e^{2}-36 B a \,b^{3} d^{3} e +7 B \,b^{4} d^{4}\right ) x^{3}}{6 e^{5}}+\frac {b^{3} \left (60 A \,a^{2} b \,e^{3}-30 A a \,b^{2} d \,e^{2}+6 A \,b^{3} d^{2} e +80 B \,a^{3} e^{3}-75 B \,a^{2} b d \,e^{2}+36 B a \,b^{2} d^{2} e -7 b^{3} B \,d^{3}\right ) x^{4}}{12 e^{4}}+\frac {b^{4} \left (30 A a b \,e^{2}-6 A \,b^{2} d e +75 B \,a^{2} e^{2}-36 B a b d e +7 b^{2} B \,d^{2}\right ) x^{5}}{20 e^{3}}+\frac {b^{5} \left (6 A b e +36 B a e -7 B b d \right ) x^{6}}{30 e^{2}}+\frac {b^{6} B \,x^{7}}{6 e}}{e x +d}+\frac {\left (6 A \,a^{5} b \,e^{6}-30 A \,a^{4} b^{2} d \,e^{5}+60 A \,a^{3} b^{3} d^{2} e^{4}-60 A \,a^{2} b^{4} d^{3} e^{3}+30 A a \,b^{5} d^{4} e^{2}-6 A \,b^{6} d^{5} e +B \,a^{6} e^{6}-12 B \,a^{5} b d \,e^{5}+45 B \,a^{4} b^{2} d^{2} e^{4}-80 B \,a^{3} b^{3} d^{3} e^{3}+75 B \,a^{2} b^{4} d^{4} e^{2}-36 B a \,b^{5} d^{5} e +7 b^{6} B \,d^{6}\right ) \ln \left (e x +d \right )}{e^{8}}\) | \(795\) |
| default | \(\frac {b \left (60 B \,a^{3} b^{2} d^{2} e^{3} x -60 B \,a^{2} b^{3} d^{3} e^{2} x +30 B a \,b^{4} d^{4} e x +\frac {45}{2} B \,a^{2} b^{3} d^{2} e^{3} x^{2}-12 B a \,b^{4} d^{3} e^{2} x^{2}-40 A \,a^{3} b^{2} d \,e^{4} x +45 A \,a^{2} b^{3} d^{2} e^{3} x -24 A a \,b^{4} d^{3} e^{2} x -30 B \,a^{4} b d \,e^{4} x +\frac {15}{4} B \,a^{2} b^{3} e^{5} x^{4}+6 B \,a^{5} e^{5} x -6 B \,b^{5} d^{5} x +\frac {6}{5} B a \,b^{4} e^{5} x^{5}-\frac {2}{5} B \,b^{5} d \,e^{4} x^{5}+\frac {3}{4} B \,b^{5} d^{2} e^{3} x^{4}+5 A \,a^{2} b^{3} e^{5} x^{3}+\frac {20}{3} B \,a^{3} b^{2} e^{5} x^{3}-\frac {4}{3} B \,b^{5} d^{3} e^{2} x^{3}-\frac {1}{2} A \,b^{5} d \,e^{4} x^{4}-2 A \,b^{5} d^{3} e^{2} x^{2}+\frac {15}{2} B \,a^{4} b \,e^{5} x^{2}+\frac {5}{2} B \,b^{5} d^{4} e \,x^{2}+15 A \,a^{4} b \,e^{5} x +5 A \,b^{5} d^{4} e x +10 A \,a^{3} b^{2} e^{5} x^{2}+\frac {3}{2} A a \,b^{4} e^{5} x^{4}-10 B \,a^{2} b^{3} d \,e^{4} x^{3}+6 B a \,b^{4} d^{2} e^{3} x^{3}-15 A \,a^{2} b^{3} d \,e^{4} x^{2}+9 A a \,b^{4} d^{2} e^{3} x^{2}-20 B \,a^{3} b^{2} d \,e^{4} x^{2}+A \,b^{5} d^{2} e^{3} x^{3}+\frac {1}{5} A \,b^{5} e^{5} x^{5}+\frac {1}{6} b^{5} B \,x^{6} e^{5}-3 B a \,b^{4} d \,e^{4} x^{4}-4 A a \,b^{4} d \,e^{4} x^{3}\right )}{e^{7}}-\frac {A \,a^{6} e^{7}-6 A \,a^{5} b d \,e^{6}+15 A \,a^{4} b^{2} d^{2} e^{5}-20 A \,a^{3} b^{3} d^{3} e^{4}+15 A \,a^{2} b^{4} d^{4} e^{3}-6 A a \,b^{5} d^{5} e^{2}+A \,b^{6} d^{6} e -B \,a^{6} d \,e^{6}+6 B \,a^{5} b \,d^{2} e^{5}-15 B \,a^{4} b^{2} d^{3} e^{4}+20 B \,a^{3} b^{3} d^{4} e^{3}-15 B \,a^{2} b^{4} d^{5} e^{2}+6 B a \,b^{5} d^{6} e -b^{6} B \,d^{7}}{e^{8} \left (e x +d \right )}+\frac {\left (6 A \,a^{5} b \,e^{6}-30 A \,a^{4} b^{2} d \,e^{5}+60 A \,a^{3} b^{3} d^{2} e^{4}-60 A \,a^{2} b^{4} d^{3} e^{3}+30 A a \,b^{5} d^{4} e^{2}-6 A \,b^{6} d^{5} e +B \,a^{6} e^{6}-12 B \,a^{5} b d \,e^{5}+45 B \,a^{4} b^{2} d^{2} e^{4}-80 B \,a^{3} b^{3} d^{3} e^{3}+75 B \,a^{2} b^{4} d^{4} e^{2}-36 B a \,b^{5} d^{5} e +7 b^{6} B \,d^{6}\right ) \ln \left (e x +d \right )}{e^{8}}\) | \(871\) |
| risch | \(\text {Expression too large to display}\) | \(1047\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1236\) |
((A*a^6*e^7-6*A*a^5*b*d*e^6+30*A*a^4*b^2*d^2*e^5-60*A*a^3*b^3*d^3*e^4+60*A *a^2*b^4*d^4*e^3-30*A*a*b^5*d^5*e^2+6*A*b^6*d^6*e-B*a^6*d*e^6+12*B*a^5*b*d ^2*e^5-45*B*a^4*b^2*d^3*e^4+80*B*a^3*b^3*d^4*e^3-75*B*a^2*b^4*d^5*e^2+36*B *a*b^5*d^6*e-7*B*b^6*d^7)/e^7/d*x+1/2*b*(30*A*a^4*b*e^5-60*A*a^3*b^2*d*e^4 +60*A*a^2*b^3*d^2*e^3-30*A*a*b^4*d^3*e^2+6*A*b^5*d^4*e+12*B*a^5*e^5-45*B*a ^4*b*d*e^4+80*B*a^3*b^2*d^2*e^3-75*B*a^2*b^3*d^3*e^2+36*B*a*b^4*d^4*e-7*B* b^5*d^5)/e^6*x^2+1/6*b^2*(60*A*a^3*b*e^4-60*A*a^2*b^2*d*e^3+30*A*a*b^3*d^2 *e^2-6*A*b^4*d^3*e+45*B*a^4*e^4-80*B*a^3*b*d*e^3+75*B*a^2*b^2*d^2*e^2-36*B *a*b^3*d^3*e+7*B*b^4*d^4)/e^5*x^3+1/12*b^3*(60*A*a^2*b*e^3-30*A*a*b^2*d*e^ 2+6*A*b^3*d^2*e+80*B*a^3*e^3-75*B*a^2*b*d*e^2+36*B*a*b^2*d^2*e-7*B*b^3*d^3 )/e^4*x^4+1/20*b^4*(30*A*a*b*e^2-6*A*b^2*d*e+75*B*a^2*e^2-36*B*a*b*d*e+7*B *b^2*d^2)/e^3*x^5+1/30*b^5*(6*A*b*e+36*B*a*e-7*B*b*d)/e^2*x^6+1/6*b^6*B/e* x^7)/(e*x+d)+1/e^8*(6*A*a^5*b*e^6-30*A*a^4*b^2*d*e^5+60*A*a^3*b^3*d^2*e^4- 60*A*a^2*b^4*d^3*e^3+30*A*a*b^5*d^4*e^2-6*A*b^6*d^5*e+B*a^6*e^6-12*B*a^5*b *d*e^5+45*B*a^4*b^2*d^2*e^4-80*B*a^3*b^3*d^3*e^3+75*B*a^2*b^4*d^4*e^2-36*B *a*b^5*d^5*e+7*B*b^6*d^6)*ln(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 1067 vs. \(2 (267) = 534\).
Time = 0.24 (sec) , antiderivative size = 1067, normalized size of antiderivative = 3.85 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^2} \, dx=\text {Too large to display} \]
1/60*(10*B*b^6*e^7*x^7 + 60*B*b^6*d^7 - 60*A*a^6*e^7 - 60*(6*B*a*b^5 + A*b ^6)*d^6*e + 180*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 - 300*(4*B*a^3*b^3 + 3*A *a^2*b^4)*d^4*e^3 + 300*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 - 180*(2*B*a^5 *b + 5*A*a^4*b^2)*d^2*e^5 + 60*(B*a^6 + 6*A*a^5*b)*d*e^6 - 2*(7*B*b^6*d*e^ 6 - 6*(6*B*a*b^5 + A*b^6)*e^7)*x^6 + 3*(7*B*b^6*d^2*e^5 - 6*(6*B*a*b^5 + A *b^6)*d*e^6 + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 - 5*(7*B*b^6*d^3*e^4 - 6*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 - 20*( 4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + 10*(7*B*b^6*d^4*e^3 - 6*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 - 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 - 30*(7*B*b^ 6*d^5*e^2 - 6*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*d ^3*e^4 - 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 15*(3*B*a^4*b^2 + 4*A*a^ 3*b^3)*d*e^6 - 6*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 - 60*(6*B*b^6*d^6*e - 5*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 12*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 - 15* (4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 10*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e ^5 - 3*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6)*x + 60*(7*B*b^6*d^7 - 6*(6*B*a*b^5 + A*b^6)*d^6*e + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 - 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 - 6*(2*B*a^ 5*b + 5*A*a^4*b^2)*d^2*e^5 + (B*a^6 + 6*A*a^5*b)*d*e^6 + (7*B*b^6*d^6*e - 6*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 - ...
Leaf count of result is larger than twice the leaf count of optimal. 782 vs. \(2 (284) = 568\).
Time = 1.95 (sec) , antiderivative size = 782, normalized size of antiderivative = 2.82 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^2} \, dx=\frac {B b^{6} x^{6}}{6 e^{2}} + x^{5} \left (\frac {A b^{6}}{5 e^{2}} + \frac {6 B a b^{5}}{5 e^{2}} - \frac {2 B b^{6} d}{5 e^{3}}\right ) + x^{4} \cdot \left (\frac {3 A a b^{5}}{2 e^{2}} - \frac {A b^{6} d}{2 e^{3}} + \frac {15 B a^{2} b^{4}}{4 e^{2}} - \frac {3 B a b^{5} d}{e^{3}} + \frac {3 B b^{6} d^{2}}{4 e^{4}}\right ) + x^{3} \cdot \left (\frac {5 A a^{2} b^{4}}{e^{2}} - \frac {4 A a b^{5} d}{e^{3}} + \frac {A b^{6} d^{2}}{e^{4}} + \frac {20 B a^{3} b^{3}}{3 e^{2}} - \frac {10 B a^{2} b^{4} d}{e^{3}} + \frac {6 B a b^{5} d^{2}}{e^{4}} - \frac {4 B b^{6} d^{3}}{3 e^{5}}\right ) + x^{2} \cdot \left (\frac {10 A a^{3} b^{3}}{e^{2}} - \frac {15 A a^{2} b^{4} d}{e^{3}} + \frac {9 A a b^{5} d^{2}}{e^{4}} - \frac {2 A b^{6} d^{3}}{e^{5}} + \frac {15 B a^{4} b^{2}}{2 e^{2}} - \frac {20 B a^{3} b^{3} d}{e^{3}} + \frac {45 B a^{2} b^{4} d^{2}}{2 e^{4}} - \frac {12 B a b^{5} d^{3}}{e^{5}} + \frac {5 B b^{6} d^{4}}{2 e^{6}}\right ) + x \left (\frac {15 A a^{4} b^{2}}{e^{2}} - \frac {40 A a^{3} b^{3} d}{e^{3}} + \frac {45 A a^{2} b^{4} d^{2}}{e^{4}} - \frac {24 A a b^{5} d^{3}}{e^{5}} + \frac {5 A b^{6} d^{4}}{e^{6}} + \frac {6 B a^{5} b}{e^{2}} - \frac {30 B a^{4} b^{2} d}{e^{3}} + \frac {60 B a^{3} b^{3} d^{2}}{e^{4}} - \frac {60 B a^{2} b^{4} d^{3}}{e^{5}} + \frac {30 B a b^{5} d^{4}}{e^{6}} - \frac {6 B b^{6} d^{5}}{e^{7}}\right ) + \frac {- A a^{6} e^{7} + 6 A a^{5} b d e^{6} - 15 A a^{4} b^{2} d^{2} e^{5} + 20 A a^{3} b^{3} d^{3} e^{4} - 15 A a^{2} b^{4} d^{4} e^{3} + 6 A a b^{5} d^{5} e^{2} - A b^{6} d^{6} e + B a^{6} d e^{6} - 6 B a^{5} b d^{2} e^{5} + 15 B a^{4} b^{2} d^{3} e^{4} - 20 B a^{3} b^{3} d^{4} e^{3} + 15 B a^{2} b^{4} d^{5} e^{2} - 6 B a b^{5} d^{6} e + B b^{6} d^{7}}{d e^{8} + e^{9} x} + \frac {\left (a e - b d\right )^{5} \cdot \left (6 A b e + B a e - 7 B b d\right ) \log {\left (d + e x \right )}}{e^{8}} \]
B*b**6*x**6/(6*e**2) + x**5*(A*b**6/(5*e**2) + 6*B*a*b**5/(5*e**2) - 2*B*b **6*d/(5*e**3)) + x**4*(3*A*a*b**5/(2*e**2) - A*b**6*d/(2*e**3) + 15*B*a** 2*b**4/(4*e**2) - 3*B*a*b**5*d/e**3 + 3*B*b**6*d**2/(4*e**4)) + x**3*(5*A* a**2*b**4/e**2 - 4*A*a*b**5*d/e**3 + A*b**6*d**2/e**4 + 20*B*a**3*b**3/(3* e**2) - 10*B*a**2*b**4*d/e**3 + 6*B*a*b**5*d**2/e**4 - 4*B*b**6*d**3/(3*e* *5)) + x**2*(10*A*a**3*b**3/e**2 - 15*A*a**2*b**4*d/e**3 + 9*A*a*b**5*d**2 /e**4 - 2*A*b**6*d**3/e**5 + 15*B*a**4*b**2/(2*e**2) - 20*B*a**3*b**3*d/e* *3 + 45*B*a**2*b**4*d**2/(2*e**4) - 12*B*a*b**5*d**3/e**5 + 5*B*b**6*d**4/ (2*e**6)) + x*(15*A*a**4*b**2/e**2 - 40*A*a**3*b**3*d/e**3 + 45*A*a**2*b** 4*d**2/e**4 - 24*A*a*b**5*d**3/e**5 + 5*A*b**6*d**4/e**6 + 6*B*a**5*b/e**2 - 30*B*a**4*b**2*d/e**3 + 60*B*a**3*b**3*d**2/e**4 - 60*B*a**2*b**4*d**3/ e**5 + 30*B*a*b**5*d**4/e**6 - 6*B*b**6*d**5/e**7) + (-A*a**6*e**7 + 6*A*a **5*b*d*e**6 - 15*A*a**4*b**2*d**2*e**5 + 20*A*a**3*b**3*d**3*e**4 - 15*A* a**2*b**4*d**4*e**3 + 6*A*a*b**5*d**5*e**2 - A*b**6*d**6*e + B*a**6*d*e**6 - 6*B*a**5*b*d**2*e**5 + 15*B*a**4*b**2*d**3*e**4 - 20*B*a**3*b**3*d**4*e **3 + 15*B*a**2*b**4*d**5*e**2 - 6*B*a*b**5*d**6*e + B*b**6*d**7)/(d*e**8 + e**9*x) + (a*e - b*d)**5*(6*A*b*e + B*a*e - 7*B*b*d)*log(d + e*x)/e**8
Leaf count of result is larger than twice the leaf count of optimal. 771 vs. \(2 (267) = 534\).
Time = 0.21 (sec) , antiderivative size = 771, normalized size of antiderivative = 2.78 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^2} \, dx=\frac {B b^{6} d^{7} - A a^{6} e^{7} - {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} - 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} - 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} + {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6}}{e^{9} x + d e^{8}} + \frac {10 \, B b^{6} e^{5} x^{6} - 12 \, {\left (2 \, B b^{6} d e^{4} - {\left (6 \, B a b^{5} + A b^{6}\right )} e^{5}\right )} x^{5} + 15 \, {\left (3 \, B b^{6} d^{2} e^{3} - 2 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{4} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{5}\right )} x^{4} - 20 \, {\left (4 \, B b^{6} d^{3} e^{2} - 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{3} + 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{4} - 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{5}\right )} x^{3} + 30 \, {\left (5 \, B b^{6} d^{4} e - 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{2} + 9 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{3} - 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{4} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{5}\right )} x^{2} - 60 \, {\left (6 \, B b^{6} d^{5} - 5 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e + 12 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{2} - 15 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{3} + 10 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{4} - 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{5}\right )} x}{60 \, e^{7}} + \frac {{\left (7 \, B b^{6} d^{6} - 6 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e + 15 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{2} - 20 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{3} + 15 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{4} - 6 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{5} + {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{6}\right )} \log \left (e x + d\right )}{e^{8}} \]
(B*b^6*d^7 - A*a^6*e^7 - (6*B*a*b^5 + A*b^6)*d^6*e + 3*(5*B*a^2*b^4 + 2*A* a*b^5)*d^5*e^2 - 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 - 3*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^5 + (B*a^6 + 6*A* a^5*b)*d*e^6)/(e^9*x + d*e^8) + 1/60*(10*B*b^6*e^5*x^6 - 12*(2*B*b^6*d*e^4 - (6*B*a*b^5 + A*b^6)*e^5)*x^5 + 15*(3*B*b^6*d^2*e^3 - 2*(6*B*a*b^5 + A*b ^6)*d*e^4 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*e^5)*x^4 - 20*(4*B*b^6*d^3*e^2 - 3 *(6*B*a*b^5 + A*b^6)*d^2*e^3 + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^4 - 5*(4*B* a^3*b^3 + 3*A*a^2*b^4)*e^5)*x^3 + 30*(5*B*b^6*d^4*e - 4*(6*B*a*b^5 + A*b^6 )*d^3*e^2 + 9*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^3 - 10*(4*B*a^3*b^3 + 3*A*a^ 2*b^4)*d*e^4 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^5)*x^2 - 60*(6*B*b^6*d^5 - 5*(6*B*a*b^5 + A*b^6)*d^4*e + 12*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^2 - 15*(4 *B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^3 + 10*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^4 - 3*(2*B*a^5*b + 5*A*a^4*b^2)*e^5)*x)/e^7 + (7*B*b^6*d^6 - 6*(6*B*a*b^5 + A *b^6)*d^5*e + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^2 - 20*(4*B*a^3*b^3 + 3*A *a^2*b^4)*d^3*e^3 + 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^4 - 6*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^5 + (B*a^6 + 6*A*a^5*b)*e^6)*log(e*x + d)/e^8
Leaf count of result is larger than twice the leaf count of optimal. 979 vs. \(2 (267) = 534\).
Time = 0.29 (sec) , antiderivative size = 979, normalized size of antiderivative = 3.53 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^2} \, dx=\frac {{\left (10 \, B b^{6} - \frac {12 \, {\left (7 \, B b^{6} d e - 6 \, B a b^{5} e^{2} - A b^{6} e^{2}\right )}}{{\left (e x + d\right )} e} + \frac {45 \, {\left (7 \, B b^{6} d^{2} e^{2} - 12 \, B a b^{5} d e^{3} - 2 \, A b^{6} d e^{3} + 5 \, B a^{2} b^{4} e^{4} + 2 \, A a b^{5} e^{4}\right )}}{{\left (e x + d\right )}^{2} e^{2}} - \frac {100 \, {\left (7 \, B b^{6} d^{3} e^{3} - 18 \, B a b^{5} d^{2} e^{4} - 3 \, A b^{6} d^{2} e^{4} + 15 \, B a^{2} b^{4} d e^{5} + 6 \, A a b^{5} d e^{5} - 4 \, B a^{3} b^{3} e^{6} - 3 \, A a^{2} b^{4} e^{6}\right )}}{{\left (e x + d\right )}^{3} e^{3}} + \frac {150 \, {\left (7 \, B b^{6} d^{4} e^{4} - 24 \, B a b^{5} d^{3} e^{5} - 4 \, A b^{6} d^{3} e^{5} + 30 \, B a^{2} b^{4} d^{2} e^{6} + 12 \, A a b^{5} d^{2} e^{6} - 16 \, B a^{3} b^{3} d e^{7} - 12 \, A a^{2} b^{4} d e^{7} + 3 \, B a^{4} b^{2} e^{8} + 4 \, A a^{3} b^{3} e^{8}\right )}}{{\left (e x + d\right )}^{4} e^{4}} - \frac {180 \, {\left (7 \, B b^{6} d^{5} e^{5} - 30 \, B a b^{5} d^{4} e^{6} - 5 \, A b^{6} d^{4} e^{6} + 50 \, B a^{2} b^{4} d^{3} e^{7} + 20 \, A a b^{5} d^{3} e^{7} - 40 \, B a^{3} b^{3} d^{2} e^{8} - 30 \, A a^{2} b^{4} d^{2} e^{8} + 15 \, B a^{4} b^{2} d e^{9} + 20 \, A a^{3} b^{3} d e^{9} - 2 \, B a^{5} b e^{10} - 5 \, A a^{4} b^{2} e^{10}\right )}}{{\left (e x + d\right )}^{5} e^{5}}\right )} {\left (e x + d\right )}^{6}}{60 \, e^{8}} - \frac {{\left (7 \, B b^{6} d^{6} - 36 \, B a b^{5} d^{5} e - 6 \, A b^{6} d^{5} e + 75 \, B a^{2} b^{4} d^{4} e^{2} + 30 \, A a b^{5} d^{4} e^{2} - 80 \, B a^{3} b^{3} d^{3} e^{3} - 60 \, A a^{2} b^{4} d^{3} e^{3} + 45 \, B a^{4} b^{2} d^{2} e^{4} + 60 \, A a^{3} b^{3} d^{2} e^{4} - 12 \, B a^{5} b d e^{5} - 30 \, A a^{4} b^{2} d e^{5} + B a^{6} e^{6} + 6 \, A a^{5} b e^{6}\right )} \log \left (\frac {{\left | e x + d \right |}}{{\left (e x + d\right )}^{2} {\left | e \right |}}\right )}{e^{8}} + \frac {\frac {B b^{6} d^{7} e^{6}}{e x + d} - \frac {6 \, B a b^{5} d^{6} e^{7}}{e x + d} - \frac {A b^{6} d^{6} e^{7}}{e x + d} + \frac {15 \, B a^{2} b^{4} d^{5} e^{8}}{e x + d} + \frac {6 \, A a b^{5} d^{5} e^{8}}{e x + d} - \frac {20 \, B a^{3} b^{3} d^{4} e^{9}}{e x + d} - \frac {15 \, A a^{2} b^{4} d^{4} e^{9}}{e x + d} + \frac {15 \, B a^{4} b^{2} d^{3} e^{10}}{e x + d} + \frac {20 \, A a^{3} b^{3} d^{3} e^{10}}{e x + d} - \frac {6 \, B a^{5} b d^{2} e^{11}}{e x + d} - \frac {15 \, A a^{4} b^{2} d^{2} e^{11}}{e x + d} + \frac {B a^{6} d e^{12}}{e x + d} + \frac {6 \, A a^{5} b d e^{12}}{e x + d} - \frac {A a^{6} e^{13}}{e x + d}}{e^{14}} \]
1/60*(10*B*b^6 - 12*(7*B*b^6*d*e - 6*B*a*b^5*e^2 - A*b^6*e^2)/((e*x + d)*e ) + 45*(7*B*b^6*d^2*e^2 - 12*B*a*b^5*d*e^3 - 2*A*b^6*d*e^3 + 5*B*a^2*b^4*e ^4 + 2*A*a*b^5*e^4)/((e*x + d)^2*e^2) - 100*(7*B*b^6*d^3*e^3 - 18*B*a*b^5* d^2*e^4 - 3*A*b^6*d^2*e^4 + 15*B*a^2*b^4*d*e^5 + 6*A*a*b^5*d*e^5 - 4*B*a^3 *b^3*e^6 - 3*A*a^2*b^4*e^6)/((e*x + d)^3*e^3) + 150*(7*B*b^6*d^4*e^4 - 24* B*a*b^5*d^3*e^5 - 4*A*b^6*d^3*e^5 + 30*B*a^2*b^4*d^2*e^6 + 12*A*a*b^5*d^2* e^6 - 16*B*a^3*b^3*d*e^7 - 12*A*a^2*b^4*d*e^7 + 3*B*a^4*b^2*e^8 + 4*A*a^3* b^3*e^8)/((e*x + d)^4*e^4) - 180*(7*B*b^6*d^5*e^5 - 30*B*a*b^5*d^4*e^6 - 5 *A*b^6*d^4*e^6 + 50*B*a^2*b^4*d^3*e^7 + 20*A*a*b^5*d^3*e^7 - 40*B*a^3*b^3* d^2*e^8 - 30*A*a^2*b^4*d^2*e^8 + 15*B*a^4*b^2*d*e^9 + 20*A*a^3*b^3*d*e^9 - 2*B*a^5*b*e^10 - 5*A*a^4*b^2*e^10)/((e*x + d)^5*e^5))*(e*x + d)^6/e^8 - ( 7*B*b^6*d^6 - 36*B*a*b^5*d^5*e - 6*A*b^6*d^5*e + 75*B*a^2*b^4*d^4*e^2 + 30 *A*a*b^5*d^4*e^2 - 80*B*a^3*b^3*d^3*e^3 - 60*A*a^2*b^4*d^3*e^3 + 45*B*a^4* b^2*d^2*e^4 + 60*A*a^3*b^3*d^2*e^4 - 12*B*a^5*b*d*e^5 - 30*A*a^4*b^2*d*e^5 + B*a^6*e^6 + 6*A*a^5*b*e^6)*log(abs(e*x + d)/((e*x + d)^2*abs(e)))/e^8 + (B*b^6*d^7*e^6/(e*x + d) - 6*B*a*b^5*d^6*e^7/(e*x + d) - A*b^6*d^6*e^7/(e *x + d) + 15*B*a^2*b^4*d^5*e^8/(e*x + d) + 6*A*a*b^5*d^5*e^8/(e*x + d) - 2 0*B*a^3*b^3*d^4*e^9/(e*x + d) - 15*A*a^2*b^4*d^4*e^9/(e*x + d) + 15*B*a^4* b^2*d^3*e^10/(e*x + d) + 20*A*a^3*b^3*d^3*e^10/(e*x + d) - 6*B*a^5*b*d^2*e ^11/(e*x + d) - 15*A*a^4*b^2*d^2*e^11/(e*x + d) + B*a^6*d*e^12/(e*x + d...
Time = 1.30 (sec) , antiderivative size = 1228, normalized size of antiderivative = 4.43 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^2} \, dx=\text {Too large to display} \]
x^2*((d^2*((2*d*((A*b^6 + 6*B*a*b^5)/e^2 - (2*B*b^6*d)/e^3))/e - (3*a*b^4* (2*A*b + 5*B*a))/e^2 + (B*b^6*d^2)/e^4))/(2*e^2) - (d*((2*d*((2*d*((A*b^6 + 6*B*a*b^5)/e^2 - (2*B*b^6*d)/e^3))/e - (3*a*b^4*(2*A*b + 5*B*a))/e^2 + ( B*b^6*d^2)/e^4))/e - (d^2*((A*b^6 + 6*B*a*b^5)/e^2 - (2*B*b^6*d)/e^3))/e^2 + (5*a^2*b^3*(3*A*b + 4*B*a))/e^2))/e + (5*a^3*b^2*(4*A*b + 3*B*a))/(2*e^ 2)) - x^4*((d*((A*b^6 + 6*B*a*b^5)/e^2 - (2*B*b^6*d)/e^3))/(2*e) - (3*a*b^ 4*(2*A*b + 5*B*a))/(4*e^2) + (B*b^6*d^2)/(4*e^4)) - x*((d^2*((2*d*((2*d*(( A*b^6 + 6*B*a*b^5)/e^2 - (2*B*b^6*d)/e^3))/e - (3*a*b^4*(2*A*b + 5*B*a))/e ^2 + (B*b^6*d^2)/e^4))/e - (d^2*((A*b^6 + 6*B*a*b^5)/e^2 - (2*B*b^6*d)/e^3 ))/e^2 + (5*a^2*b^3*(3*A*b + 4*B*a))/e^2))/e^2 + (2*d*((d^2*((2*d*((A*b^6 + 6*B*a*b^5)/e^2 - (2*B*b^6*d)/e^3))/e - (3*a*b^4*(2*A*b + 5*B*a))/e^2 + ( B*b^6*d^2)/e^4))/e^2 - (2*d*((2*d*((2*d*((A*b^6 + 6*B*a*b^5)/e^2 - (2*B*b^ 6*d)/e^3))/e - (3*a*b^4*(2*A*b + 5*B*a))/e^2 + (B*b^6*d^2)/e^4))/e - (d^2* ((A*b^6 + 6*B*a*b^5)/e^2 - (2*B*b^6*d)/e^3))/e^2 + (5*a^2*b^3*(3*A*b + 4*B *a))/e^2))/e + (5*a^3*b^2*(4*A*b + 3*B*a))/e^2))/e - (3*a^4*b*(5*A*b + 2*B *a))/e^2) + x^3*((2*d*((2*d*((A*b^6 + 6*B*a*b^5)/e^2 - (2*B*b^6*d)/e^3))/e - (3*a*b^4*(2*A*b + 5*B*a))/e^2 + (B*b^6*d^2)/e^4))/(3*e) - (d^2*((A*b^6 + 6*B*a*b^5)/e^2 - (2*B*b^6*d)/e^3))/(3*e^2) + (5*a^2*b^3*(3*A*b + 4*B*a)) /(3*e^2)) + x^5*((A*b^6 + 6*B*a*b^5)/(5*e^2) - (2*B*b^6*d)/(5*e^3)) + (log (d + e*x)*(B*a^6*e^6 + 7*B*b^6*d^6 + 6*A*a^5*b*e^6 - 6*A*b^6*d^5*e + 30...