Integrand size = 20, antiderivative size = 290 \[ \int (a+b x)^6 (A+B x) (d+e x)^6 \, dx=\frac {(A b-a B) (b d-a e)^6 (a+b x)^7}{7 b^8}+\frac {(b d-a e)^5 (b B d+6 A b e-7 a B e) (a+b x)^8}{8 b^8}+\frac {e (b d-a e)^4 (2 b B d+5 A b e-7 a B e) (a+b x)^9}{3 b^8}+\frac {e^2 (b d-a e)^3 (3 b B d+4 A b e-7 a B e) (a+b x)^{10}}{2 b^8}+\frac {5 e^3 (b d-a e)^2 (4 b B d+3 A b e-7 a B e) (a+b x)^{11}}{11 b^8}+\frac {e^4 (b d-a e) (5 b B d+2 A b e-7 a B e) (a+b x)^{12}}{4 b^8}+\frac {e^5 (6 b B d+A b e-7 a B e) (a+b x)^{13}}{13 b^8}+\frac {B e^6 (a+b x)^{14}}{14 b^8} \] Output:
1/7*(A*b-B*a)*(-a*e+b*d)^6*(b*x+a)^7/b^8+1/8*(-a*e+b*d)^5*(6*A*b*e-7*B*a*e +B*b*d)*(b*x+a)^8/b^8+1/3*e*(-a*e+b*d)^4*(5*A*b*e-7*B*a*e+2*B*b*d)*(b*x+a) ^9/b^8+1/2*e^2*(-a*e+b*d)^3*(4*A*b*e-7*B*a*e+3*B*b*d)*(b*x+a)^10/b^8+5/11* e^3*(-a*e+b*d)^2*(3*A*b*e-7*B*a*e+4*B*b*d)*(b*x+a)^11/b^8+1/4*e^4*(-a*e+b* d)*(2*A*b*e-7*B*a*e+5*B*b*d)*(b*x+a)^12/b^8+1/13*e^5*(A*b*e-7*B*a*e+6*B*b* d)*(b*x+a)^13/b^8+1/14*B*e^6*(b*x+a)^14/b^8
Leaf count is larger than twice the leaf count of optimal. \(1069\) vs. \(2(290)=580\).
Time = 0.23 (sec) , antiderivative size = 1069, normalized size of antiderivative = 3.69 \[ \int (a+b x)^6 (A+B x) (d+e x)^6 \, dx=a^6 A d^6 x+\frac {1}{2} a^5 d^5 (a B d+6 A (b d+a e)) x^2+a^4 d^4 \left (2 a B d (b d+a e)+A \left (5 b^2 d^2+12 a b d e+5 a^2 e^2\right )\right ) x^3+\frac {1}{4} a^3 d^3 \left (3 a B d \left (5 b^2 d^2+12 a b d e+5 a^2 e^2\right )+10 A \left (2 b^3 d^3+9 a b^2 d^2 e+9 a^2 b d e^2+2 a^3 e^3\right )\right ) x^4+a^2 d^2 \left (2 a B d \left (2 b^3 d^3+9 a b^2 d^2 e+9 a^2 b d e^2+2 a^3 e^3\right )+3 A \left (b^4 d^4+8 a b^3 d^3 e+15 a^2 b^2 d^2 e^2+8 a^3 b d e^3+a^4 e^4\right )\right ) x^5+\frac {1}{2} a d \left (5 a B d \left (b^4 d^4+8 a b^3 d^3 e+15 a^2 b^2 d^2 e^2+8 a^3 b d e^3+a^4 e^4\right )+2 A \left (b^5 d^5+15 a b^4 d^4 e+50 a^2 b^3 d^3 e^2+50 a^3 b^2 d^2 e^3+15 a^4 b d e^4+a^5 e^5\right )\right ) x^6+\frac {1}{7} \left (6 a B d \left (b^5 d^5+15 a b^4 d^4 e+50 a^2 b^3 d^3 e^2+50 a^3 b^2 d^2 e^3+15 a^4 b d e^4+a^5 e^5\right )+A \left (b^6 d^6+36 a b^5 d^5 e+225 a^2 b^4 d^4 e^2+400 a^3 b^3 d^3 e^3+225 a^4 b^2 d^2 e^4+36 a^5 b d e^5+a^6 e^6\right )\right ) x^7+\frac {1}{8} \left (a^6 B e^6+6 a^5 b e^5 (6 B d+A e)+45 a^4 b^2 d e^4 (5 B d+2 A e)+100 a^3 b^3 d^2 e^3 (4 B d+3 A e)+75 a^2 b^4 d^3 e^2 (3 B d+4 A e)+18 a b^5 d^4 e (2 B d+5 A e)+b^6 d^5 (B d+6 A e)\right ) x^8+\frac {1}{3} b e \left (2 a^5 B e^5+5 a^4 b e^4 (6 B d+A e)+20 a^3 b^2 d e^3 (5 B d+2 A e)+25 a^2 b^3 d^2 e^2 (4 B d+3 A e)+10 a b^4 d^3 e (3 B d+4 A e)+b^5 d^4 (2 B d+5 A e)\right ) x^9+\frac {1}{2} b^2 e^2 \left (3 a^4 B e^4+4 a^3 b e^3 (6 B d+A e)+9 a^2 b^2 d e^2 (5 B d+2 A e)+6 a b^3 d^2 e (4 B d+3 A e)+b^4 d^3 (3 B d+4 A e)\right ) x^{10}+\frac {1}{11} b^3 e^3 \left (20 a^3 B e^3+15 a^2 b e^2 (6 B d+A e)+18 a b^2 d e (5 B d+2 A e)+5 b^3 d^2 (4 B d+3 A e)\right ) x^{11}+\frac {1}{4} b^4 e^4 \left (5 a^2 B e^2+2 a b e (6 B d+A e)+b^2 d (5 B d+2 A e)\right ) x^{12}+\frac {1}{13} b^5 e^5 (6 b B d+A b e+6 a B e) x^{13}+\frac {1}{14} b^6 B e^6 x^{14} \] Input:
Integrate[(a + b*x)^6*(A + B*x)*(d + e*x)^6,x]
Output:
a^6*A*d^6*x + (a^5*d^5*(a*B*d + 6*A*(b*d + a*e))*x^2)/2 + a^4*d^4*(2*a*B*d *(b*d + a*e) + A*(5*b^2*d^2 + 12*a*b*d*e + 5*a^2*e^2))*x^3 + (a^3*d^3*(3*a *B*d*(5*b^2*d^2 + 12*a*b*d*e + 5*a^2*e^2) + 10*A*(2*b^3*d^3 + 9*a*b^2*d^2* e + 9*a^2*b*d*e^2 + 2*a^3*e^3))*x^4)/4 + a^2*d^2*(2*a*B*d*(2*b^3*d^3 + 9*a *b^2*d^2*e + 9*a^2*b*d*e^2 + 2*a^3*e^3) + 3*A*(b^4*d^4 + 8*a*b^3*d^3*e + 1 5*a^2*b^2*d^2*e^2 + 8*a^3*b*d*e^3 + a^4*e^4))*x^5 + (a*d*(5*a*B*d*(b^4*d^4 + 8*a*b^3*d^3*e + 15*a^2*b^2*d^2*e^2 + 8*a^3*b*d*e^3 + a^4*e^4) + 2*A*(b^ 5*d^5 + 15*a*b^4*d^4*e + 50*a^2*b^3*d^3*e^2 + 50*a^3*b^2*d^2*e^3 + 15*a^4* b*d*e^4 + a^5*e^5))*x^6)/2 + ((6*a*B*d*(b^5*d^5 + 15*a*b^4*d^4*e + 50*a^2* b^3*d^3*e^2 + 50*a^3*b^2*d^2*e^3 + 15*a^4*b*d*e^4 + a^5*e^5) + A*(b^6*d^6 + 36*a*b^5*d^5*e + 225*a^2*b^4*d^4*e^2 + 400*a^3*b^3*d^3*e^3 + 225*a^4*b^2 *d^2*e^4 + 36*a^5*b*d*e^5 + a^6*e^6))*x^7)/7 + ((a^6*B*e^6 + 6*a^5*b*e^5*( 6*B*d + A*e) + 45*a^4*b^2*d*e^4*(5*B*d + 2*A*e) + 100*a^3*b^3*d^2*e^3*(4*B *d + 3*A*e) + 75*a^2*b^4*d^3*e^2*(3*B*d + 4*A*e) + 18*a*b^5*d^4*e*(2*B*d + 5*A*e) + b^6*d^5*(B*d + 6*A*e))*x^8)/8 + (b*e*(2*a^5*B*e^5 + 5*a^4*b*e^4* (6*B*d + A*e) + 20*a^3*b^2*d*e^3*(5*B*d + 2*A*e) + 25*a^2*b^3*d^2*e^2*(4*B *d + 3*A*e) + 10*a*b^4*d^3*e*(3*B*d + 4*A*e) + b^5*d^4*(2*B*d + 5*A*e))*x^ 9)/3 + (b^2*e^2*(3*a^4*B*e^4 + 4*a^3*b*e^3*(6*B*d + A*e) + 9*a^2*b^2*d*e^2 *(5*B*d + 2*A*e) + 6*a*b^3*d^2*e*(4*B*d + 3*A*e) + b^4*d^3*(3*B*d + 4*A*e) )*x^10)/2 + (b^3*e^3*(20*a^3*B*e^3 + 15*a^2*b*e^2*(6*B*d + A*e) + 18*a*...
Time = 1.31 (sec) , antiderivative size = 290, 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.
| \(\displaystyle \int (a+b x)^6 (A+B x) (d+e x)^6 \, dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {e^5 (a+b x)^{12} (-7 a B e+A b e+6 b B d)}{b^7}+\frac {3 e^4 (a+b x)^{11} (b d-a e) (-7 a B e+2 A b e+5 b B d)}{b^7}+\frac {5 e^3 (a+b x)^{10} (b d-a e)^2 (-7 a B e+3 A b e+4 b B d)}{b^7}+\frac {5 e^2 (a+b x)^9 (b d-a e)^3 (-7 a B e+4 A b e+3 b B d)}{b^7}+\frac {3 e (a+b x)^8 (b d-a e)^4 (-7 a B e+5 A b e+2 b B d)}{b^7}+\frac {(a+b x)^7 (b d-a e)^5 (-7 a B e+6 A b e+b B d)}{b^7}+\frac {(a+b x)^6 (A b-a B) (b d-a e)^6}{b^7}+\frac {B e^6 (a+b x)^{13}}{b^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^5 (a+b x)^{13} (-7 a B e+A b e+6 b B d)}{13 b^8}+\frac {e^4 (a+b x)^{12} (b d-a e) (-7 a B e+2 A b e+5 b B d)}{4 b^8}+\frac {5 e^3 (a+b x)^{11} (b d-a e)^2 (-7 a B e+3 A b e+4 b B d)}{11 b^8}+\frac {e^2 (a+b x)^{10} (b d-a e)^3 (-7 a B e+4 A b e+3 b B d)}{2 b^8}+\frac {e (a+b x)^9 (b d-a e)^4 (-7 a B e+5 A b e+2 b B d)}{3 b^8}+\frac {(a+b x)^8 (b d-a e)^5 (-7 a B e+6 A b e+b B d)}{8 b^8}+\frac {(a+b x)^7 (A b-a B) (b d-a e)^6}{7 b^8}+\frac {B e^6 (a+b x)^{14}}{14 b^8}\) |
Input:
Int[(a + b*x)^6*(A + B*x)*(d + e*x)^6,x]
Output:
((A*b - a*B)*(b*d - a*e)^6*(a + b*x)^7)/(7*b^8) + ((b*d - a*e)^5*(b*B*d + 6*A*b*e - 7*a*B*e)*(a + b*x)^8)/(8*b^8) + (e*(b*d - a*e)^4*(2*b*B*d + 5*A* b*e - 7*a*B*e)*(a + b*x)^9)/(3*b^8) + (e^2*(b*d - a*e)^3*(3*b*B*d + 4*A*b* e - 7*a*B*e)*(a + b*x)^10)/(2*b^8) + (5*e^3*(b*d - a*e)^2*(4*b*B*d + 3*A*b *e - 7*a*B*e)*(a + b*x)^11)/(11*b^8) + (e^4*(b*d - a*e)*(5*b*B*d + 2*A*b*e - 7*a*B*e)*(a + b*x)^12)/(4*b^8) + (e^5*(6*b*B*d + A*b*e - 7*a*B*e)*(a + b*x)^13)/(13*b^8) + (B*e^6*(a + b*x)^14)/(14*b^8)
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. \(1172\) vs. \(2(274)=548\).
Time = 0.24 (sec) , antiderivative size = 1173, normalized size of antiderivative = 4.04
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1173\) |
| norman | \(\text {Expression too large to display}\) | \(1253\) |
| orering | \(\text {Expression too large to display}\) | \(1480\) |
| gosper | \(\text {Expression too large to display}\) | \(1481\) |
| risch | \(\text {Expression too large to display}\) | \(1481\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1481\) |
Input:
int((b*x+a)^6*(B*x+A)*(e*x+d)^6,x,method=_RETURNVERBOSE)
Output:
1/14*b^6*B*e^6*x^14+1/13*((A*b^6+6*B*a*b^5)*e^6+6*b^6*B*d*e^5)*x^13+1/12*( (6*A*a*b^5+15*B*a^2*b^4)*e^6+6*(A*b^6+6*B*a*b^5)*d*e^5+15*b^6*B*d^2*e^4)*x ^12+1/11*((15*A*a^2*b^4+20*B*a^3*b^3)*e^6+6*(6*A*a*b^5+15*B*a^2*b^4)*d*e^5 +15*(A*b^6+6*B*a*b^5)*d^2*e^4+20*b^6*B*d^3*e^3)*x^11+1/10*((20*A*a^3*b^3+1 5*B*a^4*b^2)*e^6+6*(15*A*a^2*b^4+20*B*a^3*b^3)*d*e^5+15*(6*A*a*b^5+15*B*a^ 2*b^4)*d^2*e^4+20*(A*b^6+6*B*a*b^5)*d^3*e^3+15*b^6*B*d^4*e^2)*x^10+1/9*((1 5*A*a^4*b^2+6*B*a^5*b)*e^6+6*(20*A*a^3*b^3+15*B*a^4*b^2)*d*e^5+15*(15*A*a^ 2*b^4+20*B*a^3*b^3)*d^2*e^4+20*(6*A*a*b^5+15*B*a^2*b^4)*d^3*e^3+15*(A*b^6+ 6*B*a*b^5)*d^4*e^2+6*b^6*B*d^5*e)*x^9+1/8*((6*A*a^5*b+B*a^6)*e^6+6*(15*A*a ^4*b^2+6*B*a^5*b)*d*e^5+15*(20*A*a^3*b^3+15*B*a^4*b^2)*d^2*e^4+20*(15*A*a^ 2*b^4+20*B*a^3*b^3)*d^3*e^3+15*(6*A*a*b^5+15*B*a^2*b^4)*d^4*e^2+6*(A*b^6+6 *B*a*b^5)*d^5*e+b^6*B*d^6)*x^8+1/7*(a^6*A*e^6+6*(6*A*a^5*b+B*a^6)*d*e^5+15 *(15*A*a^4*b^2+6*B*a^5*b)*d^2*e^4+20*(20*A*a^3*b^3+15*B*a^4*b^2)*d^3*e^3+1 5*(15*A*a^2*b^4+20*B*a^3*b^3)*d^4*e^2+6*(6*A*a*b^5+15*B*a^2*b^4)*d^5*e+(A* b^6+6*B*a*b^5)*d^6)*x^7+1/6*(6*a^6*A*d*e^5+15*(6*A*a^5*b+B*a^6)*d^2*e^4+20 *(15*A*a^4*b^2+6*B*a^5*b)*d^3*e^3+15*(20*A*a^3*b^3+15*B*a^4*b^2)*d^4*e^2+6 *(15*A*a^2*b^4+20*B*a^3*b^3)*d^5*e+(6*A*a*b^5+15*B*a^2*b^4)*d^6)*x^6+1/5*( 15*a^6*A*d^2*e^4+20*(6*A*a^5*b+B*a^6)*d^3*e^3+15*(15*A*a^4*b^2+6*B*a^5*b)* d^4*e^2+6*(20*A*a^3*b^3+15*B*a^4*b^2)*d^5*e+(15*A*a^2*b^4+20*B*a^3*b^3)*d^ 6)*x^5+1/4*(20*a^6*A*d^3*e^3+15*(6*A*a^5*b+B*a^6)*d^4*e^2+6*(15*A*a^4*b...
Leaf count of result is larger than twice the leaf count of optimal. 1172 vs. \(2 (274) = 548\).
Time = 0.08 (sec) , antiderivative size = 1172, normalized size of antiderivative = 4.04 \[ \int (a+b x)^6 (A+B x) (d+e x)^6 \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^6*(B*x+A)*(e*x+d)^6,x, algorithm="fricas")
Output:
1/14*B*b^6*e^6*x^14 + A*a^6*d^6*x + 1/13*(6*B*b^6*d*e^5 + (6*B*a*b^5 + A*b ^6)*e^6)*x^13 + 1/4*(5*B*b^6*d^2*e^4 + 2*(6*B*a*b^5 + A*b^6)*d*e^5 + (5*B* a^2*b^4 + 2*A*a*b^5)*e^6)*x^12 + 1/11*(20*B*b^6*d^3*e^3 + 15*(6*B*a*b^5 + A*b^6)*d^2*e^4 + 18*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^5 + 5*(4*B*a^3*b^3 + 3*A *a^2*b^4)*e^6)*x^11 + 1/2*(3*B*b^6*d^4*e^2 + 4*(6*B*a*b^5 + A*b^6)*d^3*e^3 + 9*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^4 + 6*(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^6)*x^10 + 1/3*(2*B*b^6*d^5*e + 5*(6*B*a *b^5 + A*b^6)*d^4*e^2 + 20*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^3 + 25*(4*B*a^3 *b^3 + 3*A*a^2*b^4)*d^2*e^4 + 10*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^5 + (2*B* a^5*b + 5*A*a^4*b^2)*e^6)*x^9 + 1/8*(B*b^6*d^6 + 6*(6*B*a*b^5 + A*b^6)*d^5 *e + 45*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^2 + 100*(4*B*a^3*b^3 + 3*A*a^2*b^4 )*d^3*e^3 + 75*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^4 + 18*(2*B*a^5*b + 5*A*a ^4*b^2)*d*e^5 + (B*a^6 + 6*A*a^5*b)*e^6)*x^8 + 1/7*(A*a^6*e^6 + (6*B*a*b^5 + A*b^6)*d^6 + 18*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e + 75*(4*B*a^3*b^3 + 3*A *a^2*b^4)*d^4*e^2 + 100*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^3 + 45*(2*B*a^5* b + 5*A*a^4*b^2)*d^2*e^4 + 6*(B*a^6 + 6*A*a^5*b)*d*e^5)*x^7 + 1/2*(2*A*a^6 *d*e^5 + (5*B*a^2*b^4 + 2*A*a*b^5)*d^6 + 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^ 5*e + 25*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^4*e^2 + 20*(2*B*a^5*b + 5*A*a^4*b^2 )*d^3*e^3 + 5*(B*a^6 + 6*A*a^5*b)*d^2*e^4)*x^6 + (3*A*a^6*d^2*e^4 + (4*B*a ^3*b^3 + 3*A*a^2*b^4)*d^6 + 6*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^5*e + 9*(2*...
Leaf count of result is larger than twice the leaf count of optimal. 1504 vs. \(2 (292) = 584\).
Time = 0.09 (sec) , antiderivative size = 1504, normalized size of antiderivative = 5.19 \[ \int (a+b x)^6 (A+B x) (d+e x)^6 \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)**6*(B*x+A)*(e*x+d)**6,x)
Output:
A*a**6*d**6*x + B*b**6*e**6*x**14/14 + x**13*(A*b**6*e**6/13 + 6*B*a*b**5* e**6/13 + 6*B*b**6*d*e**5/13) + x**12*(A*a*b**5*e**6/2 + A*b**6*d*e**5/2 + 5*B*a**2*b**4*e**6/4 + 3*B*a*b**5*d*e**5 + 5*B*b**6*d**2*e**4/4) + x**11* (15*A*a**2*b**4*e**6/11 + 36*A*a*b**5*d*e**5/11 + 15*A*b**6*d**2*e**4/11 + 20*B*a**3*b**3*e**6/11 + 90*B*a**2*b**4*d*e**5/11 + 90*B*a*b**5*d**2*e**4 /11 + 20*B*b**6*d**3*e**3/11) + x**10*(2*A*a**3*b**3*e**6 + 9*A*a**2*b**4* d*e**5 + 9*A*a*b**5*d**2*e**4 + 2*A*b**6*d**3*e**3 + 3*B*a**4*b**2*e**6/2 + 12*B*a**3*b**3*d*e**5 + 45*B*a**2*b**4*d**2*e**4/2 + 12*B*a*b**5*d**3*e* *3 + 3*B*b**6*d**4*e**2/2) + x**9*(5*A*a**4*b**2*e**6/3 + 40*A*a**3*b**3*d *e**5/3 + 25*A*a**2*b**4*d**2*e**4 + 40*A*a*b**5*d**3*e**3/3 + 5*A*b**6*d* *4*e**2/3 + 2*B*a**5*b*e**6/3 + 10*B*a**4*b**2*d*e**5 + 100*B*a**3*b**3*d* *2*e**4/3 + 100*B*a**2*b**4*d**3*e**3/3 + 10*B*a*b**5*d**4*e**2 + 2*B*b**6 *d**5*e/3) + x**8*(3*A*a**5*b*e**6/4 + 45*A*a**4*b**2*d*e**5/4 + 75*A*a**3 *b**3*d**2*e**4/2 + 75*A*a**2*b**4*d**3*e**3/2 + 45*A*a*b**5*d**4*e**2/4 + 3*A*b**6*d**5*e/4 + B*a**6*e**6/8 + 9*B*a**5*b*d*e**5/2 + 225*B*a**4*b**2 *d**2*e**4/8 + 50*B*a**3*b**3*d**3*e**3 + 225*B*a**2*b**4*d**4*e**2/8 + 9* B*a*b**5*d**5*e/2 + B*b**6*d**6/8) + x**7*(A*a**6*e**6/7 + 36*A*a**5*b*d*e **5/7 + 225*A*a**4*b**2*d**2*e**4/7 + 400*A*a**3*b**3*d**3*e**3/7 + 225*A* a**2*b**4*d**4*e**2/7 + 36*A*a*b**5*d**5*e/7 + A*b**6*d**6/7 + 6*B*a**6*d* e**5/7 + 90*B*a**5*b*d**2*e**4/7 + 300*B*a**4*b**2*d**3*e**3/7 + 300*B*...
Leaf count of result is larger than twice the leaf count of optimal. 1172 vs. \(2 (274) = 548\).
Time = 0.04 (sec) , antiderivative size = 1172, normalized size of antiderivative = 4.04 \[ \int (a+b x)^6 (A+B x) (d+e x)^6 \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^6*(B*x+A)*(e*x+d)^6,x, algorithm="maxima")
Output:
1/14*B*b^6*e^6*x^14 + A*a^6*d^6*x + 1/13*(6*B*b^6*d*e^5 + (6*B*a*b^5 + A*b ^6)*e^6)*x^13 + 1/4*(5*B*b^6*d^2*e^4 + 2*(6*B*a*b^5 + A*b^6)*d*e^5 + (5*B* a^2*b^4 + 2*A*a*b^5)*e^6)*x^12 + 1/11*(20*B*b^6*d^3*e^3 + 15*(6*B*a*b^5 + A*b^6)*d^2*e^4 + 18*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^5 + 5*(4*B*a^3*b^3 + 3*A *a^2*b^4)*e^6)*x^11 + 1/2*(3*B*b^6*d^4*e^2 + 4*(6*B*a*b^5 + A*b^6)*d^3*e^3 + 9*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^4 + 6*(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^6)*x^10 + 1/3*(2*B*b^6*d^5*e + 5*(6*B*a *b^5 + A*b^6)*d^4*e^2 + 20*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^3 + 25*(4*B*a^3 *b^3 + 3*A*a^2*b^4)*d^2*e^4 + 10*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^5 + (2*B* a^5*b + 5*A*a^4*b^2)*e^6)*x^9 + 1/8*(B*b^6*d^6 + 6*(6*B*a*b^5 + A*b^6)*d^5 *e + 45*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^2 + 100*(4*B*a^3*b^3 + 3*A*a^2*b^4 )*d^3*e^3 + 75*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^4 + 18*(2*B*a^5*b + 5*A*a ^4*b^2)*d*e^5 + (B*a^6 + 6*A*a^5*b)*e^6)*x^8 + 1/7*(A*a^6*e^6 + (6*B*a*b^5 + A*b^6)*d^6 + 18*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e + 75*(4*B*a^3*b^3 + 3*A *a^2*b^4)*d^4*e^2 + 100*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^3 + 45*(2*B*a^5* b + 5*A*a^4*b^2)*d^2*e^4 + 6*(B*a^6 + 6*A*a^5*b)*d*e^5)*x^7 + 1/2*(2*A*a^6 *d*e^5 + (5*B*a^2*b^4 + 2*A*a*b^5)*d^6 + 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^ 5*e + 25*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^4*e^2 + 20*(2*B*a^5*b + 5*A*a^4*b^2 )*d^3*e^3 + 5*(B*a^6 + 6*A*a^5*b)*d^2*e^4)*x^6 + (3*A*a^6*d^2*e^4 + (4*B*a ^3*b^3 + 3*A*a^2*b^4)*d^6 + 6*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^5*e + 9*(2*...
Leaf count of result is larger than twice the leaf count of optimal. 1480 vs. \(2 (274) = 548\).
Time = 0.12 (sec) , antiderivative size = 1480, normalized size of antiderivative = 5.10 \[ \int (a+b x)^6 (A+B x) (d+e x)^6 \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^6*(B*x+A)*(e*x+d)^6,x, algorithm="giac")
Output:
1/14*B*b^6*e^6*x^14 + 6/13*B*b^6*d*e^5*x^13 + 6/13*B*a*b^5*e^6*x^13 + 1/13 *A*b^6*e^6*x^13 + 5/4*B*b^6*d^2*e^4*x^12 + 3*B*a*b^5*d*e^5*x^12 + 1/2*A*b^ 6*d*e^5*x^12 + 5/4*B*a^2*b^4*e^6*x^12 + 1/2*A*a*b^5*e^6*x^12 + 20/11*B*b^6 *d^3*e^3*x^11 + 90/11*B*a*b^5*d^2*e^4*x^11 + 15/11*A*b^6*d^2*e^4*x^11 + 90 /11*B*a^2*b^4*d*e^5*x^11 + 36/11*A*a*b^5*d*e^5*x^11 + 20/11*B*a^3*b^3*e^6* x^11 + 15/11*A*a^2*b^4*e^6*x^11 + 3/2*B*b^6*d^4*e^2*x^10 + 12*B*a*b^5*d^3* e^3*x^10 + 2*A*b^6*d^3*e^3*x^10 + 45/2*B*a^2*b^4*d^2*e^4*x^10 + 9*A*a*b^5* d^2*e^4*x^10 + 12*B*a^3*b^3*d*e^5*x^10 + 9*A*a^2*b^4*d*e^5*x^10 + 3/2*B*a^ 4*b^2*e^6*x^10 + 2*A*a^3*b^3*e^6*x^10 + 2/3*B*b^6*d^5*e*x^9 + 10*B*a*b^5*d ^4*e^2*x^9 + 5/3*A*b^6*d^4*e^2*x^9 + 100/3*B*a^2*b^4*d^3*e^3*x^9 + 40/3*A* a*b^5*d^3*e^3*x^9 + 100/3*B*a^3*b^3*d^2*e^4*x^9 + 25*A*a^2*b^4*d^2*e^4*x^9 + 10*B*a^4*b^2*d*e^5*x^9 + 40/3*A*a^3*b^3*d*e^5*x^9 + 2/3*B*a^5*b*e^6*x^9 + 5/3*A*a^4*b^2*e^6*x^9 + 1/8*B*b^6*d^6*x^8 + 9/2*B*a*b^5*d^5*e*x^8 + 3/4 *A*b^6*d^5*e*x^8 + 225/8*B*a^2*b^4*d^4*e^2*x^8 + 45/4*A*a*b^5*d^4*e^2*x^8 + 50*B*a^3*b^3*d^3*e^3*x^8 + 75/2*A*a^2*b^4*d^3*e^3*x^8 + 225/8*B*a^4*b^2* d^2*e^4*x^8 + 75/2*A*a^3*b^3*d^2*e^4*x^8 + 9/2*B*a^5*b*d*e^5*x^8 + 45/4*A* a^4*b^2*d*e^5*x^8 + 1/8*B*a^6*e^6*x^8 + 3/4*A*a^5*b*e^6*x^8 + 6/7*B*a*b^5* d^6*x^7 + 1/7*A*b^6*d^6*x^7 + 90/7*B*a^2*b^4*d^5*e*x^7 + 36/7*A*a*b^5*d^5* e*x^7 + 300/7*B*a^3*b^3*d^4*e^2*x^7 + 225/7*A*a^2*b^4*d^4*e^2*x^7 + 300/7* B*a^4*b^2*d^3*e^3*x^7 + 400/7*A*a^3*b^3*d^3*e^3*x^7 + 90/7*B*a^5*b*d^2*...
Time = 1.15 (sec) , antiderivative size = 1221, normalized size of antiderivative = 4.21 \[ \int (a+b x)^6 (A+B x) (d+e x)^6 \, dx =\text {Too large to display} \] Input:
int((A + B*x)*(a + b*x)^6*(d + e*x)^6,x)
Output:
x^5*(3*A*a^2*b^4*d^6 + 4*B*a^3*b^3*d^6 + 3*A*a^6*d^2*e^4 + 4*B*a^6*d^3*e^3 + 24*A*a^3*b^3*d^5*e + 24*A*a^5*b*d^3*e^3 + 18*B*a^4*b^2*d^5*e + 18*B*a^5 *b*d^4*e^2 + 45*A*a^4*b^2*d^4*e^2) + x^10*(2*A*a^3*b^3*e^6 + (3*B*a^4*b^2* e^6)/2 + 2*A*b^6*d^3*e^3 + (3*B*b^6*d^4*e^2)/2 + 9*A*a*b^5*d^2*e^4 + 9*A*a ^2*b^4*d*e^5 + 12*B*a*b^5*d^3*e^3 + 12*B*a^3*b^3*d*e^5 + (45*B*a^2*b^4*d^2 *e^4)/2) + x^6*(A*a*b^5*d^6 + A*a^6*d*e^5 + (5*B*a^2*b^4*d^6)/2 + (5*B*a^6 *d^2*e^4)/2 + 15*A*a^2*b^4*d^5*e + 15*A*a^5*b*d^2*e^4 + 20*B*a^3*b^3*d^5*e + 20*B*a^5*b*d^3*e^3 + 50*A*a^3*b^3*d^4*e^2 + 50*A*a^4*b^2*d^3*e^3 + (75* B*a^4*b^2*d^4*e^2)/2) + x^9*((2*B*a^5*b*e^6)/3 + (2*B*b^6*d^5*e)/3 + (5*A* a^4*b^2*e^6)/3 + (5*A*b^6*d^4*e^2)/3 + (40*A*a*b^5*d^3*e^3)/3 + (40*A*a^3* b^3*d*e^5)/3 + 10*B*a*b^5*d^4*e^2 + 10*B*a^4*b^2*d*e^5 + 25*A*a^2*b^4*d^2* e^4 + (100*B*a^2*b^4*d^3*e^3)/3 + (100*B*a^3*b^3*d^2*e^4)/3) + x^7*((A*a^6 *e^6)/7 + (A*b^6*d^6)/7 + (6*B*a*b^5*d^6)/7 + (6*B*a^6*d*e^5)/7 + (90*B*a^ 2*b^4*d^5*e)/7 + (90*B*a^5*b*d^2*e^4)/7 + (225*A*a^2*b^4*d^4*e^2)/7 + (400 *A*a^3*b^3*d^3*e^3)/7 + (225*A*a^4*b^2*d^2*e^4)/7 + (300*B*a^3*b^3*d^4*e^2 )/7 + (300*B*a^4*b^2*d^3*e^3)/7 + (36*A*a*b^5*d^5*e)/7 + (36*A*a^5*b*d*e^5 )/7) + x^4*(5*A*a^3*b^3*d^6 + (15*B*a^4*b^2*d^6)/4 + 5*A*a^6*d^3*e^3 + (15 *B*a^6*d^4*e^2)/4 + (45*A*a^4*b^2*d^5*e)/2 + (45*A*a^5*b*d^4*e^2)/2 + 9*B* a^5*b*d^5*e) + x^8*((B*a^6*e^6)/8 + (B*b^6*d^6)/8 + (3*A*a^5*b*e^6)/4 + (3 *A*b^6*d^5*e)/4 + (45*A*a*b^5*d^4*e^2)/4 + (45*A*a^4*b^2*d*e^5)/4 + (75...
Time = 0.16 (sec) , antiderivative size = 799, normalized size of antiderivative = 2.76 \[ \int (a+b x)^6 (A+B x) (d+e x)^6 \, dx=\frac {x \left (1716 b^{7} e^{6} x^{13}+12936 a \,b^{6} e^{6} x^{12}+11088 b^{7} d \,e^{5} x^{12}+42042 a^{2} b^{5} e^{6} x^{11}+84084 a \,b^{6} d \,e^{5} x^{11}+30030 b^{7} d^{2} e^{4} x^{11}+76440 a^{3} b^{4} e^{6} x^{10}+275184 a^{2} b^{5} d \,e^{5} x^{10}+229320 a \,b^{6} d^{2} e^{4} x^{10}+43680 b^{7} d^{3} e^{3} x^{10}+84084 a^{4} b^{3} e^{6} x^{9}+504504 a^{3} b^{4} d \,e^{5} x^{9}+756756 a^{2} b^{5} d^{2} e^{4} x^{9}+336336 a \,b^{6} d^{3} e^{3} x^{9}+36036 b^{7} d^{4} e^{2} x^{9}+56056 a^{5} b^{2} e^{6} x^{8}+560560 a^{4} b^{3} d \,e^{5} x^{8}+1401400 a^{3} b^{4} d^{2} e^{4} x^{8}+1121120 a^{2} b^{5} d^{3} e^{3} x^{8}+280280 a \,b^{6} d^{4} e^{2} x^{8}+16016 b^{7} d^{5} e \,x^{8}+21021 a^{6} b \,e^{6} x^{7}+378378 a^{5} b^{2} d \,e^{5} x^{7}+1576575 a^{4} b^{3} d^{2} e^{4} x^{7}+2102100 a^{3} b^{4} d^{3} e^{3} x^{7}+945945 a^{2} b^{5} d^{4} e^{2} x^{7}+126126 a \,b^{6} d^{5} e \,x^{7}+3003 b^{7} d^{6} x^{7}+3432 a^{7} e^{6} x^{6}+144144 a^{6} b d \,e^{5} x^{6}+1081080 a^{5} b^{2} d^{2} e^{4} x^{6}+2402400 a^{4} b^{3} d^{3} e^{3} x^{6}+1801800 a^{3} b^{4} d^{4} e^{2} x^{6}+432432 a^{2} b^{5} d^{5} e \,x^{6}+24024 a \,b^{6} d^{6} x^{6}+24024 a^{7} d \,e^{5} x^{5}+420420 a^{6} b \,d^{2} e^{4} x^{5}+1681680 a^{5} b^{2} d^{3} e^{3} x^{5}+2102100 a^{4} b^{3} d^{4} e^{2} x^{5}+840840 a^{3} b^{4} d^{5} e \,x^{5}+84084 a^{2} b^{5} d^{6} x^{5}+72072 a^{7} d^{2} e^{4} x^{4}+672672 a^{6} b \,d^{3} e^{3} x^{4}+1513512 a^{5} b^{2} d^{4} e^{2} x^{4}+1009008 a^{4} b^{3} d^{5} e \,x^{4}+168168 a^{3} b^{4} d^{6} x^{4}+120120 a^{7} d^{3} e^{3} x^{3}+630630 a^{6} b \,d^{4} e^{2} x^{3}+756756 a^{5} b^{2} d^{5} e \,x^{3}+210210 a^{4} b^{3} d^{6} x^{3}+120120 a^{7} d^{4} e^{2} x^{2}+336336 a^{6} b \,d^{5} e \,x^{2}+168168 a^{5} b^{2} d^{6} x^{2}+72072 a^{7} d^{5} e x +84084 a^{6} b \,d^{6} x +24024 a^{7} d^{6}\right )}{24024} \] Input:
int((b*x+a)^6*(B*x+A)*(e*x+d)^6,x)
Output:
(x*(24024*a**7*d**6 + 72072*a**7*d**5*e*x + 120120*a**7*d**4*e**2*x**2 + 1 20120*a**7*d**3*e**3*x**3 + 72072*a**7*d**2*e**4*x**4 + 24024*a**7*d*e**5* x**5 + 3432*a**7*e**6*x**6 + 84084*a**6*b*d**6*x + 336336*a**6*b*d**5*e*x* *2 + 630630*a**6*b*d**4*e**2*x**3 + 672672*a**6*b*d**3*e**3*x**4 + 420420* a**6*b*d**2*e**4*x**5 + 144144*a**6*b*d*e**5*x**6 + 21021*a**6*b*e**6*x**7 + 168168*a**5*b**2*d**6*x**2 + 756756*a**5*b**2*d**5*e*x**3 + 1513512*a** 5*b**2*d**4*e**2*x**4 + 1681680*a**5*b**2*d**3*e**3*x**5 + 1081080*a**5*b* *2*d**2*e**4*x**6 + 378378*a**5*b**2*d*e**5*x**7 + 56056*a**5*b**2*e**6*x* *8 + 210210*a**4*b**3*d**6*x**3 + 1009008*a**4*b**3*d**5*e*x**4 + 2102100* a**4*b**3*d**4*e**2*x**5 + 2402400*a**4*b**3*d**3*e**3*x**6 + 1576575*a**4 *b**3*d**2*e**4*x**7 + 560560*a**4*b**3*d*e**5*x**8 + 84084*a**4*b**3*e**6 *x**9 + 168168*a**3*b**4*d**6*x**4 + 840840*a**3*b**4*d**5*e*x**5 + 180180 0*a**3*b**4*d**4*e**2*x**6 + 2102100*a**3*b**4*d**3*e**3*x**7 + 1401400*a* *3*b**4*d**2*e**4*x**8 + 504504*a**3*b**4*d*e**5*x**9 + 76440*a**3*b**4*e* *6*x**10 + 84084*a**2*b**5*d**6*x**5 + 432432*a**2*b**5*d**5*e*x**6 + 9459 45*a**2*b**5*d**4*e**2*x**7 + 1121120*a**2*b**5*d**3*e**3*x**8 + 756756*a* *2*b**5*d**2*e**4*x**9 + 275184*a**2*b**5*d*e**5*x**10 + 42042*a**2*b**5*e **6*x**11 + 24024*a*b**6*d**6*x**6 + 126126*a*b**6*d**5*e*x**7 + 280280*a* b**6*d**4*e**2*x**8 + 336336*a*b**6*d**3*e**3*x**9 + 229320*a*b**6*d**2*e* *4*x**10 + 84084*a*b**6*d*e**5*x**11 + 12936*a*b**6*e**6*x**12 + 3003*b...