Integrand size = 18, antiderivative size = 75 \[ \int (a+b x)^{10} (A+B x) (d+e x) \, dx=\frac {(A b-a B) (b d-a e) (a+b x)^{11}}{11 b^3}+\frac {(b B d+A b e-2 a B e) (a+b x)^{12}}{12 b^3}+\frac {B e (a+b x)^{13}}{13 b^3} \] Output:
1/11*(A*b-B*a)*(-a*e+b*d)*(b*x+a)^11/b^3+1/12*(A*b*e-2*B*a*e+B*b*d)*(b*x+a )^12/b^3+1/13*B*e*(b*x+a)^13/b^3
Leaf count is larger than twice the leaf count of optimal. \(383\) vs. \(2(75)=150\).
Time = 0.11 (sec) , antiderivative size = 383, normalized size of antiderivative = 5.11 \[ \int (a+b x)^{10} (A+B x) (d+e x) \, dx=\frac {1}{66} a b^9 x^{10} \left (66 A d+60 B d x+60 A e x+55 B e x^2\right )+\frac {1}{22} a^2 b^8 x^9 \left (110 A d+99 B d x+99 A e x+90 B e x^2\right )+\frac {1}{6} a^{10} x (3 A (2 d+e x)+B x (3 d+2 e x))+\frac {3}{4} a^8 b^2 x^3 (5 A (4 d+3 e x)+3 B x (5 d+4 e x))+\frac {5}{6} a^9 b x^2 (B x (4 d+3 e x)+A (6 d+4 e x))+2 a^7 b^3 x^4 (3 A (5 d+4 e x)+2 B x (6 d+5 e x))+a^6 b^4 x^5 (7 A (6 d+5 e x)+5 B x (7 d+6 e x))+\frac {3}{2} a^5 b^5 x^6 (4 A (7 d+6 e x)+3 B x (8 d+7 e x))+\frac {5}{12} a^4 b^6 x^7 (9 A (8 d+7 e x)+7 B x (9 d+8 e x))+\frac {1}{3} a^3 b^7 x^8 (5 A (9 d+8 e x)+4 B x (10 d+9 e x))+\frac {b^{10} x^{11} (13 A (12 d+11 e x)+11 B x (13 d+12 e x))}{1716} \] Input:
Integrate[(a + b*x)^10*(A + B*x)*(d + e*x),x]
Output:
(a*b^9*x^10*(66*A*d + 60*B*d*x + 60*A*e*x + 55*B*e*x^2))/66 + (a^2*b^8*x^9 *(110*A*d + 99*B*d*x + 99*A*e*x + 90*B*e*x^2))/22 + (a^10*x*(3*A*(2*d + e* x) + B*x*(3*d + 2*e*x)))/6 + (3*a^8*b^2*x^3*(5*A*(4*d + 3*e*x) + 3*B*x*(5* d + 4*e*x)))/4 + (5*a^9*b*x^2*(B*x*(4*d + 3*e*x) + A*(6*d + 4*e*x)))/6 + 2 *a^7*b^3*x^4*(3*A*(5*d + 4*e*x) + 2*B*x*(6*d + 5*e*x)) + a^6*b^4*x^5*(7*A* (6*d + 5*e*x) + 5*B*x*(7*d + 6*e*x)) + (3*a^5*b^5*x^6*(4*A*(7*d + 6*e*x) + 3*B*x*(8*d + 7*e*x)))/2 + (5*a^4*b^6*x^7*(9*A*(8*d + 7*e*x) + 7*B*x*(9*d + 8*e*x)))/12 + (a^3*b^7*x^8*(5*A*(9*d + 8*e*x) + 4*B*x*(10*d + 9*e*x)))/3 + (b^10*x^11*(13*A*(12*d + 11*e*x) + 11*B*x*(13*d + 12*e*x)))/1716
Time = 0.50 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, 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.
| \(\displaystyle \int (a+b x)^{10} (A+B x) (d+e x) \, dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {(a+b x)^{11} (-2 a B e+A b e+b B d)}{b^2}+\frac {(a+b x)^{10} (A b-a B) (b d-a e)}{b^2}+\frac {B e (a+b x)^{12}}{b^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(a+b x)^{12} (-2 a B e+A b e+b B d)}{12 b^3}+\frac {(a+b x)^{11} (A b-a B) (b d-a e)}{11 b^3}+\frac {B e (a+b x)^{13}}{13 b^3}\) |
Input:
Int[(a + b*x)^10*(A + B*x)*(d + e*x),x]
Output:
((A*b - a*B)*(b*d - a*e)*(a + b*x)^11)/(11*b^3) + ((b*B*d + A*b*e - 2*a*B* e)*(a + b*x)^12)/(12*b^3) + (B*e*(a + b*x)^13)/(13*b^3)
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. \(458\) vs. \(2(69)=138\).
Time = 0.17 (sec) , antiderivative size = 459, normalized size of antiderivative = 6.12
| method | result | size |
| norman | \(\frac {b^{10} B e \,x^{13}}{13}+\left (\frac {1}{12} A \,b^{10} e +\frac {5}{6} B a \,b^{9} e +\frac {1}{12} b^{10} B d \right ) x^{12}+\left (\frac {10}{11} A a \,b^{9} e +\frac {1}{11} A \,b^{10} d +\frac {45}{11} B \,a^{2} b^{8} e +\frac {10}{11} B a \,b^{9} d \right ) x^{11}+\left (\frac {9}{2} A \,a^{2} b^{8} e +A a \,b^{9} d +12 B \,a^{3} b^{7} e +\frac {9}{2} B \,a^{2} b^{8} d \right ) x^{10}+\left (\frac {40}{3} A \,a^{3} b^{7} e +5 A \,a^{2} b^{8} d +\frac {70}{3} B \,a^{4} b^{6} e +\frac {40}{3} B \,a^{3} b^{7} d \right ) x^{9}+\left (\frac {105}{4} A \,a^{4} b^{6} e +15 A \,a^{3} b^{7} d +\frac {63}{2} B \,a^{5} b^{5} e +\frac {105}{4} B \,a^{4} b^{6} d \right ) x^{8}+\left (36 A \,a^{5} b^{5} e +30 A \,a^{4} b^{6} d +30 B \,a^{6} b^{4} e +36 B \,a^{5} b^{5} d \right ) x^{7}+\left (35 A \,a^{6} b^{4} e +42 A \,a^{5} b^{5} d +20 B \,a^{7} b^{3} e +35 B \,a^{6} b^{4} d \right ) x^{6}+\left (24 A \,a^{7} b^{3} e +42 A \,a^{6} b^{4} d +9 B \,a^{8} b^{2} e +24 B \,a^{7} b^{3} d \right ) x^{5}+\left (\frac {45}{4} A \,a^{8} b^{2} e +30 A \,a^{7} b^{3} d +\frac {5}{2} B \,a^{9} b e +\frac {45}{4} B \,a^{8} b^{2} d \right ) x^{4}+\left (\frac {10}{3} A \,a^{9} b e +15 A \,a^{8} b^{2} d +\frac {1}{3} B \,a^{10} e +\frac {10}{3} B \,a^{9} b d \right ) x^{3}+\left (\frac {1}{2} a^{10} A e +5 A \,a^{9} b d +\frac {1}{2} B \,a^{10} d \right ) x^{2}+a^{10} A d x\) | \(459\) |
| default | \(\frac {b^{10} B e \,x^{13}}{13}+\frac {\left (\left (b^{10} A +10 a \,b^{9} B \right ) e +b^{10} B d \right ) x^{12}}{12}+\frac {\left (\left (10 a \,b^{9} A +45 a^{2} b^{8} B \right ) e +\left (b^{10} A +10 a \,b^{9} B \right ) d \right ) x^{11}}{11}+\frac {\left (\left (45 a^{2} b^{8} A +120 a^{3} b^{7} B \right ) e +\left (10 a \,b^{9} A +45 a^{2} b^{8} B \right ) d \right ) x^{10}}{10}+\frac {\left (\left (120 a^{3} b^{7} A +210 a^{4} b^{6} B \right ) e +\left (45 a^{2} b^{8} A +120 a^{3} b^{7} B \right ) d \right ) x^{9}}{9}+\frac {\left (\left (210 a^{4} b^{6} A +252 a^{5} b^{5} B \right ) e +\left (120 a^{3} b^{7} A +210 a^{4} b^{6} B \right ) d \right ) x^{8}}{8}+\frac {\left (\left (252 a^{5} b^{5} A +210 a^{6} b^{4} B \right ) e +\left (210 a^{4} b^{6} A +252 a^{5} b^{5} B \right ) d \right ) x^{7}}{7}+\frac {\left (\left (210 a^{6} b^{4} A +120 a^{7} b^{3} B \right ) e +\left (252 a^{5} b^{5} A +210 a^{6} b^{4} B \right ) d \right ) x^{6}}{6}+\frac {\left (\left (120 a^{7} b^{3} A +45 a^{8} b^{2} B \right ) e +\left (210 a^{6} b^{4} A +120 a^{7} b^{3} B \right ) d \right ) x^{5}}{5}+\frac {\left (\left (45 a^{8} b^{2} A +10 a^{9} b B \right ) e +\left (120 a^{7} b^{3} A +45 a^{8} b^{2} B \right ) d \right ) x^{4}}{4}+\frac {\left (\left (10 a^{9} b A +a^{10} B \right ) e +\left (45 a^{8} b^{2} A +10 a^{9} b B \right ) d \right ) x^{3}}{3}+\frac {\left (a^{10} A e +\left (10 a^{9} b A +a^{10} B \right ) d \right ) x^{2}}{2}+a^{10} A d x\) | \(485\) |
| orering | \(\frac {x \left (132 b^{10} B e \,x^{12}+143 A \,b^{10} e \,x^{11}+1430 B a \,b^{9} e \,x^{11}+143 B \,b^{10} d \,x^{11}+1560 A a \,b^{9} e \,x^{10}+156 A \,b^{10} d \,x^{10}+7020 B \,a^{2} b^{8} e \,x^{10}+1560 B a \,b^{9} d \,x^{10}+7722 A \,a^{2} b^{8} e \,x^{9}+1716 A a \,b^{9} d \,x^{9}+20592 B \,a^{3} b^{7} e \,x^{9}+7722 B \,a^{2} b^{8} d \,x^{9}+22880 A \,a^{3} b^{7} e \,x^{8}+8580 A \,a^{2} b^{8} d \,x^{8}+40040 B \,a^{4} b^{6} e \,x^{8}+22880 B \,a^{3} b^{7} d \,x^{8}+45045 A \,a^{4} b^{6} e \,x^{7}+25740 A \,a^{3} b^{7} d \,x^{7}+54054 B \,a^{5} b^{5} e \,x^{7}+45045 B \,a^{4} b^{6} d \,x^{7}+61776 A \,a^{5} b^{5} e \,x^{6}+51480 A \,a^{4} b^{6} d \,x^{6}+51480 B \,a^{6} b^{4} e \,x^{6}+61776 B \,a^{5} b^{5} d \,x^{6}+60060 A \,a^{6} b^{4} e \,x^{5}+72072 A \,a^{5} b^{5} d \,x^{5}+34320 B \,a^{7} b^{3} e \,x^{5}+60060 B \,a^{6} b^{4} d \,x^{5}+41184 A \,a^{7} b^{3} e \,x^{4}+72072 A \,a^{6} b^{4} d \,x^{4}+15444 B \,a^{8} b^{2} e \,x^{4}+41184 B \,a^{7} b^{3} d \,x^{4}+19305 A \,a^{8} b^{2} e \,x^{3}+51480 A \,a^{7} b^{3} d \,x^{3}+4290 B \,a^{9} b e \,x^{3}+19305 B \,a^{8} b^{2} d \,x^{3}+5720 A \,a^{9} b e \,x^{2}+25740 A \,a^{8} b^{2} d \,x^{2}+572 B \,a^{10} e \,x^{2}+5720 B \,a^{9} b d \,x^{2}+858 A \,a^{10} e x +8580 A \,a^{9} b d x +858 B \,a^{10} d x +1716 a^{10} A d \right )}{1716}\) | \(528\) |
| gosper | \(\frac {105}{4} x^{8} A \,a^{4} b^{6} e +15 x^{8} A \,a^{3} b^{7} d +\frac {63}{2} x^{8} B \,a^{5} b^{5} e +\frac {9}{2} x^{10} A \,a^{2} b^{8} e +\frac {1}{11} x^{11} A \,b^{10} d +\frac {105}{4} x^{8} B \,a^{4} b^{6} d +\frac {45}{4} x^{4} A \,a^{8} b^{2} e +\frac {1}{2} x^{2} a^{10} A e +\frac {1}{2} x^{2} B \,a^{10} d +x^{10} A a \,b^{9} d +12 x^{10} B \,a^{3} b^{7} e +\frac {9}{2} x^{10} B \,a^{2} b^{8} d +\frac {40}{3} x^{9} A \,a^{3} b^{7} e +5 x^{9} A \,a^{2} b^{8} d +\frac {70}{3} x^{9} B \,a^{4} b^{6} e +\frac {40}{3} x^{9} B \,a^{3} b^{7} d +35 A \,a^{6} b^{4} e \,x^{6}+\frac {45}{4} x^{4} B \,a^{8} b^{2} d +\frac {10}{3} x^{3} A \,a^{9} b e +\frac {5}{6} x^{12} B a \,b^{9} e +\frac {10}{11} x^{11} A a \,b^{9} e +\frac {45}{11} x^{11} B \,a^{2} b^{8} e +\frac {1}{13} b^{10} B e \,x^{13}+a^{10} A d x +\frac {1}{12} x^{12} A \,b^{10} e +\frac {1}{12} x^{12} b^{10} B d +\frac {10}{11} x^{11} B a \,b^{9} d +15 x^{3} A \,a^{8} b^{2} d +24 A \,a^{7} b^{3} e \,x^{5}+42 A \,a^{5} b^{5} d \,x^{6}+20 B \,a^{7} b^{3} e \,x^{6}+35 B \,a^{6} b^{4} d \,x^{6}+\frac {1}{3} x^{3} B \,a^{10} e +\frac {10}{3} x^{3} B \,a^{9} b d +36 B \,a^{5} b^{5} d \,x^{7}+30 B \,a^{6} b^{4} e \,x^{7}+9 B \,a^{8} b^{2} e \,x^{5}+24 B \,a^{7} b^{3} d \,x^{5}+30 x^{4} A \,a^{7} b^{3} d +5 x^{2} A \,a^{9} b d +36 A \,a^{5} b^{5} e \,x^{7}+30 A \,a^{4} b^{6} d \,x^{7}+\frac {5}{2} x^{4} B \,a^{9} b e +42 A \,a^{6} b^{4} d \,x^{5}\) | \(530\) |
| risch | \(\frac {105}{4} x^{8} A \,a^{4} b^{6} e +15 x^{8} A \,a^{3} b^{7} d +\frac {63}{2} x^{8} B \,a^{5} b^{5} e +\frac {9}{2} x^{10} A \,a^{2} b^{8} e +\frac {1}{11} x^{11} A \,b^{10} d +\frac {105}{4} x^{8} B \,a^{4} b^{6} d +\frac {45}{4} x^{4} A \,a^{8} b^{2} e +\frac {1}{2} x^{2} a^{10} A e +\frac {1}{2} x^{2} B \,a^{10} d +x^{10} A a \,b^{9} d +12 x^{10} B \,a^{3} b^{7} e +\frac {9}{2} x^{10} B \,a^{2} b^{8} d +\frac {40}{3} x^{9} A \,a^{3} b^{7} e +5 x^{9} A \,a^{2} b^{8} d +\frac {70}{3} x^{9} B \,a^{4} b^{6} e +\frac {40}{3} x^{9} B \,a^{3} b^{7} d +35 A \,a^{6} b^{4} e \,x^{6}+\frac {45}{4} x^{4} B \,a^{8} b^{2} d +\frac {10}{3} x^{3} A \,a^{9} b e +\frac {5}{6} x^{12} B a \,b^{9} e +\frac {10}{11} x^{11} A a \,b^{9} e +\frac {45}{11} x^{11} B \,a^{2} b^{8} e +\frac {1}{13} b^{10} B e \,x^{13}+a^{10} A d x +\frac {1}{12} x^{12} A \,b^{10} e +\frac {1}{12} x^{12} b^{10} B d +\frac {10}{11} x^{11} B a \,b^{9} d +15 x^{3} A \,a^{8} b^{2} d +24 A \,a^{7} b^{3} e \,x^{5}+42 A \,a^{5} b^{5} d \,x^{6}+20 B \,a^{7} b^{3} e \,x^{6}+35 B \,a^{6} b^{4} d \,x^{6}+\frac {1}{3} x^{3} B \,a^{10} e +\frac {10}{3} x^{3} B \,a^{9} b d +36 B \,a^{5} b^{5} d \,x^{7}+30 B \,a^{6} b^{4} e \,x^{7}+9 B \,a^{8} b^{2} e \,x^{5}+24 B \,a^{7} b^{3} d \,x^{5}+30 x^{4} A \,a^{7} b^{3} d +5 x^{2} A \,a^{9} b d +36 A \,a^{5} b^{5} e \,x^{7}+30 A \,a^{4} b^{6} d \,x^{7}+\frac {5}{2} x^{4} B \,a^{9} b e +42 A \,a^{6} b^{4} d \,x^{5}\) | \(530\) |
| parallelrisch | \(\frac {105}{4} x^{8} A \,a^{4} b^{6} e +15 x^{8} A \,a^{3} b^{7} d +\frac {63}{2} x^{8} B \,a^{5} b^{5} e +\frac {9}{2} x^{10} A \,a^{2} b^{8} e +\frac {1}{11} x^{11} A \,b^{10} d +\frac {105}{4} x^{8} B \,a^{4} b^{6} d +\frac {45}{4} x^{4} A \,a^{8} b^{2} e +\frac {1}{2} x^{2} a^{10} A e +\frac {1}{2} x^{2} B \,a^{10} d +x^{10} A a \,b^{9} d +12 x^{10} B \,a^{3} b^{7} e +\frac {9}{2} x^{10} B \,a^{2} b^{8} d +\frac {40}{3} x^{9} A \,a^{3} b^{7} e +5 x^{9} A \,a^{2} b^{8} d +\frac {70}{3} x^{9} B \,a^{4} b^{6} e +\frac {40}{3} x^{9} B \,a^{3} b^{7} d +35 A \,a^{6} b^{4} e \,x^{6}+\frac {45}{4} x^{4} B \,a^{8} b^{2} d +\frac {10}{3} x^{3} A \,a^{9} b e +\frac {5}{6} x^{12} B a \,b^{9} e +\frac {10}{11} x^{11} A a \,b^{9} e +\frac {45}{11} x^{11} B \,a^{2} b^{8} e +\frac {1}{13} b^{10} B e \,x^{13}+a^{10} A d x +\frac {1}{12} x^{12} A \,b^{10} e +\frac {1}{12} x^{12} b^{10} B d +\frac {10}{11} x^{11} B a \,b^{9} d +15 x^{3} A \,a^{8} b^{2} d +24 A \,a^{7} b^{3} e \,x^{5}+42 A \,a^{5} b^{5} d \,x^{6}+20 B \,a^{7} b^{3} e \,x^{6}+35 B \,a^{6} b^{4} d \,x^{6}+\frac {1}{3} x^{3} B \,a^{10} e +\frac {10}{3} x^{3} B \,a^{9} b d +36 B \,a^{5} b^{5} d \,x^{7}+30 B \,a^{6} b^{4} e \,x^{7}+9 B \,a^{8} b^{2} e \,x^{5}+24 B \,a^{7} b^{3} d \,x^{5}+30 x^{4} A \,a^{7} b^{3} d +5 x^{2} A \,a^{9} b d +36 A \,a^{5} b^{5} e \,x^{7}+30 A \,a^{4} b^{6} d \,x^{7}+\frac {5}{2} x^{4} B \,a^{9} b e +42 A \,a^{6} b^{4} d \,x^{5}\) | \(530\) |
Input:
int((b*x+a)^10*(B*x+A)*(e*x+d),x,method=_RETURNVERBOSE)
Output:
1/13*b^10*B*e*x^13+(1/12*A*b^10*e+5/6*B*a*b^9*e+1/12*b^10*B*d)*x^12+(10/11 *A*a*b^9*e+1/11*A*b^10*d+45/11*B*a^2*b^8*e+10/11*B*a*b^9*d)*x^11+(9/2*A*a^ 2*b^8*e+A*a*b^9*d+12*B*a^3*b^7*e+9/2*B*a^2*b^8*d)*x^10+(40/3*A*a^3*b^7*e+5 *A*a^2*b^8*d+70/3*B*a^4*b^6*e+40/3*B*a^3*b^7*d)*x^9+(105/4*A*a^4*b^6*e+15* A*a^3*b^7*d+63/2*B*a^5*b^5*e+105/4*B*a^4*b^6*d)*x^8+(36*A*a^5*b^5*e+30*A*a ^4*b^6*d+30*B*a^6*b^4*e+36*B*a^5*b^5*d)*x^7+(35*A*a^6*b^4*e+42*A*a^5*b^5*d +20*B*a^7*b^3*e+35*B*a^6*b^4*d)*x^6+(24*A*a^7*b^3*e+42*A*a^6*b^4*d+9*B*a^8 *b^2*e+24*B*a^7*b^3*d)*x^5+(45/4*A*a^8*b^2*e+30*A*a^7*b^3*d+5/2*B*a^9*b*e+ 45/4*B*a^8*b^2*d)*x^4+(10/3*A*a^9*b*e+15*A*a^8*b^2*d+1/3*B*a^10*e+10/3*B*a ^9*b*d)*x^3+(1/2*a^10*A*e+5*A*a^9*b*d+1/2*B*a^10*d)*x^2+a^10*A*d*x
Leaf count of result is larger than twice the leaf count of optimal. 493 vs. \(2 (69) = 138\).
Time = 0.07 (sec) , antiderivative size = 493, normalized size of antiderivative = 6.57 \[ \int (a+b x)^{10} (A+B x) (d+e x) \, dx=\frac {1}{13} \, B b^{10} e x^{13} + A a^{10} d x + \frac {1}{12} \, {\left (B b^{10} d + {\left (10 \, B a b^{9} + A b^{10}\right )} e\right )} x^{12} + \frac {1}{11} \, {\left ({\left (10 \, B a b^{9} + A b^{10}\right )} d + 5 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} e\right )} x^{11} + \frac {1}{2} \, {\left ({\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} d + 3 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} e\right )} x^{10} + \frac {5}{3} \, {\left ({\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} d + 2 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} e\right )} x^{9} + \frac {3}{4} \, {\left (5 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} d + 7 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} e\right )} x^{8} + 6 \, {\left ({\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} d + {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} e\right )} x^{7} + {\left (7 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} d + 5 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} e\right )} x^{6} + 3 \, {\left (2 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} d + {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} e\right )} x^{5} + \frac {5}{4} \, {\left (3 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} d + {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} e\right )} x^{4} + \frac {1}{3} \, {\left (5 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} d + {\left (B a^{10} + 10 \, A a^{9} b\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (A a^{10} e + {\left (B a^{10} + 10 \, A a^{9} b\right )} d\right )} x^{2} \] Input:
integrate((b*x+a)^10*(B*x+A)*(e*x+d),x, algorithm="fricas")
Output:
1/13*B*b^10*e*x^13 + A*a^10*d*x + 1/12*(B*b^10*d + (10*B*a*b^9 + A*b^10)*e )*x^12 + 1/11*((10*B*a*b^9 + A*b^10)*d + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*e)*x^ 11 + 1/2*((9*B*a^2*b^8 + 2*A*a*b^9)*d + 3*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e)*x ^10 + 5/3*((8*B*a^3*b^7 + 3*A*a^2*b^8)*d + 2*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e )*x^9 + 3/4*(5*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d + 7*(6*B*a^5*b^5 + 5*A*a^4*b^ 6)*e)*x^8 + 6*((6*B*a^5*b^5 + 5*A*a^4*b^6)*d + (5*B*a^6*b^4 + 6*A*a^5*b^5) *e)*x^7 + (7*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d + 5*(4*B*a^7*b^3 + 7*A*a^6*b^4) *e)*x^6 + 3*(2*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d + (3*B*a^8*b^2 + 8*A*a^7*b^3) *e)*x^5 + 5/4*(3*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d + (2*B*a^9*b + 9*A*a^8*b^2) *e)*x^4 + 1/3*(5*(2*B*a^9*b + 9*A*a^8*b^2)*d + (B*a^10 + 10*A*a^9*b)*e)*x^ 3 + 1/2*(A*a^10*e + (B*a^10 + 10*A*a^9*b)*d)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 549 vs. \(2 (71) = 142\).
Time = 0.07 (sec) , antiderivative size = 549, normalized size of antiderivative = 7.32 \[ \int (a+b x)^{10} (A+B x) (d+e x) \, dx=A a^{10} d x + \frac {B b^{10} e x^{13}}{13} + x^{12} \left (\frac {A b^{10} e}{12} + \frac {5 B a b^{9} e}{6} + \frac {B b^{10} d}{12}\right ) + x^{11} \cdot \left (\frac {10 A a b^{9} e}{11} + \frac {A b^{10} d}{11} + \frac {45 B a^{2} b^{8} e}{11} + \frac {10 B a b^{9} d}{11}\right ) + x^{10} \cdot \left (\frac {9 A a^{2} b^{8} e}{2} + A a b^{9} d + 12 B a^{3} b^{7} e + \frac {9 B a^{2} b^{8} d}{2}\right ) + x^{9} \cdot \left (\frac {40 A a^{3} b^{7} e}{3} + 5 A a^{2} b^{8} d + \frac {70 B a^{4} b^{6} e}{3} + \frac {40 B a^{3} b^{7} d}{3}\right ) + x^{8} \cdot \left (\frac {105 A a^{4} b^{6} e}{4} + 15 A a^{3} b^{7} d + \frac {63 B a^{5} b^{5} e}{2} + \frac {105 B a^{4} b^{6} d}{4}\right ) + x^{7} \cdot \left (36 A a^{5} b^{5} e + 30 A a^{4} b^{6} d + 30 B a^{6} b^{4} e + 36 B a^{5} b^{5} d\right ) + x^{6} \cdot \left (35 A a^{6} b^{4} e + 42 A a^{5} b^{5} d + 20 B a^{7} b^{3} e + 35 B a^{6} b^{4} d\right ) + x^{5} \cdot \left (24 A a^{7} b^{3} e + 42 A a^{6} b^{4} d + 9 B a^{8} b^{2} e + 24 B a^{7} b^{3} d\right ) + x^{4} \cdot \left (\frac {45 A a^{8} b^{2} e}{4} + 30 A a^{7} b^{3} d + \frac {5 B a^{9} b e}{2} + \frac {45 B a^{8} b^{2} d}{4}\right ) + x^{3} \cdot \left (\frac {10 A a^{9} b e}{3} + 15 A a^{8} b^{2} d + \frac {B a^{10} e}{3} + \frac {10 B a^{9} b d}{3}\right ) + x^{2} \left (\frac {A a^{10} e}{2} + 5 A a^{9} b d + \frac {B a^{10} d}{2}\right ) \] Input:
integrate((b*x+a)**10*(B*x+A)*(e*x+d),x)
Output:
A*a**10*d*x + B*b**10*e*x**13/13 + x**12*(A*b**10*e/12 + 5*B*a*b**9*e/6 + B*b**10*d/12) + x**11*(10*A*a*b**9*e/11 + A*b**10*d/11 + 45*B*a**2*b**8*e/ 11 + 10*B*a*b**9*d/11) + x**10*(9*A*a**2*b**8*e/2 + A*a*b**9*d + 12*B*a**3 *b**7*e + 9*B*a**2*b**8*d/2) + x**9*(40*A*a**3*b**7*e/3 + 5*A*a**2*b**8*d + 70*B*a**4*b**6*e/3 + 40*B*a**3*b**7*d/3) + x**8*(105*A*a**4*b**6*e/4 + 1 5*A*a**3*b**7*d + 63*B*a**5*b**5*e/2 + 105*B*a**4*b**6*d/4) + x**7*(36*A*a **5*b**5*e + 30*A*a**4*b**6*d + 30*B*a**6*b**4*e + 36*B*a**5*b**5*d) + x** 6*(35*A*a**6*b**4*e + 42*A*a**5*b**5*d + 20*B*a**7*b**3*e + 35*B*a**6*b**4 *d) + x**5*(24*A*a**7*b**3*e + 42*A*a**6*b**4*d + 9*B*a**8*b**2*e + 24*B*a **7*b**3*d) + x**4*(45*A*a**8*b**2*e/4 + 30*A*a**7*b**3*d + 5*B*a**9*b*e/2 + 45*B*a**8*b**2*d/4) + x**3*(10*A*a**9*b*e/3 + 15*A*a**8*b**2*d + B*a**1 0*e/3 + 10*B*a**9*b*d/3) + x**2*(A*a**10*e/2 + 5*A*a**9*b*d + B*a**10*d/2)
Leaf count of result is larger than twice the leaf count of optimal. 493 vs. \(2 (69) = 138\).
Time = 0.04 (sec) , antiderivative size = 493, normalized size of antiderivative = 6.57 \[ \int (a+b x)^{10} (A+B x) (d+e x) \, dx=\frac {1}{13} \, B b^{10} e x^{13} + A a^{10} d x + \frac {1}{12} \, {\left (B b^{10} d + {\left (10 \, B a b^{9} + A b^{10}\right )} e\right )} x^{12} + \frac {1}{11} \, {\left ({\left (10 \, B a b^{9} + A b^{10}\right )} d + 5 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} e\right )} x^{11} + \frac {1}{2} \, {\left ({\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} d + 3 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} e\right )} x^{10} + \frac {5}{3} \, {\left ({\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} d + 2 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} e\right )} x^{9} + \frac {3}{4} \, {\left (5 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} d + 7 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} e\right )} x^{8} + 6 \, {\left ({\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} d + {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} e\right )} x^{7} + {\left (7 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} d + 5 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} e\right )} x^{6} + 3 \, {\left (2 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} d + {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} e\right )} x^{5} + \frac {5}{4} \, {\left (3 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} d + {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} e\right )} x^{4} + \frac {1}{3} \, {\left (5 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} d + {\left (B a^{10} + 10 \, A a^{9} b\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (A a^{10} e + {\left (B a^{10} + 10 \, A a^{9} b\right )} d\right )} x^{2} \] Input:
integrate((b*x+a)^10*(B*x+A)*(e*x+d),x, algorithm="maxima")
Output:
1/13*B*b^10*e*x^13 + A*a^10*d*x + 1/12*(B*b^10*d + (10*B*a*b^9 + A*b^10)*e )*x^12 + 1/11*((10*B*a*b^9 + A*b^10)*d + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*e)*x^ 11 + 1/2*((9*B*a^2*b^8 + 2*A*a*b^9)*d + 3*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e)*x ^10 + 5/3*((8*B*a^3*b^7 + 3*A*a^2*b^8)*d + 2*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e )*x^9 + 3/4*(5*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d + 7*(6*B*a^5*b^5 + 5*A*a^4*b^ 6)*e)*x^8 + 6*((6*B*a^5*b^5 + 5*A*a^4*b^6)*d + (5*B*a^6*b^4 + 6*A*a^5*b^5) *e)*x^7 + (7*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d + 5*(4*B*a^7*b^3 + 7*A*a^6*b^4) *e)*x^6 + 3*(2*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d + (3*B*a^8*b^2 + 8*A*a^7*b^3) *e)*x^5 + 5/4*(3*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d + (2*B*a^9*b + 9*A*a^8*b^2) *e)*x^4 + 1/3*(5*(2*B*a^9*b + 9*A*a^8*b^2)*d + (B*a^10 + 10*A*a^9*b)*e)*x^ 3 + 1/2*(A*a^10*e + (B*a^10 + 10*A*a^9*b)*d)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 529 vs. \(2 (69) = 138\).
Time = 0.12 (sec) , antiderivative size = 529, normalized size of antiderivative = 7.05 \[ \int (a+b x)^{10} (A+B x) (d+e x) \, dx=\frac {1}{13} \, B b^{10} e x^{13} + \frac {1}{12} \, B b^{10} d x^{12} + \frac {5}{6} \, B a b^{9} e x^{12} + \frac {1}{12} \, A b^{10} e x^{12} + \frac {10}{11} \, B a b^{9} d x^{11} + \frac {1}{11} \, A b^{10} d x^{11} + \frac {45}{11} \, B a^{2} b^{8} e x^{11} + \frac {10}{11} \, A a b^{9} e x^{11} + \frac {9}{2} \, B a^{2} b^{8} d x^{10} + A a b^{9} d x^{10} + 12 \, B a^{3} b^{7} e x^{10} + \frac {9}{2} \, A a^{2} b^{8} e x^{10} + \frac {40}{3} \, B a^{3} b^{7} d x^{9} + 5 \, A a^{2} b^{8} d x^{9} + \frac {70}{3} \, B a^{4} b^{6} e x^{9} + \frac {40}{3} \, A a^{3} b^{7} e x^{9} + \frac {105}{4} \, B a^{4} b^{6} d x^{8} + 15 \, A a^{3} b^{7} d x^{8} + \frac {63}{2} \, B a^{5} b^{5} e x^{8} + \frac {105}{4} \, A a^{4} b^{6} e x^{8} + 36 \, B a^{5} b^{5} d x^{7} + 30 \, A a^{4} b^{6} d x^{7} + 30 \, B a^{6} b^{4} e x^{7} + 36 \, A a^{5} b^{5} e x^{7} + 35 \, B a^{6} b^{4} d x^{6} + 42 \, A a^{5} b^{5} d x^{6} + 20 \, B a^{7} b^{3} e x^{6} + 35 \, A a^{6} b^{4} e x^{6} + 24 \, B a^{7} b^{3} d x^{5} + 42 \, A a^{6} b^{4} d x^{5} + 9 \, B a^{8} b^{2} e x^{5} + 24 \, A a^{7} b^{3} e x^{5} + \frac {45}{4} \, B a^{8} b^{2} d x^{4} + 30 \, A a^{7} b^{3} d x^{4} + \frac {5}{2} \, B a^{9} b e x^{4} + \frac {45}{4} \, A a^{8} b^{2} e x^{4} + \frac {10}{3} \, B a^{9} b d x^{3} + 15 \, A a^{8} b^{2} d x^{3} + \frac {1}{3} \, B a^{10} e x^{3} + \frac {10}{3} \, A a^{9} b e x^{3} + \frac {1}{2} \, B a^{10} d x^{2} + 5 \, A a^{9} b d x^{2} + \frac {1}{2} \, A a^{10} e x^{2} + A a^{10} d x \] Input:
integrate((b*x+a)^10*(B*x+A)*(e*x+d),x, algorithm="giac")
Output:
1/13*B*b^10*e*x^13 + 1/12*B*b^10*d*x^12 + 5/6*B*a*b^9*e*x^12 + 1/12*A*b^10 *e*x^12 + 10/11*B*a*b^9*d*x^11 + 1/11*A*b^10*d*x^11 + 45/11*B*a^2*b^8*e*x^ 11 + 10/11*A*a*b^9*e*x^11 + 9/2*B*a^2*b^8*d*x^10 + A*a*b^9*d*x^10 + 12*B*a ^3*b^7*e*x^10 + 9/2*A*a^2*b^8*e*x^10 + 40/3*B*a^3*b^7*d*x^9 + 5*A*a^2*b^8* d*x^9 + 70/3*B*a^4*b^6*e*x^9 + 40/3*A*a^3*b^7*e*x^9 + 105/4*B*a^4*b^6*d*x^ 8 + 15*A*a^3*b^7*d*x^8 + 63/2*B*a^5*b^5*e*x^8 + 105/4*A*a^4*b^6*e*x^8 + 36 *B*a^5*b^5*d*x^7 + 30*A*a^4*b^6*d*x^7 + 30*B*a^6*b^4*e*x^7 + 36*A*a^5*b^5* e*x^7 + 35*B*a^6*b^4*d*x^6 + 42*A*a^5*b^5*d*x^6 + 20*B*a^7*b^3*e*x^6 + 35* A*a^6*b^4*e*x^6 + 24*B*a^7*b^3*d*x^5 + 42*A*a^6*b^4*d*x^5 + 9*B*a^8*b^2*e* x^5 + 24*A*a^7*b^3*e*x^5 + 45/4*B*a^8*b^2*d*x^4 + 30*A*a^7*b^3*d*x^4 + 5/2 *B*a^9*b*e*x^4 + 45/4*A*a^8*b^2*e*x^4 + 10/3*B*a^9*b*d*x^3 + 15*A*a^8*b^2* d*x^3 + 1/3*B*a^10*e*x^3 + 10/3*A*a^9*b*e*x^3 + 1/2*B*a^10*d*x^2 + 5*A*a^9 *b*d*x^2 + 1/2*A*a^10*e*x^2 + A*a^10*d*x
Time = 0.15 (sec) , antiderivative size = 409, normalized size of antiderivative = 5.45 \[ \int (a+b x)^{10} (A+B x) (d+e x) \, dx=x^3\,\left (\frac {B\,a^{10}\,e}{3}+\frac {10\,A\,a^9\,b\,e}{3}+\frac {10\,B\,a^9\,b\,d}{3}+15\,A\,a^8\,b^2\,d\right )+x^{11}\,\left (\frac {A\,b^{10}\,d}{11}+\frac {10\,A\,a\,b^9\,e}{11}+\frac {10\,B\,a\,b^9\,d}{11}+\frac {45\,B\,a^2\,b^8\,e}{11}\right )+x^2\,\left (\frac {A\,a^{10}\,e}{2}+\frac {B\,a^{10}\,d}{2}+5\,A\,a^9\,b\,d\right )+x^{12}\,\left (\frac {A\,b^{10}\,e}{12}+\frac {B\,b^{10}\,d}{12}+\frac {5\,B\,a\,b^9\,e}{6}\right )+6\,a^4\,b^4\,x^7\,\left (5\,A\,b^2\,d+5\,B\,a^2\,e+6\,A\,a\,b\,e+6\,B\,a\,b\,d\right )+3\,a^6\,b^2\,x^5\,\left (14\,A\,b^2\,d+3\,B\,a^2\,e+8\,A\,a\,b\,e+8\,B\,a\,b\,d\right )+\frac {5\,a^2\,b^6\,x^9\,\left (3\,A\,b^2\,d+14\,B\,a^2\,e+8\,A\,a\,b\,e+8\,B\,a\,b\,d\right )}{3}+a^5\,b^3\,x^6\,\left (42\,A\,b^2\,d+20\,B\,a^2\,e+35\,A\,a\,b\,e+35\,B\,a\,b\,d\right )+\frac {3\,a^3\,b^5\,x^8\,\left (20\,A\,b^2\,d+42\,B\,a^2\,e+35\,A\,a\,b\,e+35\,B\,a\,b\,d\right )}{4}+A\,a^{10}\,d\,x+\frac {B\,b^{10}\,e\,x^{13}}{13}+\frac {5\,a^7\,b\,x^4\,\left (24\,A\,b^2\,d+2\,B\,a^2\,e+9\,A\,a\,b\,e+9\,B\,a\,b\,d\right )}{4}+\frac {a\,b^7\,x^{10}\,\left (2\,A\,b^2\,d+24\,B\,a^2\,e+9\,A\,a\,b\,e+9\,B\,a\,b\,d\right )}{2} \] Input:
int((A + B*x)*(a + b*x)^10*(d + e*x),x)
Output:
x^3*((B*a^10*e)/3 + (10*A*a^9*b*e)/3 + (10*B*a^9*b*d)/3 + 15*A*a^8*b^2*d) + x^11*((A*b^10*d)/11 + (10*A*a*b^9*e)/11 + (10*B*a*b^9*d)/11 + (45*B*a^2* b^8*e)/11) + x^2*((A*a^10*e)/2 + (B*a^10*d)/2 + 5*A*a^9*b*d) + x^12*((A*b^ 10*e)/12 + (B*b^10*d)/12 + (5*B*a*b^9*e)/6) + 6*a^4*b^4*x^7*(5*A*b^2*d + 5 *B*a^2*e + 6*A*a*b*e + 6*B*a*b*d) + 3*a^6*b^2*x^5*(14*A*b^2*d + 3*B*a^2*e + 8*A*a*b*e + 8*B*a*b*d) + (5*a^2*b^6*x^9*(3*A*b^2*d + 14*B*a^2*e + 8*A*a* b*e + 8*B*a*b*d))/3 + a^5*b^3*x^6*(42*A*b^2*d + 20*B*a^2*e + 35*A*a*b*e + 35*B*a*b*d) + (3*a^3*b^5*x^8*(20*A*b^2*d + 42*B*a^2*e + 35*A*a*b*e + 35*B* a*b*d))/4 + A*a^10*d*x + (B*b^10*e*x^13)/13 + (5*a^7*b*x^4*(24*A*b^2*d + 2 *B*a^2*e + 9*A*a*b*e + 9*B*a*b*d))/4 + (a*b^7*x^10*(2*A*b^2*d + 24*B*a^2*e + 9*A*a*b*e + 9*B*a*b*d))/2
Time = 0.17 (sec) , antiderivative size = 265, normalized size of antiderivative = 3.53 \[ \int (a+b x)^{10} (A+B x) (d+e x) \, dx=\frac {x \left (12 b^{11} e \,x^{12}+143 a \,b^{10} e \,x^{11}+13 b^{11} d \,x^{11}+780 a^{2} b^{9} e \,x^{10}+156 a \,b^{10} d \,x^{10}+2574 a^{3} b^{8} e \,x^{9}+858 a^{2} b^{9} d \,x^{9}+5720 a^{4} b^{7} e \,x^{8}+2860 a^{3} b^{8} d \,x^{8}+9009 a^{5} b^{6} e \,x^{7}+6435 a^{4} b^{7} d \,x^{7}+10296 a^{6} b^{5} e \,x^{6}+10296 a^{5} b^{6} d \,x^{6}+8580 a^{7} b^{4} e \,x^{5}+12012 a^{6} b^{5} d \,x^{5}+5148 a^{8} b^{3} e \,x^{4}+10296 a^{7} b^{4} d \,x^{4}+2145 a^{9} b^{2} e \,x^{3}+6435 a^{8} b^{3} d \,x^{3}+572 a^{10} b e \,x^{2}+2860 a^{9} b^{2} d \,x^{2}+78 a^{11} e x +858 a^{10} b d x +156 a^{11} d \right )}{156} \] Input:
int((b*x+a)^10*(B*x+A)*(e*x+d),x)
Output:
(x*(156*a**11*d + 78*a**11*e*x + 858*a**10*b*d*x + 572*a**10*b*e*x**2 + 28 60*a**9*b**2*d*x**2 + 2145*a**9*b**2*e*x**3 + 6435*a**8*b**3*d*x**3 + 5148 *a**8*b**3*e*x**4 + 10296*a**7*b**4*d*x**4 + 8580*a**7*b**4*e*x**5 + 12012 *a**6*b**5*d*x**5 + 10296*a**6*b**5*e*x**6 + 10296*a**5*b**6*d*x**6 + 9009 *a**5*b**6*e*x**7 + 6435*a**4*b**7*d*x**7 + 5720*a**4*b**7*e*x**8 + 2860*a **3*b**8*d*x**8 + 2574*a**3*b**8*e*x**9 + 858*a**2*b**9*d*x**9 + 780*a**2* b**9*e*x**10 + 156*a*b**10*d*x**10 + 143*a*b**10*e*x**11 + 13*b**11*d*x**1 1 + 12*b**11*e*x**12))/156