Integrand size = 20, antiderivative size = 159 \[ \int (a+b x)^{10} (A+B x) (d+e x)^3 \, dx=\frac {(A b-a B) (b d-a e)^3 (a+b x)^{11}}{11 b^5}+\frac {(b d-a e)^2 (b B d+3 A b e-4 a B e) (a+b x)^{12}}{12 b^5}+\frac {3 e (b d-a e) (b B d+A b e-2 a B e) (a+b x)^{13}}{13 b^5}+\frac {e^2 (3 b B d+A b e-4 a B e) (a+b x)^{14}}{14 b^5}+\frac {B e^3 (a+b x)^{15}}{15 b^5} \] Output:
1/11*(A*b-B*a)*(-a*e+b*d)^3*(b*x+a)^11/b^5+1/12*(-a*e+b*d)^2*(3*A*b*e-4*B* a*e+B*b*d)*(b*x+a)^12/b^5+3/13*e*(-a*e+b*d)*(A*b*e-2*B*a*e+B*b*d)*(b*x+a)^ 13/b^5+1/14*e^2*(A*b*e-4*B*a*e+3*B*b*d)*(b*x+a)^14/b^5+1/15*B*e^3*(b*x+a)^ 15/b^5
Leaf count is larger than twice the leaf count of optimal. \(855\) vs. \(2(159)=318\).
Time = 0.33 (sec) , antiderivative size = 855, normalized size of antiderivative = 5.38 \[ \int (a+b x)^{10} (A+B x) (d+e x)^3 \, dx=\frac {x \left (3003 a^{10} \left (5 A \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+B x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )\right )+10010 a^9 b x \left (3 A \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )+B x \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )\right )+6435 a^8 b^2 x^2 \left (7 A \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )+3 B x \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right )\right )+25740 a^7 b^3 x^3 \left (2 A \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right )+B x \left (56 d^3+140 d^2 e x+120 d e^2 x^2+35 e^3 x^3\right )\right )+5005 a^6 b^4 x^4 \left (9 A \left (56 d^3+140 d^2 e x+120 d e^2 x^2+35 e^3 x^3\right )+5 B x \left (84 d^3+216 d^2 e x+189 d e^2 x^2+56 e^3 x^3\right )\right )+6006 a^5 b^5 x^5 \left (5 A \left (84 d^3+216 d^2 e x+189 d e^2 x^2+56 e^3 x^3\right )+3 B x \left (120 d^3+315 d^2 e x+280 d e^2 x^2+84 e^3 x^3\right )\right )+1365 a^4 b^6 x^6 \left (11 A \left (120 d^3+315 d^2 e x+280 d e^2 x^2+84 e^3 x^3\right )+7 B x \left (165 d^3+440 d^2 e x+396 d e^2 x^2+120 e^3 x^3\right )\right )+1820 a^3 b^7 x^7 \left (3 A \left (165 d^3+440 d^2 e x+396 d e^2 x^2+120 e^3 x^3\right )+2 B x \left (220 d^3+594 d^2 e x+540 d e^2 x^2+165 e^3 x^3\right )\right )+105 a^2 b^8 x^8 \left (13 A \left (220 d^3+594 d^2 e x+540 d e^2 x^2+165 e^3 x^3\right )+9 B x \left (286 d^3+780 d^2 e x+715 d e^2 x^2+220 e^3 x^3\right )\right )+30 a b^9 x^9 \left (7 A \left (286 d^3+780 d^2 e x+715 d e^2 x^2+220 e^3 x^3\right )+5 B x \left (364 d^3+1001 d^2 e x+924 d e^2 x^2+286 e^3 x^3\right )\right )+b^{10} x^{10} \left (15 A \left (364 d^3+1001 d^2 e x+924 d e^2 x^2+286 e^3 x^3\right )+11 B x \left (455 d^3+1260 d^2 e x+1170 d e^2 x^2+364 e^3 x^3\right )\right )\right )}{60060} \] Input:
Integrate[(a + b*x)^10*(A + B*x)*(d + e*x)^3,x]
Output:
(x*(3003*a^10*(5*A*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + B*x*(10*d ^3 + 20*d^2*e*x + 15*d*e^2*x^2 + 4*e^3*x^3)) + 10010*a^9*b*x*(3*A*(10*d^3 + 20*d^2*e*x + 15*d*e^2*x^2 + 4*e^3*x^3) + B*x*(20*d^3 + 45*d^2*e*x + 36*d *e^2*x^2 + 10*e^3*x^3)) + 6435*a^8*b^2*x^2*(7*A*(20*d^3 + 45*d^2*e*x + 36* d*e^2*x^2 + 10*e^3*x^3) + 3*B*x*(35*d^3 + 84*d^2*e*x + 70*d*e^2*x^2 + 20*e ^3*x^3)) + 25740*a^7*b^3*x^3*(2*A*(35*d^3 + 84*d^2*e*x + 70*d*e^2*x^2 + 20 *e^3*x^3) + B*x*(56*d^3 + 140*d^2*e*x + 120*d*e^2*x^2 + 35*e^3*x^3)) + 500 5*a^6*b^4*x^4*(9*A*(56*d^3 + 140*d^2*e*x + 120*d*e^2*x^2 + 35*e^3*x^3) + 5 *B*x*(84*d^3 + 216*d^2*e*x + 189*d*e^2*x^2 + 56*e^3*x^3)) + 6006*a^5*b^5*x ^5*(5*A*(84*d^3 + 216*d^2*e*x + 189*d*e^2*x^2 + 56*e^3*x^3) + 3*B*x*(120*d ^3 + 315*d^2*e*x + 280*d*e^2*x^2 + 84*e^3*x^3)) + 1365*a^4*b^6*x^6*(11*A*( 120*d^3 + 315*d^2*e*x + 280*d*e^2*x^2 + 84*e^3*x^3) + 7*B*x*(165*d^3 + 440 *d^2*e*x + 396*d*e^2*x^2 + 120*e^3*x^3)) + 1820*a^3*b^7*x^7*(3*A*(165*d^3 + 440*d^2*e*x + 396*d*e^2*x^2 + 120*e^3*x^3) + 2*B*x*(220*d^3 + 594*d^2*e* x + 540*d*e^2*x^2 + 165*e^3*x^3)) + 105*a^2*b^8*x^8*(13*A*(220*d^3 + 594*d ^2*e*x + 540*d*e^2*x^2 + 165*e^3*x^3) + 9*B*x*(286*d^3 + 780*d^2*e*x + 715 *d*e^2*x^2 + 220*e^3*x^3)) + 30*a*b^9*x^9*(7*A*(286*d^3 + 780*d^2*e*x + 71 5*d*e^2*x^2 + 220*e^3*x^3) + 5*B*x*(364*d^3 + 1001*d^2*e*x + 924*d*e^2*x^2 + 286*e^3*x^3)) + b^10*x^10*(15*A*(364*d^3 + 1001*d^2*e*x + 924*d*e^2*x^2 + 286*e^3*x^3) + 11*B*x*(455*d^3 + 1260*d^2*e*x + 1170*d*e^2*x^2 + 364...
Time = 0.93 (sec) , antiderivative size = 159, 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.
| \(\displaystyle \int (a+b x)^{10} (A+B x) (d+e x)^3 \, dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {e^2 (a+b x)^{13} (-4 a B e+A b e+3 b B d)}{b^4}+\frac {3 e (a+b x)^{12} (b d-a e) (-2 a B e+A b e+b B d)}{b^4}+\frac {(a+b x)^{11} (b d-a e)^2 (-4 a B e+3 A b e+b B d)}{b^4}+\frac {(a+b x)^{10} (A b-a B) (b d-a e)^3}{b^4}+\frac {B e^3 (a+b x)^{14}}{b^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^2 (a+b x)^{14} (-4 a B e+A b e+3 b B d)}{14 b^5}+\frac {3 e (a+b x)^{13} (b d-a e) (-2 a B e+A b e+b B d)}{13 b^5}+\frac {(a+b x)^{12} (b d-a e)^2 (-4 a B e+3 A b e+b B d)}{12 b^5}+\frac {(a+b x)^{11} (A b-a B) (b d-a e)^3}{11 b^5}+\frac {B e^3 (a+b x)^{15}}{15 b^5}\) |
Input:
Int[(a + b*x)^10*(A + B*x)*(d + e*x)^3,x]
Output:
((A*b - a*B)*(b*d - a*e)^3*(a + b*x)^11)/(11*b^5) + ((b*d - a*e)^2*(b*B*d + 3*A*b*e - 4*a*B*e)*(a + b*x)^12)/(12*b^5) + (3*e*(b*d - a*e)*(b*B*d + A* b*e - 2*a*B*e)*(a + b*x)^13)/(13*b^5) + (e^2*(3*b*B*d + A*b*e - 4*a*B*e)*( a + b*x)^14)/(14*b^5) + (B*e^3*(a + b*x)^15)/(15*b^5)
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. \(1052\) vs. \(2(149)=298\).
Time = 0.21 (sec) , antiderivative size = 1053, normalized size of antiderivative = 6.62
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1053\) |
| norman | \(\text {Expression too large to display}\) | \(1087\) |
| gosper | \(\text {Expression too large to display}\) | \(1280\) |
| risch | \(\text {Expression too large to display}\) | \(1280\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1280\) |
| orering | \(\text {Expression too large to display}\) | \(1280\) |
Input:
int((b*x+a)^10*(B*x+A)*(e*x+d)^3,x,method=_RETURNVERBOSE)
Output:
1/15*b^10*B*e^3*x^15+1/14*((A*b^10+10*B*a*b^9)*e^3+3*b^10*B*d*e^2)*x^14+1/ 13*((10*A*a*b^9+45*B*a^2*b^8)*e^3+3*(A*b^10+10*B*a*b^9)*d*e^2+3*b^10*B*d^2 *e)*x^13+1/12*((45*A*a^2*b^8+120*B*a^3*b^7)*e^3+3*(10*A*a*b^9+45*B*a^2*b^8 )*d*e^2+3*(A*b^10+10*B*a*b^9)*d^2*e+b^10*B*d^3)*x^12+1/11*((120*A*a^3*b^7+ 210*B*a^4*b^6)*e^3+3*(45*A*a^2*b^8+120*B*a^3*b^7)*d*e^2+3*(10*A*a*b^9+45*B *a^2*b^8)*d^2*e+(A*b^10+10*B*a*b^9)*d^3)*x^11+1/10*((210*A*a^4*b^6+252*B*a ^5*b^5)*e^3+3*(120*A*a^3*b^7+210*B*a^4*b^6)*d*e^2+3*(45*A*a^2*b^8+120*B*a^ 3*b^7)*d^2*e+(10*A*a*b^9+45*B*a^2*b^8)*d^3)*x^10+1/9*((252*A*a^5*b^5+210*B *a^6*b^4)*e^3+3*(210*A*a^4*b^6+252*B*a^5*b^5)*d*e^2+3*(120*A*a^3*b^7+210*B *a^4*b^6)*d^2*e+(45*A*a^2*b^8+120*B*a^3*b^7)*d^3)*x^9+1/8*((210*A*a^6*b^4+ 120*B*a^7*b^3)*e^3+3*(252*A*a^5*b^5+210*B*a^6*b^4)*d*e^2+3*(210*A*a^4*b^6+ 252*B*a^5*b^5)*d^2*e+(120*A*a^3*b^7+210*B*a^4*b^6)*d^3)*x^8+1/7*((120*A*a^ 7*b^3+45*B*a^8*b^2)*e^3+3*(210*A*a^6*b^4+120*B*a^7*b^3)*d*e^2+3*(252*A*a^5 *b^5+210*B*a^6*b^4)*d^2*e+(210*A*a^4*b^6+252*B*a^5*b^5)*d^3)*x^7+1/6*((45* A*a^8*b^2+10*B*a^9*b)*e^3+3*(120*A*a^7*b^3+45*B*a^8*b^2)*d*e^2+3*(210*A*a^ 6*b^4+120*B*a^7*b^3)*d^2*e+(252*A*a^5*b^5+210*B*a^6*b^4)*d^3)*x^6+1/5*((10 *A*a^9*b+B*a^10)*e^3+3*(45*A*a^8*b^2+10*B*a^9*b)*d*e^2+3*(120*A*a^7*b^3+45 *B*a^8*b^2)*d^2*e+(210*A*a^6*b^4+120*B*a^7*b^3)*d^3)*x^5+1/4*(a^10*A*e^3+3 *(10*A*a^9*b+B*a^10)*d*e^2+3*(45*A*a^8*b^2+10*B*a^9*b)*d^2*e+(120*A*a^7*b^ 3+45*B*a^8*b^2)*d^3)*x^4+1/3*(3*a^10*A*d*e^2+3*(10*A*a^9*b+B*a^10)*d^2*...
Leaf count of result is larger than twice the leaf count of optimal. 1068 vs. \(2 (149) = 298\).
Time = 0.10 (sec) , antiderivative size = 1068, normalized size of antiderivative = 6.72 \[ \int (a+b x)^{10} (A+B x) (d+e x)^3 \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^10*(B*x+A)*(e*x+d)^3,x, algorithm="fricas")
Output:
1/15*B*b^10*e^3*x^15 + A*a^10*d^3*x + 1/14*(3*B*b^10*d*e^2 + (10*B*a*b^9 + A*b^10)*e^3)*x^14 + 1/13*(3*B*b^10*d^2*e + 3*(10*B*a*b^9 + A*b^10)*d*e^2 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*e^3)*x^13 + 1/12*(B*b^10*d^3 + 3*(10*B*a*b^9 + A*b^10)*d^2*e + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^2 + 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^3)*x^12 + 1/11*((10*B*a*b^9 + A*b^10)*d^3 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e + 45*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^2 + 30*(7*B*a^4*b ^6 + 4*A*a^3*b^7)*e^3)*x^11 + 1/10*(5*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3 + 45*( 8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e + 90*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^2 + 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^3)*x^10 + 1/3*(5*(8*B*a^3*b^7 + 3*A*a^2*b ^8)*d^3 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e + 42*(6*B*a^5*b^5 + 5*A*a^4 *b^6)*d*e^2 + 14*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^3)*x^9 + 3/4*(5*(7*B*a^4*b^ 6 + 4*A*a^3*b^7)*d^3 + 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e + 21*(5*B*a^6* b^4 + 6*A*a^5*b^5)*d*e^2 + 5*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^3)*x^8 + 3/7*(1 4*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e + 30*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^2 + 5*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^3) *x^7 + 1/6*(42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3 + 90*(4*B*a^7*b^3 + 7*A*a^6 *b^4)*d^2*e + 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^2 + 5*(2*B*a^9*b + 9*A*a^ 8*b^2)*e^3)*x^6 + 1/5*(30*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3 + 45*(3*B*a^8*b^ 2 + 8*A*a^7*b^3)*d^2*e + 15*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^2 + (B*a^10 + 10 *A*a^9*b)*e^3)*x^5 + 1/4*(A*a^10*e^3 + 15*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d...
Leaf count of result is larger than twice the leaf count of optimal. 1302 vs. \(2 (156) = 312\).
Time = 0.13 (sec) , antiderivative size = 1302, normalized size of antiderivative = 8.19 \[ \int (a+b x)^{10} (A+B x) (d+e x)^3 \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)**10*(B*x+A)*(e*x+d)**3,x)
Output:
A*a**10*d**3*x + B*b**10*e**3*x**15/15 + x**14*(A*b**10*e**3/14 + 5*B*a*b* *9*e**3/7 + 3*B*b**10*d*e**2/14) + x**13*(10*A*a*b**9*e**3/13 + 3*A*b**10* d*e**2/13 + 45*B*a**2*b**8*e**3/13 + 30*B*a*b**9*d*e**2/13 + 3*B*b**10*d** 2*e/13) + x**12*(15*A*a**2*b**8*e**3/4 + 5*A*a*b**9*d*e**2/2 + A*b**10*d** 2*e/4 + 10*B*a**3*b**7*e**3 + 45*B*a**2*b**8*d*e**2/4 + 5*B*a*b**9*d**2*e/ 2 + B*b**10*d**3/12) + x**11*(120*A*a**3*b**7*e**3/11 + 135*A*a**2*b**8*d* e**2/11 + 30*A*a*b**9*d**2*e/11 + A*b**10*d**3/11 + 210*B*a**4*b**6*e**3/1 1 + 360*B*a**3*b**7*d*e**2/11 + 135*B*a**2*b**8*d**2*e/11 + 10*B*a*b**9*d* *3/11) + x**10*(21*A*a**4*b**6*e**3 + 36*A*a**3*b**7*d*e**2 + 27*A*a**2*b* *8*d**2*e/2 + A*a*b**9*d**3 + 126*B*a**5*b**5*e**3/5 + 63*B*a**4*b**6*d*e* *2 + 36*B*a**3*b**7*d**2*e + 9*B*a**2*b**8*d**3/2) + x**9*(28*A*a**5*b**5* e**3 + 70*A*a**4*b**6*d*e**2 + 40*A*a**3*b**7*d**2*e + 5*A*a**2*b**8*d**3 + 70*B*a**6*b**4*e**3/3 + 84*B*a**5*b**5*d*e**2 + 70*B*a**4*b**6*d**2*e + 40*B*a**3*b**7*d**3/3) + x**8*(105*A*a**6*b**4*e**3/4 + 189*A*a**5*b**5*d* e**2/2 + 315*A*a**4*b**6*d**2*e/4 + 15*A*a**3*b**7*d**3 + 15*B*a**7*b**3*e **3 + 315*B*a**6*b**4*d*e**2/4 + 189*B*a**5*b**5*d**2*e/2 + 105*B*a**4*b** 6*d**3/4) + x**7*(120*A*a**7*b**3*e**3/7 + 90*A*a**6*b**4*d*e**2 + 108*A*a **5*b**5*d**2*e + 30*A*a**4*b**6*d**3 + 45*B*a**8*b**2*e**3/7 + 360*B*a**7 *b**3*d*e**2/7 + 90*B*a**6*b**4*d**2*e + 36*B*a**5*b**5*d**3) + x**6*(15*A *a**8*b**2*e**3/2 + 60*A*a**7*b**3*d*e**2 + 105*A*a**6*b**4*d**2*e + 42...
Leaf count of result is larger than twice the leaf count of optimal. 1068 vs. \(2 (149) = 298\).
Time = 0.04 (sec) , antiderivative size = 1068, normalized size of antiderivative = 6.72 \[ \int (a+b x)^{10} (A+B x) (d+e x)^3 \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^10*(B*x+A)*(e*x+d)^3,x, algorithm="maxima")
Output:
1/15*B*b^10*e^3*x^15 + A*a^10*d^3*x + 1/14*(3*B*b^10*d*e^2 + (10*B*a*b^9 + A*b^10)*e^3)*x^14 + 1/13*(3*B*b^10*d^2*e + 3*(10*B*a*b^9 + A*b^10)*d*e^2 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*e^3)*x^13 + 1/12*(B*b^10*d^3 + 3*(10*B*a*b^9 + A*b^10)*d^2*e + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^2 + 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^3)*x^12 + 1/11*((10*B*a*b^9 + A*b^10)*d^3 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e + 45*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^2 + 30*(7*B*a^4*b ^6 + 4*A*a^3*b^7)*e^3)*x^11 + 1/10*(5*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3 + 45*( 8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e + 90*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^2 + 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^3)*x^10 + 1/3*(5*(8*B*a^3*b^7 + 3*A*a^2*b ^8)*d^3 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e + 42*(6*B*a^5*b^5 + 5*A*a^4 *b^6)*d*e^2 + 14*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^3)*x^9 + 3/4*(5*(7*B*a^4*b^ 6 + 4*A*a^3*b^7)*d^3 + 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e + 21*(5*B*a^6* b^4 + 6*A*a^5*b^5)*d*e^2 + 5*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^3)*x^8 + 3/7*(1 4*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^2*e + 30*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^2 + 5*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^3) *x^7 + 1/6*(42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3 + 90*(4*B*a^7*b^3 + 7*A*a^6 *b^4)*d^2*e + 45*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^2 + 5*(2*B*a^9*b + 9*A*a^ 8*b^2)*e^3)*x^6 + 1/5*(30*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3 + 45*(3*B*a^8*b^ 2 + 8*A*a^7*b^3)*d^2*e + 15*(2*B*a^9*b + 9*A*a^8*b^2)*d*e^2 + (B*a^10 + 10 *A*a^9*b)*e^3)*x^5 + 1/4*(A*a^10*e^3 + 15*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d...
Leaf count of result is larger than twice the leaf count of optimal. 1279 vs. \(2 (149) = 298\).
Time = 0.12 (sec) , antiderivative size = 1279, normalized size of antiderivative = 8.04 \[ \int (a+b x)^{10} (A+B x) (d+e x)^3 \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^10*(B*x+A)*(e*x+d)^3,x, algorithm="giac")
Output:
1/15*B*b^10*e^3*x^15 + 3/14*B*b^10*d*e^2*x^14 + 5/7*B*a*b^9*e^3*x^14 + 1/1 4*A*b^10*e^3*x^14 + 3/13*B*b^10*d^2*e*x^13 + 30/13*B*a*b^9*d*e^2*x^13 + 3/ 13*A*b^10*d*e^2*x^13 + 45/13*B*a^2*b^8*e^3*x^13 + 10/13*A*a*b^9*e^3*x^13 + 1/12*B*b^10*d^3*x^12 + 5/2*B*a*b^9*d^2*e*x^12 + 1/4*A*b^10*d^2*e*x^12 + 4 5/4*B*a^2*b^8*d*e^2*x^12 + 5/2*A*a*b^9*d*e^2*x^12 + 10*B*a^3*b^7*e^3*x^12 + 15/4*A*a^2*b^8*e^3*x^12 + 10/11*B*a*b^9*d^3*x^11 + 1/11*A*b^10*d^3*x^11 + 135/11*B*a^2*b^8*d^2*e*x^11 + 30/11*A*a*b^9*d^2*e*x^11 + 360/11*B*a^3*b^ 7*d*e^2*x^11 + 135/11*A*a^2*b^8*d*e^2*x^11 + 210/11*B*a^4*b^6*e^3*x^11 + 1 20/11*A*a^3*b^7*e^3*x^11 + 9/2*B*a^2*b^8*d^3*x^10 + A*a*b^9*d^3*x^10 + 36* B*a^3*b^7*d^2*e*x^10 + 27/2*A*a^2*b^8*d^2*e*x^10 + 63*B*a^4*b^6*d*e^2*x^10 + 36*A*a^3*b^7*d*e^2*x^10 + 126/5*B*a^5*b^5*e^3*x^10 + 21*A*a^4*b^6*e^3*x ^10 + 40/3*B*a^3*b^7*d^3*x^9 + 5*A*a^2*b^8*d^3*x^9 + 70*B*a^4*b^6*d^2*e*x^ 9 + 40*A*a^3*b^7*d^2*e*x^9 + 84*B*a^5*b^5*d*e^2*x^9 + 70*A*a^4*b^6*d*e^2*x ^9 + 70/3*B*a^6*b^4*e^3*x^9 + 28*A*a^5*b^5*e^3*x^9 + 105/4*B*a^4*b^6*d^3*x ^8 + 15*A*a^3*b^7*d^3*x^8 + 189/2*B*a^5*b^5*d^2*e*x^8 + 315/4*A*a^4*b^6*d^ 2*e*x^8 + 315/4*B*a^6*b^4*d*e^2*x^8 + 189/2*A*a^5*b^5*d*e^2*x^8 + 15*B*a^7 *b^3*e^3*x^8 + 105/4*A*a^6*b^4*e^3*x^8 + 36*B*a^5*b^5*d^3*x^7 + 30*A*a^4*b ^6*d^3*x^7 + 90*B*a^6*b^4*d^2*e*x^7 + 108*A*a^5*b^5*d^2*e*x^7 + 360/7*B*a^ 7*b^3*d*e^2*x^7 + 90*A*a^6*b^4*d*e^2*x^7 + 45/7*B*a^8*b^2*e^3*x^7 + 120/7* A*a^7*b^3*e^3*x^7 + 35*B*a^6*b^4*d^3*x^6 + 42*A*a^5*b^5*d^3*x^6 + 60*B*...
Time = 1.17 (sec) , antiderivative size = 1070, normalized size of antiderivative = 6.73 \[ \int (a+b x)^{10} (A+B x) (d+e x)^3 \, dx =\text {Too large to display} \] Input:
int((A + B*x)*(a + b*x)^10*(d + e*x)^3,x)
Output:
x^3*((10*B*a^9*b*d^3)/3 + A*a^10*d*e^2 + B*a^10*d^2*e + 15*A*a^8*b^2*d^3 + 10*A*a^9*b*d^2*e) + x^13*((10*A*a*b^9*e^3)/13 + (3*A*b^10*d*e^2)/13 + (3* B*b^10*d^2*e)/13 + (45*B*a^2*b^8*e^3)/13 + (30*B*a*b^9*d*e^2)/13) + x^4*(( A*a^10*e^3)/4 + (3*B*a^10*d*e^2)/4 + 30*A*a^7*b^3*d^3 + (45*B*a^8*b^2*d^3) /4 + (135*A*a^8*b^2*d^2*e)/4 + (15*A*a^9*b*d*e^2)/2 + (15*B*a^9*b*d^2*e)/2 ) + x^12*((B*b^10*d^3)/12 + (A*b^10*d^2*e)/4 + (15*A*a^2*b^8*e^3)/4 + 10*B *a^3*b^7*e^3 + (45*B*a^2*b^8*d*e^2)/4 + (5*A*a*b^9*d*e^2)/2 + (5*B*a*b^9*d ^2*e)/2) + x^9*(5*A*a^2*b^8*d^3 + 28*A*a^5*b^5*e^3 + (40*B*a^3*b^7*d^3)/3 + (70*B*a^6*b^4*e^3)/3 + 40*A*a^3*b^7*d^2*e + 70*A*a^4*b^6*d*e^2 + 70*B*a^ 4*b^6*d^2*e + 84*B*a^5*b^5*d*e^2) + x^7*(30*A*a^4*b^6*d^3 + (120*A*a^7*b^3 *e^3)/7 + 36*B*a^5*b^5*d^3 + (45*B*a^8*b^2*e^3)/7 + 108*A*a^5*b^5*d^2*e + 90*A*a^6*b^4*d*e^2 + 90*B*a^6*b^4*d^2*e + (360*B*a^7*b^3*d*e^2)/7) + x^8*( 15*A*a^3*b^7*d^3 + (105*A*a^6*b^4*e^3)/4 + (105*B*a^4*b^6*d^3)/4 + 15*B*a^ 7*b^3*e^3 + (315*A*a^4*b^6*d^2*e)/4 + (189*A*a^5*b^5*d*e^2)/2 + (189*B*a^5 *b^5*d^2*e)/2 + (315*B*a^6*b^4*d*e^2)/4) + x^5*((B*a^10*e^3)/5 + 2*A*a^9*b *e^3 + 42*A*a^6*b^4*d^3 + 24*B*a^7*b^3*d^3 + 72*A*a^7*b^3*d^2*e + 27*A*a^8 *b^2*d*e^2 + 27*B*a^8*b^2*d^2*e + 6*B*a^9*b*d*e^2) + x^11*((A*b^10*d^3)/11 + (10*B*a*b^9*d^3)/11 + (120*A*a^3*b^7*e^3)/11 + (210*B*a^4*b^6*e^3)/11 + (135*A*a^2*b^8*d*e^2)/11 + (135*B*a^2*b^8*d^2*e)/11 + (360*B*a^3*b^7*d*e^ 2)/11 + (30*A*a*b^9*d^2*e)/11) + x^10*(A*a*b^9*d^3 + 21*A*a^4*b^6*e^3 +...
Time = 0.16 (sec) , antiderivative size = 653, normalized size of antiderivative = 4.11 \[ \int (a+b x)^{10} (A+B x) (d+e x)^3 \, dx=\frac {x \left (364 b^{11} e^{3} x^{14}+4290 a \,b^{10} e^{3} x^{13}+1170 b^{11} d \,e^{2} x^{13}+23100 a^{2} b^{9} e^{3} x^{12}+13860 a \,b^{10} d \,e^{2} x^{12}+1260 b^{11} d^{2} e \,x^{12}+75075 a^{3} b^{8} e^{3} x^{11}+75075 a^{2} b^{9} d \,e^{2} x^{11}+15015 a \,b^{10} d^{2} e \,x^{11}+455 b^{11} d^{3} x^{11}+163800 a^{4} b^{7} e^{3} x^{10}+245700 a^{3} b^{8} d \,e^{2} x^{10}+81900 a^{2} b^{9} d^{2} e \,x^{10}+5460 a \,b^{10} d^{3} x^{10}+252252 a^{5} b^{6} e^{3} x^{9}+540540 a^{4} b^{7} d \,e^{2} x^{9}+270270 a^{3} b^{8} d^{2} e \,x^{9}+30030 a^{2} b^{9} d^{3} x^{9}+280280 a^{6} b^{5} e^{3} x^{8}+840840 a^{5} b^{6} d \,e^{2} x^{8}+600600 a^{4} b^{7} d^{2} e \,x^{8}+100100 a^{3} b^{8} d^{3} x^{8}+225225 a^{7} b^{4} e^{3} x^{7}+945945 a^{6} b^{5} d \,e^{2} x^{7}+945945 a^{5} b^{6} d^{2} e \,x^{7}+225225 a^{4} b^{7} d^{3} x^{7}+128700 a^{8} b^{3} e^{3} x^{6}+772200 a^{7} b^{4} d \,e^{2} x^{6}+1081080 a^{6} b^{5} d^{2} e \,x^{6}+360360 a^{5} b^{6} d^{3} x^{6}+50050 a^{9} b^{2} e^{3} x^{5}+450450 a^{8} b^{3} d \,e^{2} x^{5}+900900 a^{7} b^{4} d^{2} e \,x^{5}+420420 a^{6} b^{5} d^{3} x^{5}+12012 a^{10} b \,e^{3} x^{4}+180180 a^{9} b^{2} d \,e^{2} x^{4}+540540 a^{8} b^{3} d^{2} e \,x^{4}+360360 a^{7} b^{4} d^{3} x^{4}+1365 a^{11} e^{3} x^{3}+45045 a^{10} b d \,e^{2} x^{3}+225225 a^{9} b^{2} d^{2} e \,x^{3}+225225 a^{8} b^{3} d^{3} x^{3}+5460 a^{11} d \,e^{2} x^{2}+60060 a^{10} b \,d^{2} e \,x^{2}+100100 a^{9} b^{2} d^{3} x^{2}+8190 a^{11} d^{2} e x +30030 a^{10} b \,d^{3} x +5460 a^{11} d^{3}\right )}{5460} \] Input:
int((b*x+a)^10*(B*x+A)*(e*x+d)^3,x)
Output:
(x*(5460*a**11*d**3 + 8190*a**11*d**2*e*x + 5460*a**11*d*e**2*x**2 + 1365* a**11*e**3*x**3 + 30030*a**10*b*d**3*x + 60060*a**10*b*d**2*e*x**2 + 45045 *a**10*b*d*e**2*x**3 + 12012*a**10*b*e**3*x**4 + 100100*a**9*b**2*d**3*x** 2 + 225225*a**9*b**2*d**2*e*x**3 + 180180*a**9*b**2*d*e**2*x**4 + 50050*a* *9*b**2*e**3*x**5 + 225225*a**8*b**3*d**3*x**3 + 540540*a**8*b**3*d**2*e*x **4 + 450450*a**8*b**3*d*e**2*x**5 + 128700*a**8*b**3*e**3*x**6 + 360360*a **7*b**4*d**3*x**4 + 900900*a**7*b**4*d**2*e*x**5 + 772200*a**7*b**4*d*e** 2*x**6 + 225225*a**7*b**4*e**3*x**7 + 420420*a**6*b**5*d**3*x**5 + 1081080 *a**6*b**5*d**2*e*x**6 + 945945*a**6*b**5*d*e**2*x**7 + 280280*a**6*b**5*e **3*x**8 + 360360*a**5*b**6*d**3*x**6 + 945945*a**5*b**6*d**2*e*x**7 + 840 840*a**5*b**6*d*e**2*x**8 + 252252*a**5*b**6*e**3*x**9 + 225225*a**4*b**7* d**3*x**7 + 600600*a**4*b**7*d**2*e*x**8 + 540540*a**4*b**7*d*e**2*x**9 + 163800*a**4*b**7*e**3*x**10 + 100100*a**3*b**8*d**3*x**8 + 270270*a**3*b** 8*d**2*e*x**9 + 245700*a**3*b**8*d*e**2*x**10 + 75075*a**3*b**8*e**3*x**11 + 30030*a**2*b**9*d**3*x**9 + 81900*a**2*b**9*d**2*e*x**10 + 75075*a**2*b **9*d*e**2*x**11 + 23100*a**2*b**9*e**3*x**12 + 5460*a*b**10*d**3*x**10 + 15015*a*b**10*d**2*e*x**11 + 13860*a*b**10*d*e**2*x**12 + 4290*a*b**10*e** 3*x**13 + 455*b**11*d**3*x**11 + 1260*b**11*d**2*e*x**12 + 1170*b**11*d*e* *2*x**13 + 364*b**11*e**3*x**14))/5460