3.1034 \(\int (a+b x)^6 (A+B x) (d+e x)^8 \, dx\)

Optimal. Leaf size=292 \[ -\frac{b^5 (d+e x)^{15} (-6 a B e-A b e+7 b B d)}{15 e^8}+\frac{3 b^4 (d+e x)^{14} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{14 e^8}-\frac{5 b^3 (d+e x)^{13} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{13 e^8}+\frac{5 b^2 (d+e x)^{12} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{12 e^8}-\frac{3 b (d+e x)^{11} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{11 e^8}+\frac{(d+e x)^{10} (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{10 e^8}-\frac{(d+e x)^9 (b d-a e)^6 (B d-A e)}{9 e^8}+\frac{b^6 B (d+e x)^{16}}{16 e^8} \]

[Out]

-((b*d - a*e)^6*(B*d - A*e)*(d + e*x)^9)/(9*e^8) + ((b*d - a*e)^5*(7*b*B*d - 6*A
*b*e - a*B*e)*(d + e*x)^10)/(10*e^8) - (3*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2
*a*B*e)*(d + e*x)^11)/(11*e^8) + (5*b^2*(b*d - a*e)^3*(7*b*B*d - 4*A*b*e - 3*a*B
*e)*(d + e*x)^12)/(12*e^8) - (5*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4*a*B*e)*
(d + e*x)^13)/(13*e^8) + (3*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(d + e
*x)^14)/(14*e^8) - (b^5*(7*b*B*d - A*b*e - 6*a*B*e)*(d + e*x)^15)/(15*e^8) + (b^
6*B*(d + e*x)^16)/(16*e^8)

_______________________________________________________________________________________

Rubi [A]  time = 4.94768, antiderivative size = 292, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{b^5 (d+e x)^{15} (-6 a B e-A b e+7 b B d)}{15 e^8}+\frac{3 b^4 (d+e x)^{14} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{14 e^8}-\frac{5 b^3 (d+e x)^{13} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{13 e^8}+\frac{5 b^2 (d+e x)^{12} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{12 e^8}-\frac{3 b (d+e x)^{11} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{11 e^8}+\frac{(d+e x)^{10} (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{10 e^8}-\frac{(d+e x)^9 (b d-a e)^6 (B d-A e)}{9 e^8}+\frac{b^6 B (d+e x)^{16}}{16 e^8} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^6*(A + B*x)*(d + e*x)^8,x]

[Out]

-((b*d - a*e)^6*(B*d - A*e)*(d + e*x)^9)/(9*e^8) + ((b*d - a*e)^5*(7*b*B*d - 6*A
*b*e - a*B*e)*(d + e*x)^10)/(10*e^8) - (3*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2
*a*B*e)*(d + e*x)^11)/(11*e^8) + (5*b^2*(b*d - a*e)^3*(7*b*B*d - 4*A*b*e - 3*a*B
*e)*(d + e*x)^12)/(12*e^8) - (5*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4*a*B*e)*
(d + e*x)^13)/(13*e^8) + (3*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(d + e
*x)^14)/(14*e^8) - (b^5*(7*b*B*d - A*b*e - 6*a*B*e)*(d + e*x)^15)/(15*e^8) + (b^
6*B*(d + e*x)^16)/(16*e^8)

_______________________________________________________________________________________

Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**6*(B*x+A)*(e*x+d)**8,x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [B]  time = 1.07923, size = 1385, normalized size = 4.74 \[ \frac{1}{16} b^6 B e^8 x^{16}+\frac{1}{15} b^5 e^7 (8 b B d+A b e+6 a B e) x^{15}+\frac{1}{14} b^4 e^6 \left (4 d (7 B d+2 A e) b^2+6 a e (8 B d+A e) b+15 a^2 B e^2\right ) x^{14}+\frac{1}{13} b^3 e^5 \left (28 d^2 (2 B d+A e) b^3+24 a d e (7 B d+2 A e) b^2+15 a^2 e^2 (8 B d+A e) b+20 a^3 B e^3\right ) x^{13}+\frac{1}{12} b^2 e^4 \left (14 d^3 (5 B d+4 A e) b^4+168 a d^2 e (2 B d+A e) b^3+60 a^2 d e^2 (7 B d+2 A e) b^2+20 a^3 e^3 (8 B d+A e) b+15 a^4 B e^4\right ) x^{12}+\frac{1}{11} b e^3 \left (14 d^4 (4 B d+5 A e) b^5+84 a d^3 e (5 B d+4 A e) b^4+420 a^2 d^2 e^2 (2 B d+A e) b^3+80 a^3 d e^3 (7 B d+2 A e) b^2+15 a^4 e^4 (8 B d+A e) b+6 a^5 B e^5\right ) x^{11}+\frac{1}{10} e^2 \left (28 d^5 (B d+2 A e) b^6+84 a d^4 e (4 B d+5 A e) b^5+210 a^2 d^3 e^2 (5 B d+4 A e) b^4+560 a^3 d^2 e^3 (2 B d+A e) b^3+60 a^4 d e^4 (7 B d+2 A e) b^2+6 a^5 e^5 (8 B d+A e) b+a^6 B e^6\right ) x^{10}+\frac{1}{9} e \left (4 b^6 (2 B d+7 A e) d^6+168 a b^5 e (B d+2 A e) d^5+210 a^2 b^4 e^2 (4 B d+5 A e) d^4+280 a^3 b^3 e^3 (5 B d+4 A e) d^3+420 a^4 b^2 e^4 (2 B d+A e) d^2+24 a^5 b e^5 (7 B d+2 A e) d+a^6 e^6 (8 B d+A e)\right ) x^9+\frac{1}{8} d \left (b^6 (B d+8 A e) d^6+24 a b^5 e (2 B d+7 A e) d^5+420 a^2 b^4 e^2 (B d+2 A e) d^4+280 a^3 b^3 e^3 (4 B d+5 A e) d^3+210 a^4 b^2 e^4 (5 B d+4 A e) d^2+168 a^5 b e^5 (2 B d+A e) d+4 a^6 e^6 (7 B d+2 A e)\right ) x^8+\frac{1}{7} d^2 \left (2 a B d \left (3 b^5 d^5+60 a b^4 e d^4+280 a^2 b^3 e^2 d^3+420 a^3 b^2 e^3 d^2+210 a^4 b e^4 d+28 a^5 e^5\right )+A \left (b^6 d^6+48 a b^5 e d^5+420 a^2 b^4 e^2 d^4+1120 a^3 b^3 e^3 d^3+1050 a^4 b^2 e^4 d^2+336 a^5 b e^5 d+28 a^6 e^6\right )\right ) x^7+\frac{1}{6} a d^3 \left (a B d \left (15 b^4 d^4+160 a b^3 e d^3+420 a^2 b^2 e^2 d^2+336 a^3 b e^3 d+70 a^4 e^4\right )+2 A \left (3 b^5 d^5+60 a b^4 e d^4+280 a^2 b^3 e^2 d^3+420 a^3 b^2 e^3 d^2+210 a^4 b e^4 d+28 a^5 e^5\right )\right ) x^6+\frac{1}{5} a^2 d^4 \left (4 a B d \left (5 b^3 d^3+30 a b^2 e d^2+42 a^2 b e^2 d+14 a^3 e^3\right )+A \left (15 b^4 d^4+160 a b^3 e d^3+420 a^2 b^2 e^2 d^2+336 a^3 b e^3 d+70 a^4 e^4\right )\right ) x^5+\frac{1}{4} a^3 d^5 \left (a B d \left (15 b^2 d^2+48 a b e d+28 a^2 e^2\right )+4 A \left (5 b^3 d^3+30 a b^2 e d^2+42 a^2 b e^2 d+14 a^3 e^3\right )\right ) x^4+\frac{1}{3} a^4 d^6 \left (2 a B d (3 b d+4 a e)+A \left (15 b^2 d^2+48 a b e d+28 a^2 e^2\right )\right ) x^3+\frac{1}{2} a^5 d^7 (6 A b d+a B d+8 a A e) x^2+a^6 A d^8 x \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^6*(A + B*x)*(d + e*x)^8,x]

[Out]

a^6*A*d^8*x + (a^5*d^7*(6*A*b*d + a*B*d + 8*a*A*e)*x^2)/2 + (a^4*d^6*(2*a*B*d*(3
*b*d + 4*a*e) + A*(15*b^2*d^2 + 48*a*b*d*e + 28*a^2*e^2))*x^3)/3 + (a^3*d^5*(a*B
*d*(15*b^2*d^2 + 48*a*b*d*e + 28*a^2*e^2) + 4*A*(5*b^3*d^3 + 30*a*b^2*d^2*e + 42
*a^2*b*d*e^2 + 14*a^3*e^3))*x^4)/4 + (a^2*d^4*(4*a*B*d*(5*b^3*d^3 + 30*a*b^2*d^2
*e + 42*a^2*b*d*e^2 + 14*a^3*e^3) + A*(15*b^4*d^4 + 160*a*b^3*d^3*e + 420*a^2*b^
2*d^2*e^2 + 336*a^3*b*d*e^3 + 70*a^4*e^4))*x^5)/5 + (a*d^3*(a*B*d*(15*b^4*d^4 +
160*a*b^3*d^3*e + 420*a^2*b^2*d^2*e^2 + 336*a^3*b*d*e^3 + 70*a^4*e^4) + 2*A*(3*b
^5*d^5 + 60*a*b^4*d^4*e + 280*a^2*b^3*d^3*e^2 + 420*a^3*b^2*d^2*e^3 + 210*a^4*b*
d*e^4 + 28*a^5*e^5))*x^6)/6 + (d^2*(2*a*B*d*(3*b^5*d^5 + 60*a*b^4*d^4*e + 280*a^
2*b^3*d^3*e^2 + 420*a^3*b^2*d^2*e^3 + 210*a^4*b*d*e^4 + 28*a^5*e^5) + A*(b^6*d^6
+ 48*a*b^5*d^5*e + 420*a^2*b^4*d^4*e^2 + 1120*a^3*b^3*d^3*e^3 + 1050*a^4*b^2*d^
2*e^4 + 336*a^5*b*d*e^5 + 28*a^6*e^6))*x^7)/7 + (d*(168*a^5*b*d*e^5*(2*B*d + A*e
) + 420*a^2*b^4*d^4*e^2*(B*d + 2*A*e) + 4*a^6*e^6*(7*B*d + 2*A*e) + 210*a^4*b^2*
d^2*e^4*(5*B*d + 4*A*e) + 280*a^3*b^3*d^3*e^3*(4*B*d + 5*A*e) + 24*a*b^5*d^5*e*(
2*B*d + 7*A*e) + b^6*d^6*(B*d + 8*A*e))*x^8)/8 + (e*(420*a^4*b^2*d^2*e^4*(2*B*d
+ A*e) + a^6*e^6*(8*B*d + A*e) + 168*a*b^5*d^5*e*(B*d + 2*A*e) + 24*a^5*b*d*e^5*
(7*B*d + 2*A*e) + 280*a^3*b^3*d^3*e^3*(5*B*d + 4*A*e) + 210*a^2*b^4*d^4*e^2*(4*B
*d + 5*A*e) + 4*b^6*d^6*(2*B*d + 7*A*e))*x^9)/9 + (e^2*(a^6*B*e^6 + 560*a^3*b^3*
d^2*e^3*(2*B*d + A*e) + 6*a^5*b*e^5*(8*B*d + A*e) + 28*b^6*d^5*(B*d + 2*A*e) + 6
0*a^4*b^2*d*e^4*(7*B*d + 2*A*e) + 210*a^2*b^4*d^3*e^2*(5*B*d + 4*A*e) + 84*a*b^5
*d^4*e*(4*B*d + 5*A*e))*x^10)/10 + (b*e^3*(6*a^5*B*e^5 + 420*a^2*b^3*d^2*e^2*(2*
B*d + A*e) + 15*a^4*b*e^4*(8*B*d + A*e) + 80*a^3*b^2*d*e^3*(7*B*d + 2*A*e) + 84*
a*b^4*d^3*e*(5*B*d + 4*A*e) + 14*b^5*d^4*(4*B*d + 5*A*e))*x^11)/11 + (b^2*e^4*(1
5*a^4*B*e^4 + 168*a*b^3*d^2*e*(2*B*d + A*e) + 20*a^3*b*e^3*(8*B*d + A*e) + 60*a^
2*b^2*d*e^2*(7*B*d + 2*A*e) + 14*b^4*d^3*(5*B*d + 4*A*e))*x^12)/12 + (b^3*e^5*(2
0*a^3*B*e^3 + 28*b^3*d^2*(2*B*d + A*e) + 15*a^2*b*e^2*(8*B*d + A*e) + 24*a*b^2*d
*e*(7*B*d + 2*A*e))*x^13)/13 + (b^4*e^6*(15*a^2*B*e^2 + 6*a*b*e*(8*B*d + A*e) +
4*b^2*d*(7*B*d + 2*A*e))*x^14)/14 + (b^5*e^7*(8*b*B*d + A*b*e + 6*a*B*e)*x^15)/1
5 + (b^6*B*e^8*x^16)/16

_______________________________________________________________________________________

Maple [B]  time = 0.003, size = 1525, normalized size = 5.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^6*(B*x+A)*(e*x+d)^8,x)

[Out]

1/16*b^6*B*e^8*x^16+1/15*((A*b^6+6*B*a*b^5)*e^8+8*b^6*B*d*e^7)*x^15+1/14*((6*A*a
*b^5+15*B*a^2*b^4)*e^8+8*(A*b^6+6*B*a*b^5)*d*e^7+28*b^6*B*d^2*e^6)*x^14+1/13*((1
5*A*a^2*b^4+20*B*a^3*b^3)*e^8+8*(6*A*a*b^5+15*B*a^2*b^4)*d*e^7+28*(A*b^6+6*B*a*b
^5)*d^2*e^6+56*b^6*B*d^3*e^5)*x^13+1/12*((20*A*a^3*b^3+15*B*a^4*b^2)*e^8+8*(15*A
*a^2*b^4+20*B*a^3*b^3)*d*e^7+28*(6*A*a*b^5+15*B*a^2*b^4)*d^2*e^6+56*(A*b^6+6*B*a
*b^5)*d^3*e^5+70*b^6*B*d^4*e^4)*x^12+1/11*((15*A*a^4*b^2+6*B*a^5*b)*e^8+8*(20*A*
a^3*b^3+15*B*a^4*b^2)*d*e^7+28*(15*A*a^2*b^4+20*B*a^3*b^3)*d^2*e^6+56*(6*A*a*b^5
+15*B*a^2*b^4)*d^3*e^5+70*(A*b^6+6*B*a*b^5)*d^4*e^4+56*b^6*B*d^5*e^3)*x^11+1/10*
((6*A*a^5*b+B*a^6)*e^8+8*(15*A*a^4*b^2+6*B*a^5*b)*d*e^7+28*(20*A*a^3*b^3+15*B*a^
4*b^2)*d^2*e^6+56*(15*A*a^2*b^4+20*B*a^3*b^3)*d^3*e^5+70*(6*A*a*b^5+15*B*a^2*b^4
)*d^4*e^4+56*(A*b^6+6*B*a*b^5)*d^5*e^3+28*b^6*B*d^6*e^2)*x^10+1/9*(a^6*A*e^8+8*(
6*A*a^5*b+B*a^6)*d*e^7+28*(15*A*a^4*b^2+6*B*a^5*b)*d^2*e^6+56*(20*A*a^3*b^3+15*B
*a^4*b^2)*d^3*e^5+70*(15*A*a^2*b^4+20*B*a^3*b^3)*d^4*e^4+56*(6*A*a*b^5+15*B*a^2*
b^4)*d^5*e^3+28*(A*b^6+6*B*a*b^5)*d^6*e^2+8*b^6*B*d^7*e)*x^9+1/8*(8*a^6*A*d*e^7+
28*(6*A*a^5*b+B*a^6)*d^2*e^6+56*(15*A*a^4*b^2+6*B*a^5*b)*d^3*e^5+70*(20*A*a^3*b^
3+15*B*a^4*b^2)*d^4*e^4+56*(15*A*a^2*b^4+20*B*a^3*b^3)*d^5*e^3+28*(6*A*a*b^5+15*
B*a^2*b^4)*d^6*e^2+8*(A*b^6+6*B*a*b^5)*d^7*e+b^6*B*d^8)*x^8+1/7*(28*a^6*A*d^2*e^
6+56*(6*A*a^5*b+B*a^6)*d^3*e^5+70*(15*A*a^4*b^2+6*B*a^5*b)*d^4*e^4+56*(20*A*a^3*
b^3+15*B*a^4*b^2)*d^5*e^3+28*(15*A*a^2*b^4+20*B*a^3*b^3)*d^6*e^2+8*(6*A*a*b^5+15
*B*a^2*b^4)*d^7*e+(A*b^6+6*B*a*b^5)*d^8)*x^7+1/6*(56*a^6*A*d^3*e^5+70*(6*A*a^5*b
+B*a^6)*d^4*e^4+56*(15*A*a^4*b^2+6*B*a^5*b)*d^5*e^3+28*(20*A*a^3*b^3+15*B*a^4*b^
2)*d^6*e^2+8*(15*A*a^2*b^4+20*B*a^3*b^3)*d^7*e+(6*A*a*b^5+15*B*a^2*b^4)*d^8)*x^6
+1/5*(70*a^6*A*d^4*e^4+56*(6*A*a^5*b+B*a^6)*d^5*e^3+28*(15*A*a^4*b^2+6*B*a^5*b)*
d^6*e^2+8*(20*A*a^3*b^3+15*B*a^4*b^2)*d^7*e+(15*A*a^2*b^4+20*B*a^3*b^3)*d^8)*x^5
+1/4*(56*a^6*A*d^5*e^3+28*(6*A*a^5*b+B*a^6)*d^6*e^2+8*(15*A*a^4*b^2+6*B*a^5*b)*d
^7*e+(20*A*a^3*b^3+15*B*a^4*b^2)*d^8)*x^4+1/3*(28*a^6*A*d^6*e^2+8*(6*A*a^5*b+B*a
^6)*d^7*e+(15*A*a^4*b^2+6*B*a^5*b)*d^8)*x^3+1/2*(8*a^6*A*d^7*e+(6*A*a^5*b+B*a^6)
*d^8)*x^2+a^6*A*d^8*x

_______________________________________________________________________________________

Maxima [A]  time = 1.38791, size = 2068, normalized size = 7.08 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^6*(e*x + d)^8,x, algorithm="maxima")

[Out]

1/16*B*b^6*e^8*x^16 + A*a^6*d^8*x + 1/15*(8*B*b^6*d*e^7 + (6*B*a*b^5 + A*b^6)*e^
8)*x^15 + 1/14*(28*B*b^6*d^2*e^6 + 8*(6*B*a*b^5 + A*b^6)*d*e^7 + 3*(5*B*a^2*b^4
+ 2*A*a*b^5)*e^8)*x^14 + 1/13*(56*B*b^6*d^3*e^5 + 28*(6*B*a*b^5 + A*b^6)*d^2*e^6
+ 24*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^7 + 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^8)*x^13
+ 1/12*(70*B*b^6*d^4*e^4 + 56*(6*B*a*b^5 + A*b^6)*d^3*e^5 + 84*(5*B*a^2*b^4 + 2*
A*a*b^5)*d^2*e^6 + 40*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^7 + 5*(3*B*a^4*b^2 + 4*A*a
^3*b^3)*e^8)*x^12 + 1/11*(56*B*b^6*d^5*e^3 + 70*(6*B*a*b^5 + A*b^6)*d^4*e^4 + 16
8*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^5 + 140*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^6 +
40*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^7 + 3*(2*B*a^5*b + 5*A*a^4*b^2)*e^8)*x^11 + 1
/10*(28*B*b^6*d^6*e^2 + 56*(6*B*a*b^5 + A*b^6)*d^5*e^3 + 210*(5*B*a^2*b^4 + 2*A*
a*b^5)*d^4*e^4 + 280*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^5 + 140*(3*B*a^4*b^2 + 4*
A*a^3*b^3)*d^2*e^6 + 24*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^7 + (B*a^6 + 6*A*a^5*b)*e^
8)*x^10 + 1/9*(8*B*b^6*d^7*e + A*a^6*e^8 + 28*(6*B*a*b^5 + A*b^6)*d^6*e^2 + 168*
(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^3 + 350*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^4 + 28
0*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^5 + 84*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^6 + 8
*(B*a^6 + 6*A*a^5*b)*d*e^7)*x^9 + 1/8*(B*b^6*d^8 + 8*A*a^6*d*e^7 + 8*(6*B*a*b^5
+ A*b^6)*d^7*e + 84*(5*B*a^2*b^4 + 2*A*a*b^5)*d^6*e^2 + 280*(4*B*a^3*b^3 + 3*A*a
^2*b^4)*d^5*e^3 + 350*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^4*e^4 + 168*(2*B*a^5*b + 5*A
*a^4*b^2)*d^3*e^5 + 28*(B*a^6 + 6*A*a^5*b)*d^2*e^6)*x^8 + 1/7*(28*A*a^6*d^2*e^6
+ (6*B*a*b^5 + A*b^6)*d^8 + 24*(5*B*a^2*b^4 + 2*A*a*b^5)*d^7*e + 140*(4*B*a^3*b^
3 + 3*A*a^2*b^4)*d^6*e^2 + 280*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^5*e^3 + 210*(2*B*a^
5*b + 5*A*a^4*b^2)*d^4*e^4 + 56*(B*a^6 + 6*A*a^5*b)*d^3*e^5)*x^7 + 1/6*(56*A*a^6
*d^3*e^5 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^8 + 40*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^7*
e + 140*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^6*e^2 + 168*(2*B*a^5*b + 5*A*a^4*b^2)*d^5*
e^3 + 70*(B*a^6 + 6*A*a^5*b)*d^4*e^4)*x^6 + 1/5*(70*A*a^6*d^4*e^4 + 5*(4*B*a^3*b
^3 + 3*A*a^2*b^4)*d^8 + 40*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^7*e + 84*(2*B*a^5*b + 5
*A*a^4*b^2)*d^6*e^2 + 56*(B*a^6 + 6*A*a^5*b)*d^5*e^3)*x^5 + 1/4*(56*A*a^6*d^5*e^
3 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^8 + 24*(2*B*a^5*b + 5*A*a^4*b^2)*d^7*e + 28*
(B*a^6 + 6*A*a^5*b)*d^6*e^2)*x^4 + 1/3*(28*A*a^6*d^6*e^2 + 3*(2*B*a^5*b + 5*A*a^
4*b^2)*d^8 + 8*(B*a^6 + 6*A*a^5*b)*d^7*e)*x^3 + 1/2*(8*A*a^6*d^7*e + (B*a^6 + 6*
A*a^5*b)*d^8)*x^2

_______________________________________________________________________________________

Fricas [A]  time = 0.208038, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^6*(e*x + d)^8,x, algorithm="fricas")

[Out]

1/16*x^16*e^8*b^6*B + 8/15*x^15*e^7*d*b^6*B + 2/5*x^15*e^8*b^5*a*B + 1/15*x^15*e
^8*b^6*A + 2*x^14*e^6*d^2*b^6*B + 24/7*x^14*e^7*d*b^5*a*B + 15/14*x^14*e^8*b^4*a
^2*B + 4/7*x^14*e^7*d*b^6*A + 3/7*x^14*e^8*b^5*a*A + 56/13*x^13*e^5*d^3*b^6*B +
168/13*x^13*e^6*d^2*b^5*a*B + 120/13*x^13*e^7*d*b^4*a^2*B + 20/13*x^13*e^8*b^3*a
^3*B + 28/13*x^13*e^6*d^2*b^6*A + 48/13*x^13*e^7*d*b^5*a*A + 15/13*x^13*e^8*b^4*
a^2*A + 35/6*x^12*e^4*d^4*b^6*B + 28*x^12*e^5*d^3*b^5*a*B + 35*x^12*e^6*d^2*b^4*
a^2*B + 40/3*x^12*e^7*d*b^3*a^3*B + 5/4*x^12*e^8*b^2*a^4*B + 14/3*x^12*e^5*d^3*b
^6*A + 14*x^12*e^6*d^2*b^5*a*A + 10*x^12*e^7*d*b^4*a^2*A + 5/3*x^12*e^8*b^3*a^3*
A + 56/11*x^11*e^3*d^5*b^6*B + 420/11*x^11*e^4*d^4*b^5*a*B + 840/11*x^11*e^5*d^3
*b^4*a^2*B + 560/11*x^11*e^6*d^2*b^3*a^3*B + 120/11*x^11*e^7*d*b^2*a^4*B + 6/11*
x^11*e^8*b*a^5*B + 70/11*x^11*e^4*d^4*b^6*A + 336/11*x^11*e^5*d^3*b^5*a*A + 420/
11*x^11*e^6*d^2*b^4*a^2*A + 160/11*x^11*e^7*d*b^3*a^3*A + 15/11*x^11*e^8*b^2*a^4
*A + 14/5*x^10*e^2*d^6*b^6*B + 168/5*x^10*e^3*d^5*b^5*a*B + 105*x^10*e^4*d^4*b^4
*a^2*B + 112*x^10*e^5*d^3*b^3*a^3*B + 42*x^10*e^6*d^2*b^2*a^4*B + 24/5*x^10*e^7*
d*b*a^5*B + 1/10*x^10*e^8*a^6*B + 28/5*x^10*e^3*d^5*b^6*A + 42*x^10*e^4*d^4*b^5*
a*A + 84*x^10*e^5*d^3*b^4*a^2*A + 56*x^10*e^6*d^2*b^3*a^3*A + 12*x^10*e^7*d*b^2*
a^4*A + 3/5*x^10*e^8*b*a^5*A + 8/9*x^9*e*d^7*b^6*B + 56/3*x^9*e^2*d^6*b^5*a*B +
280/3*x^9*e^3*d^5*b^4*a^2*B + 1400/9*x^9*e^4*d^4*b^3*a^3*B + 280/3*x^9*e^5*d^3*b
^2*a^4*B + 56/3*x^9*e^6*d^2*b*a^5*B + 8/9*x^9*e^7*d*a^6*B + 28/9*x^9*e^2*d^6*b^6
*A + 112/3*x^9*e^3*d^5*b^5*a*A + 350/3*x^9*e^4*d^4*b^4*a^2*A + 1120/9*x^9*e^5*d^
3*b^3*a^3*A + 140/3*x^9*e^6*d^2*b^2*a^4*A + 16/3*x^9*e^7*d*b*a^5*A + 1/9*x^9*e^8
*a^6*A + 1/8*x^8*d^8*b^6*B + 6*x^8*e*d^7*b^5*a*B + 105/2*x^8*e^2*d^6*b^4*a^2*B +
140*x^8*e^3*d^5*b^3*a^3*B + 525/4*x^8*e^4*d^4*b^2*a^4*B + 42*x^8*e^5*d^3*b*a^5*
B + 7/2*x^8*e^6*d^2*a^6*B + x^8*e*d^7*b^6*A + 21*x^8*e^2*d^6*b^5*a*A + 105*x^8*e
^3*d^5*b^4*a^2*A + 175*x^8*e^4*d^4*b^3*a^3*A + 105*x^8*e^5*d^3*b^2*a^4*A + 21*x^
8*e^6*d^2*b*a^5*A + x^8*e^7*d*a^6*A + 6/7*x^7*d^8*b^5*a*B + 120/7*x^7*e*d^7*b^4*
a^2*B + 80*x^7*e^2*d^6*b^3*a^3*B + 120*x^7*e^3*d^5*b^2*a^4*B + 60*x^7*e^4*d^4*b*
a^5*B + 8*x^7*e^5*d^3*a^6*B + 1/7*x^7*d^8*b^6*A + 48/7*x^7*e*d^7*b^5*a*A + 60*x^
7*e^2*d^6*b^4*a^2*A + 160*x^7*e^3*d^5*b^3*a^3*A + 150*x^7*e^4*d^4*b^2*a^4*A + 48
*x^7*e^5*d^3*b*a^5*A + 4*x^7*e^6*d^2*a^6*A + 5/2*x^6*d^8*b^4*a^2*B + 80/3*x^6*e*
d^7*b^3*a^3*B + 70*x^6*e^2*d^6*b^2*a^4*B + 56*x^6*e^3*d^5*b*a^5*B + 35/3*x^6*e^4
*d^4*a^6*B + x^6*d^8*b^5*a*A + 20*x^6*e*d^7*b^4*a^2*A + 280/3*x^6*e^2*d^6*b^3*a^
3*A + 140*x^6*e^3*d^5*b^2*a^4*A + 70*x^6*e^4*d^4*b*a^5*A + 28/3*x^6*e^5*d^3*a^6*
A + 4*x^5*d^8*b^3*a^3*B + 24*x^5*e*d^7*b^2*a^4*B + 168/5*x^5*e^2*d^6*b*a^5*B + 5
6/5*x^5*e^3*d^5*a^6*B + 3*x^5*d^8*b^4*a^2*A + 32*x^5*e*d^7*b^3*a^3*A + 84*x^5*e^
2*d^6*b^2*a^4*A + 336/5*x^5*e^3*d^5*b*a^5*A + 14*x^5*e^4*d^4*a^6*A + 15/4*x^4*d^
8*b^2*a^4*B + 12*x^4*e*d^7*b*a^5*B + 7*x^4*e^2*d^6*a^6*B + 5*x^4*d^8*b^3*a^3*A +
30*x^4*e*d^7*b^2*a^4*A + 42*x^4*e^2*d^6*b*a^5*A + 14*x^4*e^3*d^5*a^6*A + 2*x^3*
d^8*b*a^5*B + 8/3*x^3*e*d^7*a^6*B + 5*x^3*d^8*b^2*a^4*A + 16*x^3*e*d^7*b*a^5*A +
28/3*x^3*e^2*d^6*a^6*A + 1/2*x^2*d^8*a^6*B + 3*x^2*d^8*b*a^5*A + 4*x^2*e*d^7*a^
6*A + x*d^8*a^6*A

_______________________________________________________________________________________

Sympy [A]  time = 0.837158, size = 1969, normalized size = 6.74 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**6*(B*x+A)*(e*x+d)**8,x)

[Out]

A*a**6*d**8*x + B*b**6*e**8*x**16/16 + x**15*(A*b**6*e**8/15 + 2*B*a*b**5*e**8/5
+ 8*B*b**6*d*e**7/15) + x**14*(3*A*a*b**5*e**8/7 + 4*A*b**6*d*e**7/7 + 15*B*a**
2*b**4*e**8/14 + 24*B*a*b**5*d*e**7/7 + 2*B*b**6*d**2*e**6) + x**13*(15*A*a**2*b
**4*e**8/13 + 48*A*a*b**5*d*e**7/13 + 28*A*b**6*d**2*e**6/13 + 20*B*a**3*b**3*e*
*8/13 + 120*B*a**2*b**4*d*e**7/13 + 168*B*a*b**5*d**2*e**6/13 + 56*B*b**6*d**3*e
**5/13) + x**12*(5*A*a**3*b**3*e**8/3 + 10*A*a**2*b**4*d*e**7 + 14*A*a*b**5*d**2
*e**6 + 14*A*b**6*d**3*e**5/3 + 5*B*a**4*b**2*e**8/4 + 40*B*a**3*b**3*d*e**7/3 +
35*B*a**2*b**4*d**2*e**6 + 28*B*a*b**5*d**3*e**5 + 35*B*b**6*d**4*e**4/6) + x**
11*(15*A*a**4*b**2*e**8/11 + 160*A*a**3*b**3*d*e**7/11 + 420*A*a**2*b**4*d**2*e*
*6/11 + 336*A*a*b**5*d**3*e**5/11 + 70*A*b**6*d**4*e**4/11 + 6*B*a**5*b*e**8/11
+ 120*B*a**4*b**2*d*e**7/11 + 560*B*a**3*b**3*d**2*e**6/11 + 840*B*a**2*b**4*d**
3*e**5/11 + 420*B*a*b**5*d**4*e**4/11 + 56*B*b**6*d**5*e**3/11) + x**10*(3*A*a**
5*b*e**8/5 + 12*A*a**4*b**2*d*e**7 + 56*A*a**3*b**3*d**2*e**6 + 84*A*a**2*b**4*d
**3*e**5 + 42*A*a*b**5*d**4*e**4 + 28*A*b**6*d**5*e**3/5 + B*a**6*e**8/10 + 24*B
*a**5*b*d*e**7/5 + 42*B*a**4*b**2*d**2*e**6 + 112*B*a**3*b**3*d**3*e**5 + 105*B*
a**2*b**4*d**4*e**4 + 168*B*a*b**5*d**5*e**3/5 + 14*B*b**6*d**6*e**2/5) + x**9*(
A*a**6*e**8/9 + 16*A*a**5*b*d*e**7/3 + 140*A*a**4*b**2*d**2*e**6/3 + 1120*A*a**3
*b**3*d**3*e**5/9 + 350*A*a**2*b**4*d**4*e**4/3 + 112*A*a*b**5*d**5*e**3/3 + 28*
A*b**6*d**6*e**2/9 + 8*B*a**6*d*e**7/9 + 56*B*a**5*b*d**2*e**6/3 + 280*B*a**4*b*
*2*d**3*e**5/3 + 1400*B*a**3*b**3*d**4*e**4/9 + 280*B*a**2*b**4*d**5*e**3/3 + 56
*B*a*b**5*d**6*e**2/3 + 8*B*b**6*d**7*e/9) + x**8*(A*a**6*d*e**7 + 21*A*a**5*b*d
**2*e**6 + 105*A*a**4*b**2*d**3*e**5 + 175*A*a**3*b**3*d**4*e**4 + 105*A*a**2*b*
*4*d**5*e**3 + 21*A*a*b**5*d**6*e**2 + A*b**6*d**7*e + 7*B*a**6*d**2*e**6/2 + 42
*B*a**5*b*d**3*e**5 + 525*B*a**4*b**2*d**4*e**4/4 + 140*B*a**3*b**3*d**5*e**3 +
105*B*a**2*b**4*d**6*e**2/2 + 6*B*a*b**5*d**7*e + B*b**6*d**8/8) + x**7*(4*A*a**
6*d**2*e**6 + 48*A*a**5*b*d**3*e**5 + 150*A*a**4*b**2*d**4*e**4 + 160*A*a**3*b**
3*d**5*e**3 + 60*A*a**2*b**4*d**6*e**2 + 48*A*a*b**5*d**7*e/7 + A*b**6*d**8/7 +
8*B*a**6*d**3*e**5 + 60*B*a**5*b*d**4*e**4 + 120*B*a**4*b**2*d**5*e**3 + 80*B*a*
*3*b**3*d**6*e**2 + 120*B*a**2*b**4*d**7*e/7 + 6*B*a*b**5*d**8/7) + x**6*(28*A*a
**6*d**3*e**5/3 + 70*A*a**5*b*d**4*e**4 + 140*A*a**4*b**2*d**5*e**3 + 280*A*a**3
*b**3*d**6*e**2/3 + 20*A*a**2*b**4*d**7*e + A*a*b**5*d**8 + 35*B*a**6*d**4*e**4/
3 + 56*B*a**5*b*d**5*e**3 + 70*B*a**4*b**2*d**6*e**2 + 80*B*a**3*b**3*d**7*e/3 +
5*B*a**2*b**4*d**8/2) + x**5*(14*A*a**6*d**4*e**4 + 336*A*a**5*b*d**5*e**3/5 +
84*A*a**4*b**2*d**6*e**2 + 32*A*a**3*b**3*d**7*e + 3*A*a**2*b**4*d**8 + 56*B*a**
6*d**5*e**3/5 + 168*B*a**5*b*d**6*e**2/5 + 24*B*a**4*b**2*d**7*e + 4*B*a**3*b**3
*d**8) + x**4*(14*A*a**6*d**5*e**3 + 42*A*a**5*b*d**6*e**2 + 30*A*a**4*b**2*d**7
*e + 5*A*a**3*b**3*d**8 + 7*B*a**6*d**6*e**2 + 12*B*a**5*b*d**7*e + 15*B*a**4*b*
*2*d**8/4) + x**3*(28*A*a**6*d**6*e**2/3 + 16*A*a**5*b*d**7*e + 5*A*a**4*b**2*d*
*8 + 8*B*a**6*d**7*e/3 + 2*B*a**5*b*d**8) + x**2*(4*A*a**6*d**7*e + 3*A*a**5*b*d
**8 + B*a**6*d**8/2)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.213142, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^6*(e*x + d)^8,x, algorithm="giac")

[Out]

Done