Integrand size = 20, antiderivative size = 118 \[ \int (a+b x)^6 (A+B x) (d+e x)^2 \, dx=\frac {(A b-a B) (b d-a e)^2 (a+b x)^7}{7 b^4}+\frac {(b d-a e) (b B d+2 A b e-3 a B e) (a+b x)^8}{8 b^4}+\frac {e (2 b B d+A b e-3 a B e) (a+b x)^9}{9 b^4}+\frac {B e^2 (a+b x)^{10}}{10 b^4} \]
1/7*(A*b-B*a)*(-a*e+b*d)^2*(b*x+a)^7/b^4+1/8*(-a*e+b*d)*(2*A*b*e-3*B*a*e+B *b*d)*(b*x+a)^8/b^4+1/9*e*(A*b*e-3*B*a*e+2*B*b*d)*(b*x+a)^9/b^4+1/10*B*e^2 *(b*x+a)^10/b^4
Leaf count is larger than twice the leaf count of optimal. \(386\) vs. \(2(118)=236\).
Time = 0.15 (sec) , antiderivative size = 386, normalized size of antiderivative = 3.27 \[ \int (a+b x)^6 (A+B x) (d+e x)^2 \, dx=\frac {x \left (210 a^6 \left (4 A \left (3 d^2+3 d e x+e^2 x^2\right )+B x \left (6 d^2+8 d e x+3 e^2 x^2\right )\right )+252 a^5 b x \left (5 A \left (6 d^2+8 d e x+3 e^2 x^2\right )+2 B x \left (10 d^2+15 d e x+6 e^2 x^2\right )\right )+630 a^4 b^2 x^2 \left (2 A \left (10 d^2+15 d e x+6 e^2 x^2\right )+B x \left (15 d^2+24 d e x+10 e^2 x^2\right )\right )+120 a^3 b^3 x^3 \left (7 A \left (15 d^2+24 d e x+10 e^2 x^2\right )+4 B x \left (21 d^2+35 d e x+15 e^2 x^2\right )\right )+45 a^2 b^4 x^4 \left (8 A \left (21 d^2+35 d e x+15 e^2 x^2\right )+5 B x \left (28 d^2+48 d e x+21 e^2 x^2\right )\right )+30 a b^5 x^5 \left (3 A \left (28 d^2+48 d e x+21 e^2 x^2\right )+2 B x \left (36 d^2+63 d e x+28 e^2 x^2\right )\right )+b^6 x^6 \left (10 A \left (36 d^2+63 d e x+28 e^2 x^2\right )+7 B x \left (45 d^2+80 d e x+36 e^2 x^2\right )\right )\right )}{2520} \]
(x*(210*a^6*(4*A*(3*d^2 + 3*d*e*x + e^2*x^2) + B*x*(6*d^2 + 8*d*e*x + 3*e^ 2*x^2)) + 252*a^5*b*x*(5*A*(6*d^2 + 8*d*e*x + 3*e^2*x^2) + 2*B*x*(10*d^2 + 15*d*e*x + 6*e^2*x^2)) + 630*a^4*b^2*x^2*(2*A*(10*d^2 + 15*d*e*x + 6*e^2* x^2) + B*x*(15*d^2 + 24*d*e*x + 10*e^2*x^2)) + 120*a^3*b^3*x^3*(7*A*(15*d^ 2 + 24*d*e*x + 10*e^2*x^2) + 4*B*x*(21*d^2 + 35*d*e*x + 15*e^2*x^2)) + 45* a^2*b^4*x^4*(8*A*(21*d^2 + 35*d*e*x + 15*e^2*x^2) + 5*B*x*(28*d^2 + 48*d*e *x + 21*e^2*x^2)) + 30*a*b^5*x^5*(3*A*(28*d^2 + 48*d*e*x + 21*e^2*x^2) + 2 *B*x*(36*d^2 + 63*d*e*x + 28*e^2*x^2)) + b^6*x^6*(10*A*(36*d^2 + 63*d*e*x + 28*e^2*x^2) + 7*B*x*(45*d^2 + 80*d*e*x + 36*e^2*x^2))))/2520
Time = 0.46 (sec) , antiderivative size = 118, 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.
| \(\displaystyle \int (a+b x)^6 (A+B x) (d+e x)^2 \, dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {e (a+b x)^8 (-3 a B e+A b e+2 b B d)}{b^3}+\frac {(a+b x)^7 (b d-a e) (-3 a B e+2 A b e+b B d)}{b^3}+\frac {(a+b x)^6 (A b-a B) (b d-a e)^2}{b^3}+\frac {B e^2 (a+b x)^9}{b^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e (a+b x)^9 (-3 a B e+A b e+2 b B d)}{9 b^4}+\frac {(a+b x)^8 (b d-a e) (-3 a B e+2 A b e+b B d)}{8 b^4}+\frac {(a+b x)^7 (A b-a B) (b d-a e)^2}{7 b^4}+\frac {B e^2 (a+b x)^{10}}{10 b^4}\) |
((A*b - a*B)*(b*d - a*e)^2*(a + b*x)^7)/(7*b^4) + ((b*d - a*e)*(b*B*d + 2* A*b*e - 3*a*B*e)*(a + b*x)^8)/(8*b^4) + (e*(2*b*B*d + A*b*e - 3*a*B*e)*(a + b*x)^9)/(9*b^4) + (B*e^2*(a + b*x)^10)/(10*b^4)
3.11.57.3.1 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. \(468\) vs. \(2(110)=220\).
Time = 0.68 (sec) , antiderivative size = 469, normalized size of antiderivative = 3.97
| method | result | size |
| default | \(\frac {b^{6} B \,e^{2} x^{10}}{10}+\frac {\left (\left (b^{6} A +6 a \,b^{5} B \right ) e^{2}+2 b^{6} B d e \right ) x^{9}}{9}+\frac {\left (\left (6 a \,b^{5} A +15 a^{2} b^{4} B \right ) e^{2}+2 \left (b^{6} A +6 a \,b^{5} B \right ) d e +b^{6} B \,d^{2}\right ) x^{8}}{8}+\frac {\left (\left (15 a^{2} b^{4} A +20 a^{3} b^{3} B \right ) e^{2}+2 \left (6 a \,b^{5} A +15 a^{2} b^{4} B \right ) d e +\left (b^{6} A +6 a \,b^{5} B \right ) d^{2}\right ) x^{7}}{7}+\frac {\left (\left (20 a^{3} b^{3} A +15 a^{4} b^{2} B \right ) e^{2}+2 \left (15 a^{2} b^{4} A +20 a^{3} b^{3} B \right ) d e +\left (6 a \,b^{5} A +15 a^{2} b^{4} B \right ) d^{2}\right ) x^{6}}{6}+\frac {\left (\left (15 a^{4} b^{2} A +6 a^{5} b B \right ) e^{2}+2 \left (20 a^{3} b^{3} A +15 a^{4} b^{2} B \right ) d e +\left (15 a^{2} b^{4} A +20 a^{3} b^{3} B \right ) d^{2}\right ) x^{5}}{5}+\frac {\left (\left (6 A \,a^{5} b +B \,a^{6}\right ) e^{2}+2 \left (15 a^{4} b^{2} A +6 a^{5} b B \right ) d e +\left (20 a^{3} b^{3} A +15 a^{4} b^{2} B \right ) d^{2}\right ) x^{4}}{4}+\frac {\left (A \,a^{6} e^{2}+2 \left (6 A \,a^{5} b +B \,a^{6}\right ) d e +\left (15 a^{4} b^{2} A +6 a^{5} b B \right ) d^{2}\right ) x^{3}}{3}+\frac {\left (2 A \,a^{6} d e +\left (6 A \,a^{5} b +B \,a^{6}\right ) d^{2}\right ) x^{2}}{2}+A \,a^{6} d^{2} x\) | \(469\) |
| norman | \(\frac {b^{6} B \,e^{2} x^{10}}{10}+\left (\frac {1}{9} A \,b^{6} e^{2}+\frac {2}{3} B a \,b^{5} e^{2}+\frac {2}{9} b^{6} B d e \right ) x^{9}+\left (\frac {3}{4} A a \,b^{5} e^{2}+\frac {1}{4} A \,b^{6} d e +\frac {15}{8} B \,a^{2} b^{4} e^{2}+\frac {3}{2} B a \,b^{5} d e +\frac {1}{8} b^{6} B \,d^{2}\right ) x^{8}+\left (\frac {15}{7} A \,a^{2} b^{4} e^{2}+\frac {12}{7} A a \,b^{5} d e +\frac {1}{7} A \,b^{6} d^{2}+\frac {20}{7} B \,a^{3} b^{3} e^{2}+\frac {30}{7} B \,a^{2} b^{4} d e +\frac {6}{7} B a \,b^{5} d^{2}\right ) x^{7}+\left (\frac {10}{3} A \,a^{3} b^{3} e^{2}+5 A \,a^{2} b^{4} d e +A a \,b^{5} d^{2}+\frac {5}{2} B \,a^{4} b^{2} e^{2}+\frac {20}{3} B \,a^{3} b^{3} d e +\frac {5}{2} B \,a^{2} b^{4} d^{2}\right ) x^{6}+\left (3 A \,a^{4} b^{2} e^{2}+8 A \,a^{3} b^{3} d e +3 A \,a^{2} b^{4} d^{2}+\frac {6}{5} B \,a^{5} b \,e^{2}+6 B \,a^{4} b^{2} d e +4 B \,a^{3} b^{3} d^{2}\right ) x^{5}+\left (\frac {3}{2} A \,a^{5} b \,e^{2}+\frac {15}{2} A \,a^{4} b^{2} d e +5 A \,a^{3} b^{3} d^{2}+\frac {1}{4} B \,a^{6} e^{2}+3 B \,a^{5} b d e +\frac {15}{4} B \,a^{4} b^{2} d^{2}\right ) x^{4}+\left (\frac {1}{3} A \,a^{6} e^{2}+4 A \,a^{5} b d e +5 A \,a^{4} b^{2} d^{2}+\frac {2}{3} B \,a^{6} d e +2 B \,a^{5} b \,d^{2}\right ) x^{3}+\left (A \,a^{6} d e +3 A \,a^{5} b \,d^{2}+\frac {1}{2} B \,a^{6} d^{2}\right ) x^{2}+A \,a^{6} d^{2} x\) | \(473\) |
| gosper | \(\frac {12}{7} x^{7} A a \,b^{5} d e +\frac {30}{7} x^{7} B \,a^{2} b^{4} d e +\frac {15}{4} x^{4} B \,a^{4} b^{2} d^{2}+5 x^{3} A \,a^{4} b^{2} d^{2}+\frac {2}{3} x^{3} B \,a^{6} d e +2 x^{3} B \,a^{5} b \,d^{2}+x^{2} A \,a^{6} d e +3 x^{2} A \,a^{5} b \,d^{2}+\frac {6}{5} x^{5} B \,a^{5} b \,e^{2}+4 x^{5} B \,a^{3} b^{3} d^{2}+\frac {3}{2} x^{4} A \,a^{5} b \,e^{2}+5 x^{4} A \,a^{3} b^{3} d^{2}+6 x^{5} B \,a^{4} b^{2} d e +\frac {15}{2} x^{4} A \,a^{4} b^{2} d e +3 x^{4} B \,a^{5} b d e +4 x^{3} A \,a^{5} b d e +\frac {2}{9} x^{9} b^{6} B d e +5 x^{6} A \,a^{2} b^{4} d e +\frac {20}{3} x^{6} B \,a^{3} b^{3} d e +\frac {1}{10} b^{6} B \,e^{2} x^{10}+A \,a^{6} d^{2} x +\frac {1}{9} x^{9} A \,b^{6} e^{2}+\frac {1}{8} x^{8} b^{6} B \,d^{2}+\frac {1}{7} x^{7} A \,b^{6} d^{2}+\frac {1}{4} x^{4} B \,a^{6} e^{2}+\frac {1}{2} x^{2} B \,a^{6} d^{2}+\frac {1}{3} x^{3} A \,a^{6} e^{2}+\frac {2}{3} x^{9} B a \,b^{5} e^{2}+\frac {3}{2} x^{8} B a \,b^{5} d e +\frac {10}{3} x^{6} A \,a^{3} b^{3} e^{2}+x^{6} A a \,b^{5} d^{2}+\frac {5}{2} x^{6} B \,a^{4} b^{2} e^{2}+\frac {5}{2} x^{6} B \,a^{2} b^{4} d^{2}+3 x^{5} A \,a^{4} b^{2} e^{2}+3 x^{5} A \,a^{2} b^{4} d^{2}+\frac {3}{4} x^{8} A a \,b^{5} e^{2}+\frac {1}{4} x^{8} A \,b^{6} d e +\frac {15}{8} x^{8} B \,a^{2} b^{4} e^{2}+\frac {15}{7} x^{7} A \,a^{2} b^{4} e^{2}+\frac {20}{7} x^{7} B \,a^{3} b^{3} e^{2}+\frac {6}{7} x^{7} B a \,b^{5} d^{2}+8 x^{5} A \,a^{3} b^{3} d e\) | \(553\) |
| risch | \(\frac {12}{7} x^{7} A a \,b^{5} d e +\frac {30}{7} x^{7} B \,a^{2} b^{4} d e +\frac {15}{4} x^{4} B \,a^{4} b^{2} d^{2}+5 x^{3} A \,a^{4} b^{2} d^{2}+\frac {2}{3} x^{3} B \,a^{6} d e +2 x^{3} B \,a^{5} b \,d^{2}+x^{2} A \,a^{6} d e +3 x^{2} A \,a^{5} b \,d^{2}+\frac {6}{5} x^{5} B \,a^{5} b \,e^{2}+4 x^{5} B \,a^{3} b^{3} d^{2}+\frac {3}{2} x^{4} A \,a^{5} b \,e^{2}+5 x^{4} A \,a^{3} b^{3} d^{2}+6 x^{5} B \,a^{4} b^{2} d e +\frac {15}{2} x^{4} A \,a^{4} b^{2} d e +3 x^{4} B \,a^{5} b d e +4 x^{3} A \,a^{5} b d e +\frac {2}{9} x^{9} b^{6} B d e +5 x^{6} A \,a^{2} b^{4} d e +\frac {20}{3} x^{6} B \,a^{3} b^{3} d e +\frac {1}{10} b^{6} B \,e^{2} x^{10}+A \,a^{6} d^{2} x +\frac {1}{9} x^{9} A \,b^{6} e^{2}+\frac {1}{8} x^{8} b^{6} B \,d^{2}+\frac {1}{7} x^{7} A \,b^{6} d^{2}+\frac {1}{4} x^{4} B \,a^{6} e^{2}+\frac {1}{2} x^{2} B \,a^{6} d^{2}+\frac {1}{3} x^{3} A \,a^{6} e^{2}+\frac {2}{3} x^{9} B a \,b^{5} e^{2}+\frac {3}{2} x^{8} B a \,b^{5} d e +\frac {10}{3} x^{6} A \,a^{3} b^{3} e^{2}+x^{6} A a \,b^{5} d^{2}+\frac {5}{2} x^{6} B \,a^{4} b^{2} e^{2}+\frac {5}{2} x^{6} B \,a^{2} b^{4} d^{2}+3 x^{5} A \,a^{4} b^{2} e^{2}+3 x^{5} A \,a^{2} b^{4} d^{2}+\frac {3}{4} x^{8} A a \,b^{5} e^{2}+\frac {1}{4} x^{8} A \,b^{6} d e +\frac {15}{8} x^{8} B \,a^{2} b^{4} e^{2}+\frac {15}{7} x^{7} A \,a^{2} b^{4} e^{2}+\frac {20}{7} x^{7} B \,a^{3} b^{3} e^{2}+\frac {6}{7} x^{7} B a \,b^{5} d^{2}+8 x^{5} A \,a^{3} b^{3} d e\) | \(553\) |
| parallelrisch | \(\frac {12}{7} x^{7} A a \,b^{5} d e +\frac {30}{7} x^{7} B \,a^{2} b^{4} d e +\frac {15}{4} x^{4} B \,a^{4} b^{2} d^{2}+5 x^{3} A \,a^{4} b^{2} d^{2}+\frac {2}{3} x^{3} B \,a^{6} d e +2 x^{3} B \,a^{5} b \,d^{2}+x^{2} A \,a^{6} d e +3 x^{2} A \,a^{5} b \,d^{2}+\frac {6}{5} x^{5} B \,a^{5} b \,e^{2}+4 x^{5} B \,a^{3} b^{3} d^{2}+\frac {3}{2} x^{4} A \,a^{5} b \,e^{2}+5 x^{4} A \,a^{3} b^{3} d^{2}+6 x^{5} B \,a^{4} b^{2} d e +\frac {15}{2} x^{4} A \,a^{4} b^{2} d e +3 x^{4} B \,a^{5} b d e +4 x^{3} A \,a^{5} b d e +\frac {2}{9} x^{9} b^{6} B d e +5 x^{6} A \,a^{2} b^{4} d e +\frac {20}{3} x^{6} B \,a^{3} b^{3} d e +\frac {1}{10} b^{6} B \,e^{2} x^{10}+A \,a^{6} d^{2} x +\frac {1}{9} x^{9} A \,b^{6} e^{2}+\frac {1}{8} x^{8} b^{6} B \,d^{2}+\frac {1}{7} x^{7} A \,b^{6} d^{2}+\frac {1}{4} x^{4} B \,a^{6} e^{2}+\frac {1}{2} x^{2} B \,a^{6} d^{2}+\frac {1}{3} x^{3} A \,a^{6} e^{2}+\frac {2}{3} x^{9} B a \,b^{5} e^{2}+\frac {3}{2} x^{8} B a \,b^{5} d e +\frac {10}{3} x^{6} A \,a^{3} b^{3} e^{2}+x^{6} A a \,b^{5} d^{2}+\frac {5}{2} x^{6} B \,a^{4} b^{2} e^{2}+\frac {5}{2} x^{6} B \,a^{2} b^{4} d^{2}+3 x^{5} A \,a^{4} b^{2} e^{2}+3 x^{5} A \,a^{2} b^{4} d^{2}+\frac {3}{4} x^{8} A a \,b^{5} e^{2}+\frac {1}{4} x^{8} A \,b^{6} d e +\frac {15}{8} x^{8} B \,a^{2} b^{4} e^{2}+\frac {15}{7} x^{7} A \,a^{2} b^{4} e^{2}+\frac {20}{7} x^{7} B \,a^{3} b^{3} e^{2}+\frac {6}{7} x^{7} B a \,b^{5} d^{2}+8 x^{5} A \,a^{3} b^{3} d e\) | \(553\) |
1/10*b^6*B*e^2*x^10+1/9*((A*b^6+6*B*a*b^5)*e^2+2*b^6*B*d*e)*x^9+1/8*((6*A* a*b^5+15*B*a^2*b^4)*e^2+2*(A*b^6+6*B*a*b^5)*d*e+b^6*B*d^2)*x^8+1/7*((15*A* a^2*b^4+20*B*a^3*b^3)*e^2+2*(6*A*a*b^5+15*B*a^2*b^4)*d*e+(A*b^6+6*B*a*b^5) *d^2)*x^7+1/6*((20*A*a^3*b^3+15*B*a^4*b^2)*e^2+2*(15*A*a^2*b^4+20*B*a^3*b^ 3)*d*e+(6*A*a*b^5+15*B*a^2*b^4)*d^2)*x^6+1/5*((15*A*a^4*b^2+6*B*a^5*b)*e^2 +2*(20*A*a^3*b^3+15*B*a^4*b^2)*d*e+(15*A*a^2*b^4+20*B*a^3*b^3)*d^2)*x^5+1/ 4*((6*A*a^5*b+B*a^6)*e^2+2*(15*A*a^4*b^2+6*B*a^5*b)*d*e+(20*A*a^3*b^3+15*B *a^4*b^2)*d^2)*x^4+1/3*(A*a^6*e^2+2*(6*A*a^5*b+B*a^6)*d*e+(15*A*a^4*b^2+6* B*a^5*b)*d^2)*x^3+1/2*(2*A*a^6*d*e+(6*A*a^5*b+B*a^6)*d^2)*x^2+A*a^6*d^2*x
Leaf count of result is larger than twice the leaf count of optimal. 476 vs. \(2 (110) = 220\).
Time = 0.21 (sec) , antiderivative size = 476, normalized size of antiderivative = 4.03 \[ \int (a+b x)^6 (A+B x) (d+e x)^2 \, dx=\frac {1}{10} \, B b^{6} e^{2} x^{10} + A a^{6} d^{2} x + \frac {1}{9} \, {\left (2 \, B b^{6} d e + {\left (6 \, B a b^{5} + A b^{6}\right )} e^{2}\right )} x^{9} + \frac {1}{8} \, {\left (B b^{6} d^{2} + 2 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{2}\right )} x^{8} + \frac {1}{7} \, {\left ({\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} + 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e + 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{2}\right )} x^{7} + \frac {1}{6} \, {\left (3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} + 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} + 10 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e + 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} + 6 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e + {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (A a^{6} e^{2} + 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} + 2 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d e\right )} x^{3} + \frac {1}{2} \, {\left (2 \, A a^{6} d e + {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{2}\right )} x^{2} \]
1/10*B*b^6*e^2*x^10 + A*a^6*d^2*x + 1/9*(2*B*b^6*d*e + (6*B*a*b^5 + A*b^6) *e^2)*x^9 + 1/8*(B*b^6*d^2 + 2*(6*B*a*b^5 + A*b^6)*d*e + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*e^2)*x^8 + 1/7*((6*B*a*b^5 + A*b^6)*d^2 + 6*(5*B*a^2*b^4 + 2*A* a*b^5)*d*e + 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^2)*x^7 + 1/6*(3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2 + 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e + 5*(3*B*a^4*b^2 + 4 *A*a^3*b^3)*e^2)*x^6 + 1/5*(5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2 + 10*(3*B*a^ 4*b^2 + 4*A*a^3*b^3)*d*e + 3*(2*B*a^5*b + 5*A*a^4*b^2)*e^2)*x^5 + 1/4*(5*( 3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2 + 6*(2*B*a^5*b + 5*A*a^4*b^2)*d*e + (B*a^6 + 6*A*a^5*b)*e^2)*x^4 + 1/3*(A*a^6*e^2 + 3*(2*B*a^5*b + 5*A*a^4*b^2)*d^2 + 2*(B*a^6 + 6*A*a^5*b)*d*e)*x^3 + 1/2*(2*A*a^6*d*e + (B*a^6 + 6*A*a^5*b)*d ^2)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 568 vs. \(2 (116) = 232\).
Time = 0.05 (sec) , antiderivative size = 568, normalized size of antiderivative = 4.81 \[ \int (a+b x)^6 (A+B x) (d+e x)^2 \, dx=A a^{6} d^{2} x + \frac {B b^{6} e^{2} x^{10}}{10} + x^{9} \left (\frac {A b^{6} e^{2}}{9} + \frac {2 B a b^{5} e^{2}}{3} + \frac {2 B b^{6} d e}{9}\right ) + x^{8} \cdot \left (\frac {3 A a b^{5} e^{2}}{4} + \frac {A b^{6} d e}{4} + \frac {15 B a^{2} b^{4} e^{2}}{8} + \frac {3 B a b^{5} d e}{2} + \frac {B b^{6} d^{2}}{8}\right ) + x^{7} \cdot \left (\frac {15 A a^{2} b^{4} e^{2}}{7} + \frac {12 A a b^{5} d e}{7} + \frac {A b^{6} d^{2}}{7} + \frac {20 B a^{3} b^{3} e^{2}}{7} + \frac {30 B a^{2} b^{4} d e}{7} + \frac {6 B a b^{5} d^{2}}{7}\right ) + x^{6} \cdot \left (\frac {10 A a^{3} b^{3} e^{2}}{3} + 5 A a^{2} b^{4} d e + A a b^{5} d^{2} + \frac {5 B a^{4} b^{2} e^{2}}{2} + \frac {20 B a^{3} b^{3} d e}{3} + \frac {5 B a^{2} b^{4} d^{2}}{2}\right ) + x^{5} \cdot \left (3 A a^{4} b^{2} e^{2} + 8 A a^{3} b^{3} d e + 3 A a^{2} b^{4} d^{2} + \frac {6 B a^{5} b e^{2}}{5} + 6 B a^{4} b^{2} d e + 4 B a^{3} b^{3} d^{2}\right ) + x^{4} \cdot \left (\frac {3 A a^{5} b e^{2}}{2} + \frac {15 A a^{4} b^{2} d e}{2} + 5 A a^{3} b^{3} d^{2} + \frac {B a^{6} e^{2}}{4} + 3 B a^{5} b d e + \frac {15 B a^{4} b^{2} d^{2}}{4}\right ) + x^{3} \left (\frac {A a^{6} e^{2}}{3} + 4 A a^{5} b d e + 5 A a^{4} b^{2} d^{2} + \frac {2 B a^{6} d e}{3} + 2 B a^{5} b d^{2}\right ) + x^{2} \left (A a^{6} d e + 3 A a^{5} b d^{2} + \frac {B a^{6} d^{2}}{2}\right ) \]
A*a**6*d**2*x + B*b**6*e**2*x**10/10 + x**9*(A*b**6*e**2/9 + 2*B*a*b**5*e* *2/3 + 2*B*b**6*d*e/9) + x**8*(3*A*a*b**5*e**2/4 + A*b**6*d*e/4 + 15*B*a** 2*b**4*e**2/8 + 3*B*a*b**5*d*e/2 + B*b**6*d**2/8) + x**7*(15*A*a**2*b**4*e **2/7 + 12*A*a*b**5*d*e/7 + A*b**6*d**2/7 + 20*B*a**3*b**3*e**2/7 + 30*B*a **2*b**4*d*e/7 + 6*B*a*b**5*d**2/7) + x**6*(10*A*a**3*b**3*e**2/3 + 5*A*a* *2*b**4*d*e + A*a*b**5*d**2 + 5*B*a**4*b**2*e**2/2 + 20*B*a**3*b**3*d*e/3 + 5*B*a**2*b**4*d**2/2) + x**5*(3*A*a**4*b**2*e**2 + 8*A*a**3*b**3*d*e + 3 *A*a**2*b**4*d**2 + 6*B*a**5*b*e**2/5 + 6*B*a**4*b**2*d*e + 4*B*a**3*b**3* d**2) + x**4*(3*A*a**5*b*e**2/2 + 15*A*a**4*b**2*d*e/2 + 5*A*a**3*b**3*d** 2 + B*a**6*e**2/4 + 3*B*a**5*b*d*e + 15*B*a**4*b**2*d**2/4) + x**3*(A*a**6 *e**2/3 + 4*A*a**5*b*d*e + 5*A*a**4*b**2*d**2 + 2*B*a**6*d*e/3 + 2*B*a**5* b*d**2) + x**2*(A*a**6*d*e + 3*A*a**5*b*d**2 + B*a**6*d**2/2)
Leaf count of result is larger than twice the leaf count of optimal. 476 vs. \(2 (110) = 220\).
Time = 0.24 (sec) , antiderivative size = 476, normalized size of antiderivative = 4.03 \[ \int (a+b x)^6 (A+B x) (d+e x)^2 \, dx=\frac {1}{10} \, B b^{6} e^{2} x^{10} + A a^{6} d^{2} x + \frac {1}{9} \, {\left (2 \, B b^{6} d e + {\left (6 \, B a b^{5} + A b^{6}\right )} e^{2}\right )} x^{9} + \frac {1}{8} \, {\left (B b^{6} d^{2} + 2 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{2}\right )} x^{8} + \frac {1}{7} \, {\left ({\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} + 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e + 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{2}\right )} x^{7} + \frac {1}{6} \, {\left (3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} + 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} + 10 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e + 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} + 6 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e + {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (A a^{6} e^{2} + 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} + 2 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d e\right )} x^{3} + \frac {1}{2} \, {\left (2 \, A a^{6} d e + {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{2}\right )} x^{2} \]
1/10*B*b^6*e^2*x^10 + A*a^6*d^2*x + 1/9*(2*B*b^6*d*e + (6*B*a*b^5 + A*b^6) *e^2)*x^9 + 1/8*(B*b^6*d^2 + 2*(6*B*a*b^5 + A*b^6)*d*e + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*e^2)*x^8 + 1/7*((6*B*a*b^5 + A*b^6)*d^2 + 6*(5*B*a^2*b^4 + 2*A* a*b^5)*d*e + 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^2)*x^7 + 1/6*(3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2 + 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e + 5*(3*B*a^4*b^2 + 4 *A*a^3*b^3)*e^2)*x^6 + 1/5*(5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2 + 10*(3*B*a^ 4*b^2 + 4*A*a^3*b^3)*d*e + 3*(2*B*a^5*b + 5*A*a^4*b^2)*e^2)*x^5 + 1/4*(5*( 3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2 + 6*(2*B*a^5*b + 5*A*a^4*b^2)*d*e + (B*a^6 + 6*A*a^5*b)*e^2)*x^4 + 1/3*(A*a^6*e^2 + 3*(2*B*a^5*b + 5*A*a^4*b^2)*d^2 + 2*(B*a^6 + 6*A*a^5*b)*d*e)*x^3 + 1/2*(2*A*a^6*d*e + (B*a^6 + 6*A*a^5*b)*d ^2)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 552 vs. \(2 (110) = 220\).
Time = 0.30 (sec) , antiderivative size = 552, normalized size of antiderivative = 4.68 \[ \int (a+b x)^6 (A+B x) (d+e x)^2 \, dx=\frac {1}{10} \, B b^{6} e^{2} x^{10} + \frac {2}{9} \, B b^{6} d e x^{9} + \frac {2}{3} \, B a b^{5} e^{2} x^{9} + \frac {1}{9} \, A b^{6} e^{2} x^{9} + \frac {1}{8} \, B b^{6} d^{2} x^{8} + \frac {3}{2} \, B a b^{5} d e x^{8} + \frac {1}{4} \, A b^{6} d e x^{8} + \frac {15}{8} \, B a^{2} b^{4} e^{2} x^{8} + \frac {3}{4} \, A a b^{5} e^{2} x^{8} + \frac {6}{7} \, B a b^{5} d^{2} x^{7} + \frac {1}{7} \, A b^{6} d^{2} x^{7} + \frac {30}{7} \, B a^{2} b^{4} d e x^{7} + \frac {12}{7} \, A a b^{5} d e x^{7} + \frac {20}{7} \, B a^{3} b^{3} e^{2} x^{7} + \frac {15}{7} \, A a^{2} b^{4} e^{2} x^{7} + \frac {5}{2} \, B a^{2} b^{4} d^{2} x^{6} + A a b^{5} d^{2} x^{6} + \frac {20}{3} \, B a^{3} b^{3} d e x^{6} + 5 \, A a^{2} b^{4} d e x^{6} + \frac {5}{2} \, B a^{4} b^{2} e^{2} x^{6} + \frac {10}{3} \, A a^{3} b^{3} e^{2} x^{6} + 4 \, B a^{3} b^{3} d^{2} x^{5} + 3 \, A a^{2} b^{4} d^{2} x^{5} + 6 \, B a^{4} b^{2} d e x^{5} + 8 \, A a^{3} b^{3} d e x^{5} + \frac {6}{5} \, B a^{5} b e^{2} x^{5} + 3 \, A a^{4} b^{2} e^{2} x^{5} + \frac {15}{4} \, B a^{4} b^{2} d^{2} x^{4} + 5 \, A a^{3} b^{3} d^{2} x^{4} + 3 \, B a^{5} b d e x^{4} + \frac {15}{2} \, A a^{4} b^{2} d e x^{4} + \frac {1}{4} \, B a^{6} e^{2} x^{4} + \frac {3}{2} \, A a^{5} b e^{2} x^{4} + 2 \, B a^{5} b d^{2} x^{3} + 5 \, A a^{4} b^{2} d^{2} x^{3} + \frac {2}{3} \, B a^{6} d e x^{3} + 4 \, A a^{5} b d e x^{3} + \frac {1}{3} \, A a^{6} e^{2} x^{3} + \frac {1}{2} \, B a^{6} d^{2} x^{2} + 3 \, A a^{5} b d^{2} x^{2} + A a^{6} d e x^{2} + A a^{6} d^{2} x \]
1/10*B*b^6*e^2*x^10 + 2/9*B*b^6*d*e*x^9 + 2/3*B*a*b^5*e^2*x^9 + 1/9*A*b^6* e^2*x^9 + 1/8*B*b^6*d^2*x^8 + 3/2*B*a*b^5*d*e*x^8 + 1/4*A*b^6*d*e*x^8 + 15 /8*B*a^2*b^4*e^2*x^8 + 3/4*A*a*b^5*e^2*x^8 + 6/7*B*a*b^5*d^2*x^7 + 1/7*A*b ^6*d^2*x^7 + 30/7*B*a^2*b^4*d*e*x^7 + 12/7*A*a*b^5*d*e*x^7 + 20/7*B*a^3*b^ 3*e^2*x^7 + 15/7*A*a^2*b^4*e^2*x^7 + 5/2*B*a^2*b^4*d^2*x^6 + A*a*b^5*d^2*x ^6 + 20/3*B*a^3*b^3*d*e*x^6 + 5*A*a^2*b^4*d*e*x^6 + 5/2*B*a^4*b^2*e^2*x^6 + 10/3*A*a^3*b^3*e^2*x^6 + 4*B*a^3*b^3*d^2*x^5 + 3*A*a^2*b^4*d^2*x^5 + 6*B *a^4*b^2*d*e*x^5 + 8*A*a^3*b^3*d*e*x^5 + 6/5*B*a^5*b*e^2*x^5 + 3*A*a^4*b^2 *e^2*x^5 + 15/4*B*a^4*b^2*d^2*x^4 + 5*A*a^3*b^3*d^2*x^4 + 3*B*a^5*b*d*e*x^ 4 + 15/2*A*a^4*b^2*d*e*x^4 + 1/4*B*a^6*e^2*x^4 + 3/2*A*a^5*b*e^2*x^4 + 2*B *a^5*b*d^2*x^3 + 5*A*a^4*b^2*d^2*x^3 + 2/3*B*a^6*d*e*x^3 + 4*A*a^5*b*d*e*x ^3 + 1/3*A*a^6*e^2*x^3 + 1/2*B*a^6*d^2*x^2 + 3*A*a^5*b*d^2*x^2 + A*a^6*d*e *x^2 + A*a^6*d^2*x
Time = 1.37 (sec) , antiderivative size = 457, normalized size of antiderivative = 3.87 \[ \int (a+b x)^6 (A+B x) (d+e x)^2 \, dx=x^4\,\left (\frac {B\,a^6\,e^2}{4}+3\,B\,a^5\,b\,d\,e+\frac {3\,A\,a^5\,b\,e^2}{2}+\frac {15\,B\,a^4\,b^2\,d^2}{4}+\frac {15\,A\,a^4\,b^2\,d\,e}{2}+5\,A\,a^3\,b^3\,d^2\right )+x^7\,\left (\frac {20\,B\,a^3\,b^3\,e^2}{7}+\frac {30\,B\,a^2\,b^4\,d\,e}{7}+\frac {15\,A\,a^2\,b^4\,e^2}{7}+\frac {6\,B\,a\,b^5\,d^2}{7}+\frac {12\,A\,a\,b^5\,d\,e}{7}+\frac {A\,b^6\,d^2}{7}\right )+x^5\,\left (\frac {6\,B\,a^5\,b\,e^2}{5}+6\,B\,a^4\,b^2\,d\,e+3\,A\,a^4\,b^2\,e^2+4\,B\,a^3\,b^3\,d^2+8\,A\,a^3\,b^3\,d\,e+3\,A\,a^2\,b^4\,d^2\right )+x^6\,\left (\frac {5\,B\,a^4\,b^2\,e^2}{2}+\frac {20\,B\,a^3\,b^3\,d\,e}{3}+\frac {10\,A\,a^3\,b^3\,e^2}{3}+\frac {5\,B\,a^2\,b^4\,d^2}{2}+5\,A\,a^2\,b^4\,d\,e+A\,a\,b^5\,d^2\right )+x^3\,\left (\frac {2\,B\,a^6\,d\,e}{3}+\frac {A\,a^6\,e^2}{3}+2\,B\,a^5\,b\,d^2+4\,A\,a^5\,b\,d\,e+5\,A\,a^4\,b^2\,d^2\right )+x^8\,\left (\frac {15\,B\,a^2\,b^4\,e^2}{8}+\frac {3\,B\,a\,b^5\,d\,e}{2}+\frac {3\,A\,a\,b^5\,e^2}{4}+\frac {B\,b^6\,d^2}{8}+\frac {A\,b^6\,d\,e}{4}\right )+A\,a^6\,d^2\,x+\frac {a^5\,d\,x^2\,\left (2\,A\,a\,e+6\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b^5\,e\,x^9\,\left (A\,b\,e+6\,B\,a\,e+2\,B\,b\,d\right )}{9}+\frac {B\,b^6\,e^2\,x^{10}}{10} \]
x^4*((B*a^6*e^2)/4 + (3*A*a^5*b*e^2)/2 + 5*A*a^3*b^3*d^2 + (15*B*a^4*b^2*d ^2)/4 + 3*B*a^5*b*d*e + (15*A*a^4*b^2*d*e)/2) + x^7*((A*b^6*d^2)/7 + (6*B* a*b^5*d^2)/7 + (15*A*a^2*b^4*e^2)/7 + (20*B*a^3*b^3*e^2)/7 + (12*A*a*b^5*d *e)/7 + (30*B*a^2*b^4*d*e)/7) + x^5*((6*B*a^5*b*e^2)/5 + 3*A*a^2*b^4*d^2 + 3*A*a^4*b^2*e^2 + 4*B*a^3*b^3*d^2 + 8*A*a^3*b^3*d*e + 6*B*a^4*b^2*d*e) + x^6*(A*a*b^5*d^2 + (10*A*a^3*b^3*e^2)/3 + (5*B*a^2*b^4*d^2)/2 + (5*B*a^4*b ^2*e^2)/2 + 5*A*a^2*b^4*d*e + (20*B*a^3*b^3*d*e)/3) + x^3*((A*a^6*e^2)/3 + (2*B*a^6*d*e)/3 + 2*B*a^5*b*d^2 + 5*A*a^4*b^2*d^2 + 4*A*a^5*b*d*e) + x^8* ((B*b^6*d^2)/8 + (A*b^6*d*e)/4 + (3*A*a*b^5*e^2)/4 + (15*B*a^2*b^4*e^2)/8 + (3*B*a*b^5*d*e)/2) + A*a^6*d^2*x + (a^5*d*x^2*(2*A*a*e + 6*A*b*d + B*a*d ))/2 + (b^5*e*x^9*(A*b*e + 6*B*a*e + 2*B*b*d))/9 + (B*b^6*e^2*x^10)/10