Integrand size = 20, antiderivative size = 118 \[ \int (a+b x)^{10} (A+B x) (d+e x)^2 \, dx=\frac {(A b-a B) (b d-a e)^2 (a+b x)^{11}}{11 b^4}+\frac {(b d-a e) (b B d+2 A b e-3 a B e) (a+b x)^{12}}{12 b^4}+\frac {e (2 b B d+A b e-3 a B e) (a+b x)^{13}}{13 b^4}+\frac {B e^2 (a+b x)^{14}}{14 b^4} \] Output:
1/11*(A*b-B*a)*(-a*e+b*d)^2*(b*x+a)^11/b^4+1/12*(-a*e+b*d)*(2*A*b*e-3*B*a* e+B*b*d)*(b*x+a)^12/b^4+1/13*e*(A*b*e-3*B*a*e+2*B*b*d)*(b*x+a)^13/b^4+1/14 *B*e^2*(b*x+a)^14/b^4
Leaf count is larger than twice the leaf count of optimal. \(614\) vs. \(2(118)=236\).
Time = 0.23 (sec) , antiderivative size = 614, normalized size of antiderivative = 5.20 \[ \int (a+b x)^{10} (A+B x) (d+e x)^2 \, dx=\frac {x \left (1001 a^{10} \left (4 A \left (3 d^2+3 d e x+e^2 x^2\right )+B x \left (6 d^2+8 d e x+3 e^2 x^2\right )\right )+2002 a^9 b x \left (5 A \left (6 d^2+8 d e x+3 e^2 x^2\right )+2 B x \left (10 d^2+15 d e x+6 e^2 x^2\right )\right )+9009 a^8 b^2 x^2 \left (2 A \left (10 d^2+15 d e x+6 e^2 x^2\right )+B x \left (15 d^2+24 d e x+10 e^2 x^2\right )\right )+3432 a^7 b^3 x^3 \left (7 A \left (15 d^2+24 d e x+10 e^2 x^2\right )+4 B x \left (21 d^2+35 d e x+15 e^2 x^2\right )\right )+3003 a^6 b^4 x^4 \left (8 A \left (21 d^2+35 d e x+15 e^2 x^2\right )+5 B x \left (28 d^2+48 d e x+21 e^2 x^2\right )\right )+6006 a^5 b^5 x^5 \left (3 A \left (28 d^2+48 d e x+21 e^2 x^2\right )+2 B x \left (36 d^2+63 d e x+28 e^2 x^2\right )\right )+1001 a^4 b^6 x^6 \left (10 A \left (36 d^2+63 d e x+28 e^2 x^2\right )+7 B x \left (45 d^2+80 d e x+36 e^2 x^2\right )\right )+364 a^3 b^7 x^7 \left (11 A \left (45 d^2+80 d e x+36 e^2 x^2\right )+8 B x \left (55 d^2+99 d e x+45 e^2 x^2\right )\right )+273 a^2 b^8 x^8 \left (4 A \left (55 d^2+99 d e x+45 e^2 x^2\right )+3 B x \left (66 d^2+120 d e x+55 e^2 x^2\right )\right )+14 a b^9 x^9 \left (13 A \left (66 d^2+120 d e x+55 e^2 x^2\right )+10 B x \left (78 d^2+143 d e x+66 e^2 x^2\right )\right )+b^{10} x^{10} \left (14 A \left (78 d^2+143 d e x+66 e^2 x^2\right )+11 B x \left (91 d^2+168 d e x+78 e^2 x^2\right )\right )\right )}{12012} \] Input:
Integrate[(a + b*x)^10*(A + B*x)*(d + e*x)^2,x]
Output:
(x*(1001*a^10*(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)) + 2002*a^9*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)) + 9009*a^8*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)) + 3432*a^7*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)) + 3003*a^6*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)) + 6006*a^5*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)) + 1001*a^4*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) ) + 364*a^3*b^7*x^7*(11*A*(45*d^2 + 80*d*e*x + 36*e^2*x^2) + 8*B*x*(55*d^2 + 99*d*e*x + 45*e^2*x^2)) + 273*a^2*b^8*x^8*(4*A*(55*d^2 + 99*d*e*x + 45* e^2*x^2) + 3*B*x*(66*d^2 + 120*d*e*x + 55*e^2*x^2)) + 14*a*b^9*x^9*(13*A*( 66*d^2 + 120*d*e*x + 55*e^2*x^2) + 10*B*x*(78*d^2 + 143*d*e*x + 66*e^2*x^2 )) + b^10*x^10*(14*A*(78*d^2 + 143*d*e*x + 66*e^2*x^2) + 11*B*x*(91*d^2 + 168*d*e*x + 78*e^2*x^2))))/12012
Time = 0.71 (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.
| \(\displaystyle \int (a+b x)^{10} (A+B x) (d+e x)^2 \, dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {e (a+b x)^{12} (-3 a B e+A b e+2 b B d)}{b^3}+\frac {(a+b x)^{11} (b d-a e) (-3 a B e+2 A b e+b B d)}{b^3}+\frac {(a+b x)^{10} (A b-a B) (b d-a e)^2}{b^3}+\frac {B e^2 (a+b x)^{13}}{b^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e (a+b x)^{13} (-3 a B e+A b e+2 b B d)}{13 b^4}+\frac {(a+b x)^{12} (b d-a e) (-3 a B e+2 A b e+b B d)}{12 b^4}+\frac {(a+b x)^{11} (A b-a B) (b d-a e)^2}{11 b^4}+\frac {B e^2 (a+b x)^{14}}{14 b^4}\) |
Input:
Int[(a + b*x)^10*(A + B*x)*(d + e*x)^2,x]
Output:
((A*b - a*B)*(b*d - a*e)^2*(a + b*x)^11)/(11*b^4) + ((b*d - a*e)*(b*B*d + 2*A*b*e - 3*a*B*e)*(a + b*x)^12)/(12*b^4) + (e*(2*b*B*d + A*b*e - 3*a*B*e) *(a + b*x)^13)/(13*b^4) + (B*e^2*(a + b*x)^14)/(14*b^4)
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. \(768\) vs. \(2(110)=220\).
Time = 0.20 (sec) , antiderivative size = 769, normalized size of antiderivative = 6.52
| method | result | size |
| default | \(\frac {b^{10} B \,e^{2} x^{14}}{14}+\frac {\left (\left (b^{10} A +10 a \,b^{9} B \right ) e^{2}+2 b^{10} B d e \right ) x^{13}}{13}+\frac {\left (\left (10 a \,b^{9} A +45 a^{2} b^{8} B \right ) e^{2}+2 \left (b^{10} A +10 a \,b^{9} B \right ) d e +b^{10} B \,d^{2}\right ) x^{12}}{12}+\frac {\left (\left (45 a^{2} b^{8} A +120 a^{3} b^{7} B \right ) e^{2}+2 \left (10 a \,b^{9} A +45 a^{2} b^{8} B \right ) d e +\left (b^{10} A +10 a \,b^{9} B \right ) d^{2}\right ) x^{11}}{11}+\frac {\left (\left (120 a^{3} b^{7} A +210 a^{4} b^{6} B \right ) e^{2}+2 \left (45 a^{2} b^{8} A +120 a^{3} b^{7} B \right ) d e +\left (10 a \,b^{9} A +45 a^{2} b^{8} B \right ) d^{2}\right ) x^{10}}{10}+\frac {\left (\left (210 a^{4} b^{6} A +252 a^{5} b^{5} B \right ) e^{2}+2 \left (120 a^{3} b^{7} A +210 a^{4} b^{6} B \right ) d e +\left (45 a^{2} b^{8} A +120 a^{3} b^{7} B \right ) d^{2}\right ) x^{9}}{9}+\frac {\left (\left (252 a^{5} b^{5} A +210 a^{6} b^{4} B \right ) e^{2}+2 \left (210 a^{4} b^{6} A +252 a^{5} b^{5} B \right ) d e +\left (120 a^{3} b^{7} A +210 a^{4} b^{6} B \right ) d^{2}\right ) x^{8}}{8}+\frac {\left (\left (210 a^{6} b^{4} A +120 a^{7} b^{3} B \right ) e^{2}+2 \left (252 a^{5} b^{5} A +210 a^{6} b^{4} B \right ) d e +\left (210 a^{4} b^{6} A +252 a^{5} b^{5} B \right ) d^{2}\right ) x^{7}}{7}+\frac {\left (\left (120 a^{7} b^{3} A +45 a^{8} b^{2} B \right ) e^{2}+2 \left (210 a^{6} b^{4} A +120 a^{7} b^{3} B \right ) d e +\left (252 a^{5} b^{5} A +210 a^{6} b^{4} B \right ) d^{2}\right ) x^{6}}{6}+\frac {\left (\left (45 a^{8} b^{2} A +10 a^{9} b B \right ) e^{2}+2 \left (120 a^{7} b^{3} A +45 a^{8} b^{2} B \right ) d e +\left (210 a^{6} b^{4} A +120 a^{7} b^{3} B \right ) d^{2}\right ) x^{5}}{5}+\frac {\left (\left (10 a^{9} b A +a^{10} B \right ) e^{2}+2 \left (45 a^{8} b^{2} A +10 a^{9} b B \right ) d e +\left (120 a^{7} b^{3} A +45 a^{8} b^{2} B \right ) d^{2}\right ) x^{4}}{4}+\frac {\left (a^{10} A \,e^{2}+2 \left (10 a^{9} b A +a^{10} B \right ) d e +\left (45 a^{8} b^{2} A +10 a^{9} b B \right ) d^{2}\right ) x^{3}}{3}+\frac {\left (2 a^{10} A d e +\left (10 a^{9} b A +a^{10} B \right ) d^{2}\right ) x^{2}}{2}+a^{10} A \,d^{2} x\) | \(769\) |
| norman | \(a^{10} A \,d^{2} x +\left (a^{10} A d e +5 A \,a^{9} b \,d^{2}+\frac {1}{2} B \,a^{10} d^{2}\right ) x^{2}+\left (\frac {1}{3} a^{10} A \,e^{2}+\frac {20}{3} A \,a^{9} b d e +15 A \,a^{8} b^{2} d^{2}+\frac {2}{3} B \,a^{10} d e +\frac {10}{3} B \,a^{9} b \,d^{2}\right ) x^{3}+\left (\frac {5}{2} A \,a^{9} b \,e^{2}+\frac {45}{2} A \,a^{8} b^{2} d e +30 A \,a^{7} b^{3} d^{2}+\frac {1}{4} B \,a^{10} e^{2}+5 B \,a^{9} b d e +\frac {45}{4} B \,a^{8} b^{2} d^{2}\right ) x^{4}+\left (9 A \,a^{8} b^{2} e^{2}+48 A \,a^{7} b^{3} d e +42 A \,a^{6} b^{4} d^{2}+2 B \,a^{9} b \,e^{2}+18 B \,a^{8} b^{2} d e +24 B \,a^{7} b^{3} d^{2}\right ) x^{5}+\left (20 A \,a^{7} b^{3} e^{2}+70 A \,a^{6} b^{4} d e +42 A \,a^{5} b^{5} d^{2}+\frac {15}{2} B \,a^{8} b^{2} e^{2}+40 B \,a^{7} b^{3} d e +35 B \,a^{6} b^{4} d^{2}\right ) x^{6}+\left (30 A \,a^{6} b^{4} e^{2}+72 A \,a^{5} b^{5} d e +30 A \,a^{4} b^{6} d^{2}+\frac {120}{7} B \,a^{7} b^{3} e^{2}+60 B \,a^{6} b^{4} d e +36 B \,a^{5} b^{5} d^{2}\right ) x^{7}+\left (\frac {63}{2} A \,a^{5} b^{5} e^{2}+\frac {105}{2} A \,a^{4} b^{6} d e +15 A \,a^{3} b^{7} d^{2}+\frac {105}{4} B \,a^{6} b^{4} e^{2}+63 B \,a^{5} b^{5} d e +\frac {105}{4} B \,a^{4} b^{6} d^{2}\right ) x^{8}+\left (\frac {70}{3} A \,a^{4} b^{6} e^{2}+\frac {80}{3} A \,a^{3} b^{7} d e +5 A \,a^{2} b^{8} d^{2}+28 B \,a^{5} b^{5} e^{2}+\frac {140}{3} B \,a^{4} b^{6} d e +\frac {40}{3} B \,a^{3} b^{7} d^{2}\right ) x^{9}+\left (12 A \,a^{3} b^{7} e^{2}+9 A \,a^{2} b^{8} d e +A a \,b^{9} d^{2}+21 B \,a^{4} b^{6} e^{2}+24 B \,a^{3} b^{7} d e +\frac {9}{2} B \,a^{2} b^{8} d^{2}\right ) x^{10}+\left (\frac {45}{11} A \,a^{2} b^{8} e^{2}+\frac {20}{11} A a \,b^{9} d e +\frac {1}{11} A \,b^{10} d^{2}+\frac {120}{11} B \,a^{3} b^{7} e^{2}+\frac {90}{11} B \,a^{2} b^{8} d e +\frac {10}{11} B a \,b^{9} d^{2}\right ) x^{11}+\left (\frac {5}{6} A a \,b^{9} e^{2}+\frac {1}{6} A \,b^{10} d e +\frac {15}{4} B \,a^{2} b^{8} e^{2}+\frac {5}{3} B a \,b^{9} d e +\frac {1}{12} b^{10} B \,d^{2}\right ) x^{12}+\left (\frac {1}{13} A \,b^{10} e^{2}+\frac {10}{13} B a \,b^{9} e^{2}+\frac {2}{13} b^{10} B d e \right ) x^{13}+\frac {b^{10} B \,e^{2} x^{14}}{14}\) | \(773\) |
| orering | \(\frac {x \left (858 b^{10} B \,e^{2} x^{13}+924 A \,b^{10} e^{2} x^{12}+9240 B a \,b^{9} e^{2} x^{12}+1848 B \,b^{10} d e \,x^{12}+10010 A a \,b^{9} e^{2} x^{11}+2002 A \,b^{10} d e \,x^{11}+45045 B \,a^{2} b^{8} e^{2} x^{11}+20020 B a \,b^{9} d e \,x^{11}+1001 B \,b^{10} d^{2} x^{11}+49140 A \,a^{2} b^{8} e^{2} x^{10}+21840 A a \,b^{9} d e \,x^{10}+1092 A \,b^{10} d^{2} x^{10}+131040 B \,a^{3} b^{7} e^{2} x^{10}+98280 B \,a^{2} b^{8} d e \,x^{10}+10920 B a \,b^{9} d^{2} x^{10}+144144 A \,a^{3} b^{7} e^{2} x^{9}+108108 A \,a^{2} b^{8} d e \,x^{9}+12012 A a \,b^{9} d^{2} x^{9}+252252 B \,a^{4} b^{6} e^{2} x^{9}+288288 B \,a^{3} b^{7} d e \,x^{9}+54054 B \,a^{2} b^{8} d^{2} x^{9}+280280 A \,a^{4} b^{6} e^{2} x^{8}+320320 A \,a^{3} b^{7} d e \,x^{8}+60060 A \,a^{2} b^{8} d^{2} x^{8}+336336 B \,a^{5} b^{5} e^{2} x^{8}+560560 B \,a^{4} b^{6} d e \,x^{8}+160160 B \,a^{3} b^{7} d^{2} x^{8}+378378 A \,a^{5} b^{5} e^{2} x^{7}+630630 A \,a^{4} b^{6} d e \,x^{7}+180180 A \,a^{3} b^{7} d^{2} x^{7}+315315 B \,a^{6} b^{4} e^{2} x^{7}+756756 B \,a^{5} b^{5} d e \,x^{7}+315315 B \,a^{4} b^{6} d^{2} x^{7}+360360 A \,a^{6} b^{4} e^{2} x^{6}+864864 A \,a^{5} b^{5} d e \,x^{6}+360360 A \,a^{4} b^{6} d^{2} x^{6}+205920 B \,a^{7} b^{3} e^{2} x^{6}+720720 B \,a^{6} b^{4} d e \,x^{6}+432432 B \,a^{5} b^{5} d^{2} x^{6}+240240 A \,a^{7} b^{3} e^{2} x^{5}+840840 A \,a^{6} b^{4} d e \,x^{5}+504504 A \,a^{5} b^{5} d^{2} x^{5}+90090 B \,a^{8} b^{2} e^{2} x^{5}+480480 B \,a^{7} b^{3} d e \,x^{5}+420420 B \,a^{6} b^{4} d^{2} x^{5}+108108 A \,a^{8} b^{2} e^{2} x^{4}+576576 A \,a^{7} b^{3} d e \,x^{4}+504504 A \,a^{6} b^{4} d^{2} x^{4}+24024 B \,a^{9} b \,e^{2} x^{4}+216216 B \,a^{8} b^{2} d e \,x^{4}+288288 B \,a^{7} b^{3} d^{2} x^{4}+30030 A \,a^{9} b \,e^{2} x^{3}+270270 A \,a^{8} b^{2} d e \,x^{3}+360360 A \,a^{7} b^{3} d^{2} x^{3}+3003 B \,a^{10} e^{2} x^{3}+60060 B \,a^{9} b d e \,x^{3}+135135 B \,a^{8} b^{2} d^{2} x^{3}+4004 A \,a^{10} e^{2} x^{2}+80080 A \,a^{9} b d e \,x^{2}+180180 A \,a^{8} b^{2} d^{2} x^{2}+8008 B \,a^{10} d e \,x^{2}+40040 B \,a^{9} b \,d^{2} x^{2}+12012 A \,a^{10} d e x +60060 A \,a^{9} b \,d^{2} x +6006 B \,a^{10} d^{2} x +12012 a^{10} A \,d^{2}\right )}{12012}\) | \(904\) |
| gosper | \(70 x^{6} A \,a^{6} b^{4} d e +40 x^{6} B \,a^{7} b^{3} d e +72 x^{7} A \,a^{5} b^{5} d e +60 x^{7} B \,a^{6} b^{4} d e +\frac {105}{2} x^{8} A \,a^{4} b^{6} d e +63 x^{8} B \,a^{5} b^{5} d e +\frac {80}{3} x^{9} A \,a^{3} b^{7} d e +\frac {140}{3} x^{9} B \,a^{4} b^{6} d e +9 x^{10} A \,a^{2} b^{8} d e +24 x^{10} B \,a^{3} b^{7} d e +\frac {20}{11} x^{11} A a \,b^{9} d e +\frac {90}{11} x^{11} B \,a^{2} b^{8} d e +\frac {5}{3} x^{12} B a \,b^{9} d e +48 A \,a^{7} b^{3} d e \,x^{5}+18 B \,a^{8} b^{2} d e \,x^{5}+\frac {45}{4} x^{4} B \,a^{8} b^{2} d^{2}+20 x^{6} A \,a^{7} b^{3} e^{2}+42 x^{6} A \,a^{5} b^{5} d^{2}+\frac {15}{2} x^{6} B \,a^{8} b^{2} e^{2}+35 x^{6} B \,a^{6} b^{4} d^{2}+30 x^{7} A \,a^{6} b^{4} e^{2}+30 x^{7} A \,a^{4} b^{6} d^{2}+\frac {120}{7} x^{7} B \,a^{7} b^{3} e^{2}+36 x^{7} B \,a^{5} b^{5} d^{2}+\frac {63}{2} x^{8} A \,a^{5} b^{5} e^{2}+15 x^{8} A \,a^{3} b^{7} d^{2}+\frac {105}{4} x^{8} B \,a^{6} b^{4} e^{2}+\frac {105}{4} x^{8} B \,a^{4} b^{6} d^{2}+\frac {70}{3} x^{9} A \,a^{4} b^{6} e^{2}+5 x^{9} A \,a^{2} b^{8} d^{2}+28 x^{9} B \,a^{5} b^{5} e^{2}+\frac {40}{3} x^{9} B \,a^{3} b^{7} d^{2}+12 x^{10} A \,a^{3} b^{7} e^{2}+x^{10} A a \,b^{9} d^{2}+21 x^{10} B \,a^{4} b^{6} e^{2}+\frac {9}{2} x^{10} B \,a^{2} b^{8} d^{2}+\frac {45}{11} x^{11} A \,a^{2} b^{8} e^{2}+\frac {120}{11} x^{11} B \,a^{3} b^{7} e^{2}+\frac {10}{11} x^{11} B a \,b^{9} d^{2}+\frac {5}{6} x^{12} A a \,b^{9} e^{2}+\frac {1}{6} x^{12} A \,b^{10} d e +\frac {15}{4} x^{12} B \,a^{2} b^{8} e^{2}+\frac {10}{13} x^{13} B a \,b^{9} e^{2}+\frac {2}{13} x^{13} b^{10} B d e +9 A \,a^{8} b^{2} e^{2} x^{5}+42 A \,a^{6} b^{4} d^{2} x^{5}+2 B \,a^{9} b \,e^{2} x^{5}+24 B \,a^{7} b^{3} d^{2} x^{5}+a^{10} A \,d^{2} x +\frac {1}{14} b^{10} B \,e^{2} x^{14}+\frac {1}{2} x^{2} B \,a^{10} d^{2}+\frac {1}{3} x^{3} a^{10} A \,e^{2}+\frac {1}{4} x^{4} B \,a^{10} e^{2}+\frac {1}{11} x^{11} A \,b^{10} d^{2}+\frac {1}{12} x^{12} b^{10} B \,d^{2}+\frac {1}{13} x^{13} A \,b^{10} e^{2}+x^{2} a^{10} A d e +5 x^{2} A \,a^{9} b \,d^{2}+15 x^{3} A \,a^{8} b^{2} d^{2}+\frac {20}{3} x^{3} A \,a^{9} b d e +\frac {45}{2} x^{4} A \,a^{8} b^{2} d e +\frac {2}{3} x^{3} B \,a^{10} d e +\frac {10}{3} x^{3} B \,a^{9} b \,d^{2}+\frac {5}{2} x^{4} A \,a^{9} b \,e^{2}+30 x^{4} A \,a^{7} b^{3} d^{2}+5 x^{4} B \,a^{9} b d e\) | \(905\) |
| risch | \(70 x^{6} A \,a^{6} b^{4} d e +40 x^{6} B \,a^{7} b^{3} d e +72 x^{7} A \,a^{5} b^{5} d e +60 x^{7} B \,a^{6} b^{4} d e +\frac {105}{2} x^{8} A \,a^{4} b^{6} d e +63 x^{8} B \,a^{5} b^{5} d e +\frac {80}{3} x^{9} A \,a^{3} b^{7} d e +\frac {140}{3} x^{9} B \,a^{4} b^{6} d e +9 x^{10} A \,a^{2} b^{8} d e +24 x^{10} B \,a^{3} b^{7} d e +\frac {20}{11} x^{11} A a \,b^{9} d e +\frac {90}{11} x^{11} B \,a^{2} b^{8} d e +\frac {5}{3} x^{12} B a \,b^{9} d e +48 A \,a^{7} b^{3} d e \,x^{5}+18 B \,a^{8} b^{2} d e \,x^{5}+\frac {45}{4} x^{4} B \,a^{8} b^{2} d^{2}+20 x^{6} A \,a^{7} b^{3} e^{2}+42 x^{6} A \,a^{5} b^{5} d^{2}+\frac {15}{2} x^{6} B \,a^{8} b^{2} e^{2}+35 x^{6} B \,a^{6} b^{4} d^{2}+30 x^{7} A \,a^{6} b^{4} e^{2}+30 x^{7} A \,a^{4} b^{6} d^{2}+\frac {120}{7} x^{7} B \,a^{7} b^{3} e^{2}+36 x^{7} B \,a^{5} b^{5} d^{2}+\frac {63}{2} x^{8} A \,a^{5} b^{5} e^{2}+15 x^{8} A \,a^{3} b^{7} d^{2}+\frac {105}{4} x^{8} B \,a^{6} b^{4} e^{2}+\frac {105}{4} x^{8} B \,a^{4} b^{6} d^{2}+\frac {70}{3} x^{9} A \,a^{4} b^{6} e^{2}+5 x^{9} A \,a^{2} b^{8} d^{2}+28 x^{9} B \,a^{5} b^{5} e^{2}+\frac {40}{3} x^{9} B \,a^{3} b^{7} d^{2}+12 x^{10} A \,a^{3} b^{7} e^{2}+x^{10} A a \,b^{9} d^{2}+21 x^{10} B \,a^{4} b^{6} e^{2}+\frac {9}{2} x^{10} B \,a^{2} b^{8} d^{2}+\frac {45}{11} x^{11} A \,a^{2} b^{8} e^{2}+\frac {120}{11} x^{11} B \,a^{3} b^{7} e^{2}+\frac {10}{11} x^{11} B a \,b^{9} d^{2}+\frac {5}{6} x^{12} A a \,b^{9} e^{2}+\frac {1}{6} x^{12} A \,b^{10} d e +\frac {15}{4} x^{12} B \,a^{2} b^{8} e^{2}+\frac {10}{13} x^{13} B a \,b^{9} e^{2}+\frac {2}{13} x^{13} b^{10} B d e +9 A \,a^{8} b^{2} e^{2} x^{5}+42 A \,a^{6} b^{4} d^{2} x^{5}+2 B \,a^{9} b \,e^{2} x^{5}+24 B \,a^{7} b^{3} d^{2} x^{5}+a^{10} A \,d^{2} x +\frac {1}{14} b^{10} B \,e^{2} x^{14}+\frac {1}{2} x^{2} B \,a^{10} d^{2}+\frac {1}{3} x^{3} a^{10} A \,e^{2}+\frac {1}{4} x^{4} B \,a^{10} e^{2}+\frac {1}{11} x^{11} A \,b^{10} d^{2}+\frac {1}{12} x^{12} b^{10} B \,d^{2}+\frac {1}{13} x^{13} A \,b^{10} e^{2}+x^{2} a^{10} A d e +5 x^{2} A \,a^{9} b \,d^{2}+15 x^{3} A \,a^{8} b^{2} d^{2}+\frac {20}{3} x^{3} A \,a^{9} b d e +\frac {45}{2} x^{4} A \,a^{8} b^{2} d e +\frac {2}{3} x^{3} B \,a^{10} d e +\frac {10}{3} x^{3} B \,a^{9} b \,d^{2}+\frac {5}{2} x^{4} A \,a^{9} b \,e^{2}+30 x^{4} A \,a^{7} b^{3} d^{2}+5 x^{4} B \,a^{9} b d e\) | \(905\) |
| parallelrisch | \(70 x^{6} A \,a^{6} b^{4} d e +40 x^{6} B \,a^{7} b^{3} d e +72 x^{7} A \,a^{5} b^{5} d e +60 x^{7} B \,a^{6} b^{4} d e +\frac {105}{2} x^{8} A \,a^{4} b^{6} d e +63 x^{8} B \,a^{5} b^{5} d e +\frac {80}{3} x^{9} A \,a^{3} b^{7} d e +\frac {140}{3} x^{9} B \,a^{4} b^{6} d e +9 x^{10} A \,a^{2} b^{8} d e +24 x^{10} B \,a^{3} b^{7} d e +\frac {20}{11} x^{11} A a \,b^{9} d e +\frac {90}{11} x^{11} B \,a^{2} b^{8} d e +\frac {5}{3} x^{12} B a \,b^{9} d e +48 A \,a^{7} b^{3} d e \,x^{5}+18 B \,a^{8} b^{2} d e \,x^{5}+\frac {45}{4} x^{4} B \,a^{8} b^{2} d^{2}+20 x^{6} A \,a^{7} b^{3} e^{2}+42 x^{6} A \,a^{5} b^{5} d^{2}+\frac {15}{2} x^{6} B \,a^{8} b^{2} e^{2}+35 x^{6} B \,a^{6} b^{4} d^{2}+30 x^{7} A \,a^{6} b^{4} e^{2}+30 x^{7} A \,a^{4} b^{6} d^{2}+\frac {120}{7} x^{7} B \,a^{7} b^{3} e^{2}+36 x^{7} B \,a^{5} b^{5} d^{2}+\frac {63}{2} x^{8} A \,a^{5} b^{5} e^{2}+15 x^{8} A \,a^{3} b^{7} d^{2}+\frac {105}{4} x^{8} B \,a^{6} b^{4} e^{2}+\frac {105}{4} x^{8} B \,a^{4} b^{6} d^{2}+\frac {70}{3} x^{9} A \,a^{4} b^{6} e^{2}+5 x^{9} A \,a^{2} b^{8} d^{2}+28 x^{9} B \,a^{5} b^{5} e^{2}+\frac {40}{3} x^{9} B \,a^{3} b^{7} d^{2}+12 x^{10} A \,a^{3} b^{7} e^{2}+x^{10} A a \,b^{9} d^{2}+21 x^{10} B \,a^{4} b^{6} e^{2}+\frac {9}{2} x^{10} B \,a^{2} b^{8} d^{2}+\frac {45}{11} x^{11} A \,a^{2} b^{8} e^{2}+\frac {120}{11} x^{11} B \,a^{3} b^{7} e^{2}+\frac {10}{11} x^{11} B a \,b^{9} d^{2}+\frac {5}{6} x^{12} A a \,b^{9} e^{2}+\frac {1}{6} x^{12} A \,b^{10} d e +\frac {15}{4} x^{12} B \,a^{2} b^{8} e^{2}+\frac {10}{13} x^{13} B a \,b^{9} e^{2}+\frac {2}{13} x^{13} b^{10} B d e +9 A \,a^{8} b^{2} e^{2} x^{5}+42 A \,a^{6} b^{4} d^{2} x^{5}+2 B \,a^{9} b \,e^{2} x^{5}+24 B \,a^{7} b^{3} d^{2} x^{5}+a^{10} A \,d^{2} x +\frac {1}{14} b^{10} B \,e^{2} x^{14}+\frac {1}{2} x^{2} B \,a^{10} d^{2}+\frac {1}{3} x^{3} a^{10} A \,e^{2}+\frac {1}{4} x^{4} B \,a^{10} e^{2}+\frac {1}{11} x^{11} A \,b^{10} d^{2}+\frac {1}{12} x^{12} b^{10} B \,d^{2}+\frac {1}{13} x^{13} A \,b^{10} e^{2}+x^{2} a^{10} A d e +5 x^{2} A \,a^{9} b \,d^{2}+15 x^{3} A \,a^{8} b^{2} d^{2}+\frac {20}{3} x^{3} A \,a^{9} b d e +\frac {45}{2} x^{4} A \,a^{8} b^{2} d e +\frac {2}{3} x^{3} B \,a^{10} d e +\frac {10}{3} x^{3} B \,a^{9} b \,d^{2}+\frac {5}{2} x^{4} A \,a^{9} b \,e^{2}+30 x^{4} A \,a^{7} b^{3} d^{2}+5 x^{4} B \,a^{9} b d e\) | \(905\) |
Input:
int((b*x+a)^10*(B*x+A)*(e*x+d)^2,x,method=_RETURNVERBOSE)
Output:
1/14*b^10*B*e^2*x^14+1/13*((A*b^10+10*B*a*b^9)*e^2+2*b^10*B*d*e)*x^13+1/12 *((10*A*a*b^9+45*B*a^2*b^8)*e^2+2*(A*b^10+10*B*a*b^9)*d*e+b^10*B*d^2)*x^12 +1/11*((45*A*a^2*b^8+120*B*a^3*b^7)*e^2+2*(10*A*a*b^9+45*B*a^2*b^8)*d*e+(A *b^10+10*B*a*b^9)*d^2)*x^11+1/10*((120*A*a^3*b^7+210*B*a^4*b^6)*e^2+2*(45* A*a^2*b^8+120*B*a^3*b^7)*d*e+(10*A*a*b^9+45*B*a^2*b^8)*d^2)*x^10+1/9*((210 *A*a^4*b^6+252*B*a^5*b^5)*e^2+2*(120*A*a^3*b^7+210*B*a^4*b^6)*d*e+(45*A*a^ 2*b^8+120*B*a^3*b^7)*d^2)*x^9+1/8*((252*A*a^5*b^5+210*B*a^6*b^4)*e^2+2*(21 0*A*a^4*b^6+252*B*a^5*b^5)*d*e+(120*A*a^3*b^7+210*B*a^4*b^6)*d^2)*x^8+1/7* ((210*A*a^6*b^4+120*B*a^7*b^3)*e^2+2*(252*A*a^5*b^5+210*B*a^6*b^4)*d*e+(21 0*A*a^4*b^6+252*B*a^5*b^5)*d^2)*x^7+1/6*((120*A*a^7*b^3+45*B*a^8*b^2)*e^2+ 2*(210*A*a^6*b^4+120*B*a^7*b^3)*d*e+(252*A*a^5*b^5+210*B*a^6*b^4)*d^2)*x^6 +1/5*((45*A*a^8*b^2+10*B*a^9*b)*e^2+2*(120*A*a^7*b^3+45*B*a^8*b^2)*d*e+(21 0*A*a^6*b^4+120*B*a^7*b^3)*d^2)*x^5+1/4*((10*A*a^9*b+B*a^10)*e^2+2*(45*A*a ^8*b^2+10*B*a^9*b)*d*e+(120*A*a^7*b^3+45*B*a^8*b^2)*d^2)*x^4+1/3*(a^10*A*e ^2+2*(10*A*a^9*b+B*a^10)*d*e+(45*A*a^8*b^2+10*B*a^9*b)*d^2)*x^3+1/2*(2*a^1 0*A*d*e+(10*A*a^9*b+B*a^10)*d^2)*x^2+a^10*A*d^2*x
Leaf count of result is larger than twice the leaf count of optimal. 781 vs. \(2 (110) = 220\).
Time = 0.09 (sec) , antiderivative size = 781, normalized size of antiderivative = 6.62 \[ \int (a+b x)^{10} (A+B x) (d+e x)^2 \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)^10*(B*x+A)*(e*x+d)^2,x, algorithm="fricas")
Output:
1/14*B*b^10*e^2*x^14 + A*a^10*d^2*x + 1/13*(2*B*b^10*d*e + (10*B*a*b^9 + A *b^10)*e^2)*x^13 + 1/12*(B*b^10*d^2 + 2*(10*B*a*b^9 + A*b^10)*d*e + 5*(9*B *a^2*b^8 + 2*A*a*b^9)*e^2)*x^12 + 1/11*((10*B*a*b^9 + A*b^10)*d^2 + 10*(9* B*a^2*b^8 + 2*A*a*b^9)*d*e + 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^2)*x^11 + 1/ 2*((9*B*a^2*b^8 + 2*A*a*b^9)*d^2 + 6*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e + 6*( 7*B*a^4*b^6 + 4*A*a^3*b^7)*e^2)*x^10 + 1/3*(5*(8*B*a^3*b^7 + 3*A*a^2*b^8)* d^2 + 20*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e + 14*(6*B*a^5*b^5 + 5*A*a^4*b^6)* e^2)*x^9 + 3/4*(5*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2 + 14*(6*B*a^5*b^5 + 5*A* a^4*b^6)*d*e + 7*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^2)*x^8 + 6/7*(7*(6*B*a^5*b^ 5 + 5*A*a^4*b^6)*d^2 + 14*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e + 5*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^2)*x^7 + 1/2*(14*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2 + 20*(4 *B*a^7*b^3 + 7*A*a^6*b^4)*d*e + 5*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^2)*x^6 + ( 6*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2 + 6*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e + (2 *B*a^9*b + 9*A*a^8*b^2)*e^2)*x^5 + 1/4*(15*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2 + 10*(2*B*a^9*b + 9*A*a^8*b^2)*d*e + (B*a^10 + 10*A*a^9*b)*e^2)*x^4 + 1/3 *(A*a^10*e^2 + 5*(2*B*a^9*b + 9*A*a^8*b^2)*d^2 + 2*(B*a^10 + 10*A*a^9*b)*d *e)*x^3 + 1/2*(2*A*a^10*d*e + (B*a^10 + 10*A*a^9*b)*d^2)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 921 vs. \(2 (116) = 232\).
Time = 0.09 (sec) , antiderivative size = 921, normalized size of antiderivative = 7.81 \[ \int (a+b x)^{10} (A+B x) (d+e x)^2 \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)**10*(B*x+A)*(e*x+d)**2,x)
Output:
A*a**10*d**2*x + B*b**10*e**2*x**14/14 + x**13*(A*b**10*e**2/13 + 10*B*a*b **9*e**2/13 + 2*B*b**10*d*e/13) + x**12*(5*A*a*b**9*e**2/6 + A*b**10*d*e/6 + 15*B*a**2*b**8*e**2/4 + 5*B*a*b**9*d*e/3 + B*b**10*d**2/12) + x**11*(45 *A*a**2*b**8*e**2/11 + 20*A*a*b**9*d*e/11 + A*b**10*d**2/11 + 120*B*a**3*b **7*e**2/11 + 90*B*a**2*b**8*d*e/11 + 10*B*a*b**9*d**2/11) + x**10*(12*A*a **3*b**7*e**2 + 9*A*a**2*b**8*d*e + A*a*b**9*d**2 + 21*B*a**4*b**6*e**2 + 24*B*a**3*b**7*d*e + 9*B*a**2*b**8*d**2/2) + x**9*(70*A*a**4*b**6*e**2/3 + 80*A*a**3*b**7*d*e/3 + 5*A*a**2*b**8*d**2 + 28*B*a**5*b**5*e**2 + 140*B*a **4*b**6*d*e/3 + 40*B*a**3*b**7*d**2/3) + x**8*(63*A*a**5*b**5*e**2/2 + 10 5*A*a**4*b**6*d*e/2 + 15*A*a**3*b**7*d**2 + 105*B*a**6*b**4*e**2/4 + 63*B* a**5*b**5*d*e + 105*B*a**4*b**6*d**2/4) + x**7*(30*A*a**6*b**4*e**2 + 72*A *a**5*b**5*d*e + 30*A*a**4*b**6*d**2 + 120*B*a**7*b**3*e**2/7 + 60*B*a**6* b**4*d*e + 36*B*a**5*b**5*d**2) + x**6*(20*A*a**7*b**3*e**2 + 70*A*a**6*b* *4*d*e + 42*A*a**5*b**5*d**2 + 15*B*a**8*b**2*e**2/2 + 40*B*a**7*b**3*d*e + 35*B*a**6*b**4*d**2) + x**5*(9*A*a**8*b**2*e**2 + 48*A*a**7*b**3*d*e + 4 2*A*a**6*b**4*d**2 + 2*B*a**9*b*e**2 + 18*B*a**8*b**2*d*e + 24*B*a**7*b**3 *d**2) + x**4*(5*A*a**9*b*e**2/2 + 45*A*a**8*b**2*d*e/2 + 30*A*a**7*b**3*d **2 + B*a**10*e**2/4 + 5*B*a**9*b*d*e + 45*B*a**8*b**2*d**2/4) + x**3*(A*a **10*e**2/3 + 20*A*a**9*b*d*e/3 + 15*A*a**8*b**2*d**2 + 2*B*a**10*d*e/3 + 10*B*a**9*b*d**2/3) + x**2*(A*a**10*d*e + 5*A*a**9*b*d**2 + B*a**10*d**...
Leaf count of result is larger than twice the leaf count of optimal. 781 vs. \(2 (110) = 220\).
Time = 0.05 (sec) , antiderivative size = 781, normalized size of antiderivative = 6.62 \[ \int (a+b x)^{10} (A+B x) (d+e x)^2 \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)^10*(B*x+A)*(e*x+d)^2,x, algorithm="maxima")
Output:
1/14*B*b^10*e^2*x^14 + A*a^10*d^2*x + 1/13*(2*B*b^10*d*e + (10*B*a*b^9 + A *b^10)*e^2)*x^13 + 1/12*(B*b^10*d^2 + 2*(10*B*a*b^9 + A*b^10)*d*e + 5*(9*B *a^2*b^8 + 2*A*a*b^9)*e^2)*x^12 + 1/11*((10*B*a*b^9 + A*b^10)*d^2 + 10*(9* B*a^2*b^8 + 2*A*a*b^9)*d*e + 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^2)*x^11 + 1/ 2*((9*B*a^2*b^8 + 2*A*a*b^9)*d^2 + 6*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e + 6*( 7*B*a^4*b^6 + 4*A*a^3*b^7)*e^2)*x^10 + 1/3*(5*(8*B*a^3*b^7 + 3*A*a^2*b^8)* d^2 + 20*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e + 14*(6*B*a^5*b^5 + 5*A*a^4*b^6)* e^2)*x^9 + 3/4*(5*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2 + 14*(6*B*a^5*b^5 + 5*A* a^4*b^6)*d*e + 7*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^2)*x^8 + 6/7*(7*(6*B*a^5*b^ 5 + 5*A*a^4*b^6)*d^2 + 14*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e + 5*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^2)*x^7 + 1/2*(14*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2 + 20*(4 *B*a^7*b^3 + 7*A*a^6*b^4)*d*e + 5*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^2)*x^6 + ( 6*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2 + 6*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e + (2 *B*a^9*b + 9*A*a^8*b^2)*e^2)*x^5 + 1/4*(15*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2 + 10*(2*B*a^9*b + 9*A*a^8*b^2)*d*e + (B*a^10 + 10*A*a^9*b)*e^2)*x^4 + 1/3 *(A*a^10*e^2 + 5*(2*B*a^9*b + 9*A*a^8*b^2)*d^2 + 2*(B*a^10 + 10*A*a^9*b)*d *e)*x^3 + 1/2*(2*A*a^10*d*e + (B*a^10 + 10*A*a^9*b)*d^2)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 904 vs. \(2 (110) = 220\).
Time = 0.13 (sec) , antiderivative size = 904, normalized size of antiderivative = 7.66 \[ \int (a+b x)^{10} (A+B x) (d+e x)^2 \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)^10*(B*x+A)*(e*x+d)^2,x, algorithm="giac")
Output:
1/14*B*b^10*e^2*x^14 + 2/13*B*b^10*d*e*x^13 + 10/13*B*a*b^9*e^2*x^13 + 1/1 3*A*b^10*e^2*x^13 + 1/12*B*b^10*d^2*x^12 + 5/3*B*a*b^9*d*e*x^12 + 1/6*A*b^ 10*d*e*x^12 + 15/4*B*a^2*b^8*e^2*x^12 + 5/6*A*a*b^9*e^2*x^12 + 10/11*B*a*b ^9*d^2*x^11 + 1/11*A*b^10*d^2*x^11 + 90/11*B*a^2*b^8*d*e*x^11 + 20/11*A*a* b^9*d*e*x^11 + 120/11*B*a^3*b^7*e^2*x^11 + 45/11*A*a^2*b^8*e^2*x^11 + 9/2* B*a^2*b^8*d^2*x^10 + A*a*b^9*d^2*x^10 + 24*B*a^3*b^7*d*e*x^10 + 9*A*a^2*b^ 8*d*e*x^10 + 21*B*a^4*b^6*e^2*x^10 + 12*A*a^3*b^7*e^2*x^10 + 40/3*B*a^3*b^ 7*d^2*x^9 + 5*A*a^2*b^8*d^2*x^9 + 140/3*B*a^4*b^6*d*e*x^9 + 80/3*A*a^3*b^7 *d*e*x^9 + 28*B*a^5*b^5*e^2*x^9 + 70/3*A*a^4*b^6*e^2*x^9 + 105/4*B*a^4*b^6 *d^2*x^8 + 15*A*a^3*b^7*d^2*x^8 + 63*B*a^5*b^5*d*e*x^8 + 105/2*A*a^4*b^6*d *e*x^8 + 105/4*B*a^6*b^4*e^2*x^8 + 63/2*A*a^5*b^5*e^2*x^8 + 36*B*a^5*b^5*d ^2*x^7 + 30*A*a^4*b^6*d^2*x^7 + 60*B*a^6*b^4*d*e*x^7 + 72*A*a^5*b^5*d*e*x^ 7 + 120/7*B*a^7*b^3*e^2*x^7 + 30*A*a^6*b^4*e^2*x^7 + 35*B*a^6*b^4*d^2*x^6 + 42*A*a^5*b^5*d^2*x^6 + 40*B*a^7*b^3*d*e*x^6 + 70*A*a^6*b^4*d*e*x^6 + 15/ 2*B*a^8*b^2*e^2*x^6 + 20*A*a^7*b^3*e^2*x^6 + 24*B*a^7*b^3*d^2*x^5 + 42*A*a ^6*b^4*d^2*x^5 + 18*B*a^8*b^2*d*e*x^5 + 48*A*a^7*b^3*d*e*x^5 + 2*B*a^9*b*e ^2*x^5 + 9*A*a^8*b^2*e^2*x^5 + 45/4*B*a^8*b^2*d^2*x^4 + 30*A*a^7*b^3*d^2*x ^4 + 5*B*a^9*b*d*e*x^4 + 45/2*A*a^8*b^2*d*e*x^4 + 1/4*B*a^10*e^2*x^4 + 5/2 *A*a^9*b*e^2*x^4 + 10/3*B*a^9*b*d^2*x^3 + 15*A*a^8*b^2*d^2*x^3 + 2/3*B*a^1 0*d*e*x^3 + 20/3*A*a^9*b*d*e*x^3 + 1/3*A*a^10*e^2*x^3 + 1/2*B*a^10*d^2*...
Time = 1.16 (sec) , antiderivative size = 757, normalized size of antiderivative = 6.42 \[ \int (a+b x)^{10} (A+B x) (d+e x)^2 \, dx=x^6\,\left (\frac {15\,B\,a^8\,b^2\,e^2}{2}+40\,B\,a^7\,b^3\,d\,e+20\,A\,a^7\,b^3\,e^2+35\,B\,a^6\,b^4\,d^2+70\,A\,a^6\,b^4\,d\,e+42\,A\,a^5\,b^5\,d^2\right )+x^7\,\left (\frac {120\,B\,a^7\,b^3\,e^2}{7}+60\,B\,a^6\,b^4\,d\,e+30\,A\,a^6\,b^4\,e^2+36\,B\,a^5\,b^5\,d^2+72\,A\,a^5\,b^5\,d\,e+30\,A\,a^4\,b^6\,d^2\right )+x^9\,\left (28\,B\,a^5\,b^5\,e^2+\frac {140\,B\,a^4\,b^6\,d\,e}{3}+\frac {70\,A\,a^4\,b^6\,e^2}{3}+\frac {40\,B\,a^3\,b^7\,d^2}{3}+\frac {80\,A\,a^3\,b^7\,d\,e}{3}+5\,A\,a^2\,b^8\,d^2\right )+x^8\,\left (\frac {105\,B\,a^6\,b^4\,e^2}{4}+63\,B\,a^5\,b^5\,d\,e+\frac {63\,A\,a^5\,b^5\,e^2}{2}+\frac {105\,B\,a^4\,b^6\,d^2}{4}+\frac {105\,A\,a^4\,b^6\,d\,e}{2}+15\,A\,a^3\,b^7\,d^2\right )+x^4\,\left (\frac {B\,a^{10}\,e^2}{4}+5\,B\,a^9\,b\,d\,e+\frac {5\,A\,a^9\,b\,e^2}{2}+\frac {45\,B\,a^8\,b^2\,d^2}{4}+\frac {45\,A\,a^8\,b^2\,d\,e}{2}+30\,A\,a^7\,b^3\,d^2\right )+x^{11}\,\left (\frac {120\,B\,a^3\,b^7\,e^2}{11}+\frac {90\,B\,a^2\,b^8\,d\,e}{11}+\frac {45\,A\,a^2\,b^8\,e^2}{11}+\frac {10\,B\,a\,b^9\,d^2}{11}+\frac {20\,A\,a\,b^9\,d\,e}{11}+\frac {A\,b^{10}\,d^2}{11}\right )+x^{10}\,\left (21\,B\,a^4\,b^6\,e^2+24\,B\,a^3\,b^7\,d\,e+12\,A\,a^3\,b^7\,e^2+\frac {9\,B\,a^2\,b^8\,d^2}{2}+9\,A\,a^2\,b^8\,d\,e+A\,a\,b^9\,d^2\right )+x^5\,\left (2\,B\,a^9\,b\,e^2+18\,B\,a^8\,b^2\,d\,e+9\,A\,a^8\,b^2\,e^2+24\,B\,a^7\,b^3\,d^2+48\,A\,a^7\,b^3\,d\,e+42\,A\,a^6\,b^4\,d^2\right )+x^3\,\left (\frac {2\,B\,a^{10}\,d\,e}{3}+\frac {A\,a^{10}\,e^2}{3}+\frac {10\,B\,a^9\,b\,d^2}{3}+\frac {20\,A\,a^9\,b\,d\,e}{3}+15\,A\,a^8\,b^2\,d^2\right )+x^{12}\,\left (\frac {15\,B\,a^2\,b^8\,e^2}{4}+\frac {5\,B\,a\,b^9\,d\,e}{3}+\frac {5\,A\,a\,b^9\,e^2}{6}+\frac {B\,b^{10}\,d^2}{12}+\frac {A\,b^{10}\,d\,e}{6}\right )+A\,a^{10}\,d^2\,x+\frac {a^9\,d\,x^2\,\left (2\,A\,a\,e+10\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b^9\,e\,x^{13}\,\left (A\,b\,e+10\,B\,a\,e+2\,B\,b\,d\right )}{13}+\frac {B\,b^{10}\,e^2\,x^{14}}{14} \] Input:
int((A + B*x)*(a + b*x)^10*(d + e*x)^2,x)
Output:
x^6*(42*A*a^5*b^5*d^2 + 20*A*a^7*b^3*e^2 + 35*B*a^6*b^4*d^2 + (15*B*a^8*b^ 2*e^2)/2 + 70*A*a^6*b^4*d*e + 40*B*a^7*b^3*d*e) + x^7*(30*A*a^4*b^6*d^2 + 30*A*a^6*b^4*e^2 + 36*B*a^5*b^5*d^2 + (120*B*a^7*b^3*e^2)/7 + 72*A*a^5*b^5 *d*e + 60*B*a^6*b^4*d*e) + x^9*(5*A*a^2*b^8*d^2 + (70*A*a^4*b^6*e^2)/3 + ( 40*B*a^3*b^7*d^2)/3 + 28*B*a^5*b^5*e^2 + (80*A*a^3*b^7*d*e)/3 + (140*B*a^4 *b^6*d*e)/3) + x^8*(15*A*a^3*b^7*d^2 + (63*A*a^5*b^5*e^2)/2 + (105*B*a^4*b ^6*d^2)/4 + (105*B*a^6*b^4*e^2)/4 + (105*A*a^4*b^6*d*e)/2 + 63*B*a^5*b^5*d *e) + x^4*((B*a^10*e^2)/4 + (5*A*a^9*b*e^2)/2 + 30*A*a^7*b^3*d^2 + (45*B*a ^8*b^2*d^2)/4 + 5*B*a^9*b*d*e + (45*A*a^8*b^2*d*e)/2) + x^11*((A*b^10*d^2) /11 + (10*B*a*b^9*d^2)/11 + (45*A*a^2*b^8*e^2)/11 + (120*B*a^3*b^7*e^2)/11 + (20*A*a*b^9*d*e)/11 + (90*B*a^2*b^8*d*e)/11) + x^10*(A*a*b^9*d^2 + 12*A *a^3*b^7*e^2 + (9*B*a^2*b^8*d^2)/2 + 21*B*a^4*b^6*e^2 + 9*A*a^2*b^8*d*e + 24*B*a^3*b^7*d*e) + x^5*(2*B*a^9*b*e^2 + 42*A*a^6*b^4*d^2 + 9*A*a^8*b^2*e^ 2 + 24*B*a^7*b^3*d^2 + 48*A*a^7*b^3*d*e + 18*B*a^8*b^2*d*e) + x^3*((A*a^10 *e^2)/3 + (2*B*a^10*d*e)/3 + (10*B*a^9*b*d^2)/3 + 15*A*a^8*b^2*d^2 + (20*A *a^9*b*d*e)/3) + x^12*((B*b^10*d^2)/12 + (A*b^10*d*e)/6 + (5*A*a*b^9*e^2)/ 6 + (15*B*a^2*b^8*e^2)/4 + (5*B*a*b^9*d*e)/3) + A*a^10*d^2*x + (a^9*d*x^2* (2*A*a*e + 10*A*b*d + B*a*d))/2 + (b^9*e*x^13*(A*b*e + 10*B*a*e + 2*B*b*d) )/13 + (B*b^10*e^2*x^14)/14
Time = 0.16 (sec) , antiderivative size = 459, normalized size of antiderivative = 3.89 \[ \int (a+b x)^{10} (A+B x) (d+e x)^2 \, dx=\frac {x \left (78 b^{11} e^{2} x^{13}+924 a \,b^{10} e^{2} x^{12}+168 b^{11} d e \,x^{12}+5005 a^{2} b^{9} e^{2} x^{11}+2002 a \,b^{10} d e \,x^{11}+91 b^{11} d^{2} x^{11}+16380 a^{3} b^{8} e^{2} x^{10}+10920 a^{2} b^{9} d e \,x^{10}+1092 a \,b^{10} d^{2} x^{10}+36036 a^{4} b^{7} e^{2} x^{9}+36036 a^{3} b^{8} d e \,x^{9}+6006 a^{2} b^{9} d^{2} x^{9}+56056 a^{5} b^{6} e^{2} x^{8}+80080 a^{4} b^{7} d e \,x^{8}+20020 a^{3} b^{8} d^{2} x^{8}+63063 a^{6} b^{5} e^{2} x^{7}+126126 a^{5} b^{6} d e \,x^{7}+45045 a^{4} b^{7} d^{2} x^{7}+51480 a^{7} b^{4} e^{2} x^{6}+144144 a^{6} b^{5} d e \,x^{6}+72072 a^{5} b^{6} d^{2} x^{6}+30030 a^{8} b^{3} e^{2} x^{5}+120120 a^{7} b^{4} d e \,x^{5}+84084 a^{6} b^{5} d^{2} x^{5}+12012 a^{9} b^{2} e^{2} x^{4}+72072 a^{8} b^{3} d e \,x^{4}+72072 a^{7} b^{4} d^{2} x^{4}+3003 a^{10} b \,e^{2} x^{3}+30030 a^{9} b^{2} d e \,x^{3}+45045 a^{8} b^{3} d^{2} x^{3}+364 a^{11} e^{2} x^{2}+8008 a^{10} b d e \,x^{2}+20020 a^{9} b^{2} d^{2} x^{2}+1092 a^{11} d e x +6006 a^{10} b \,d^{2} x +1092 a^{11} d^{2}\right )}{1092} \] Input:
int((b*x+a)^10*(B*x+A)*(e*x+d)^2,x)
Output:
(x*(1092*a**11*d**2 + 1092*a**11*d*e*x + 364*a**11*e**2*x**2 + 6006*a**10* b*d**2*x + 8008*a**10*b*d*e*x**2 + 3003*a**10*b*e**2*x**3 + 20020*a**9*b** 2*d**2*x**2 + 30030*a**9*b**2*d*e*x**3 + 12012*a**9*b**2*e**2*x**4 + 45045 *a**8*b**3*d**2*x**3 + 72072*a**8*b**3*d*e*x**4 + 30030*a**8*b**3*e**2*x** 5 + 72072*a**7*b**4*d**2*x**4 + 120120*a**7*b**4*d*e*x**5 + 51480*a**7*b** 4*e**2*x**6 + 84084*a**6*b**5*d**2*x**5 + 144144*a**6*b**5*d*e*x**6 + 6306 3*a**6*b**5*e**2*x**7 + 72072*a**5*b**6*d**2*x**6 + 126126*a**5*b**6*d*e*x **7 + 56056*a**5*b**6*e**2*x**8 + 45045*a**4*b**7*d**2*x**7 + 80080*a**4*b **7*d*e*x**8 + 36036*a**4*b**7*e**2*x**9 + 20020*a**3*b**8*d**2*x**8 + 360 36*a**3*b**8*d*e*x**9 + 16380*a**3*b**8*e**2*x**10 + 6006*a**2*b**9*d**2*x **9 + 10920*a**2*b**9*d*e*x**10 + 5005*a**2*b**9*e**2*x**11 + 1092*a*b**10 *d**2*x**10 + 2002*a*b**10*d*e*x**11 + 924*a*b**10*e**2*x**12 + 91*b**11*d **2*x**11 + 168*b**11*d*e*x**12 + 78*b**11*e**2*x**13))/1092