3.1470 \(\int (d+e x)^8 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx\)

Optimal. Leaf size=173 \[ -\frac{3 b^5 (d+e x)^{14} (b d-a e)}{7 e^7}+\frac{15 b^4 (d+e x)^{13} (b d-a e)^2}{13 e^7}-\frac{5 b^3 (d+e x)^{12} (b d-a e)^3}{3 e^7}+\frac{15 b^2 (d+e x)^{11} (b d-a e)^4}{11 e^7}-\frac{3 b (d+e x)^{10} (b d-a e)^5}{5 e^7}+\frac{(d+e x)^9 (b d-a e)^6}{9 e^7}+\frac{b^6 (d+e x)^{15}}{15 e^7} \]

[Out]

((b*d - a*e)^6*(d + e*x)^9)/(9*e^7) - (3*b*(b*d - a*e)^5*(d + e*x)^10)/(5*e^7) +
(15*b^2*(b*d - a*e)^4*(d + e*x)^11)/(11*e^7) - (5*b^3*(b*d - a*e)^3*(d + e*x)^1
2)/(3*e^7) + (15*b^4*(b*d - a*e)^2*(d + e*x)^13)/(13*e^7) - (3*b^5*(b*d - a*e)*(
d + e*x)^14)/(7*e^7) + (b^6*(d + e*x)^15)/(15*e^7)

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Rubi [A]  time = 1.19138, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{3 b^5 (d+e x)^{14} (b d-a e)}{7 e^7}+\frac{15 b^4 (d+e x)^{13} (b d-a e)^2}{13 e^7}-\frac{5 b^3 (d+e x)^{12} (b d-a e)^3}{3 e^7}+\frac{15 b^2 (d+e x)^{11} (b d-a e)^4}{11 e^7}-\frac{3 b (d+e x)^{10} (b d-a e)^5}{5 e^7}+\frac{(d+e x)^9 (b d-a e)^6}{9 e^7}+\frac{b^6 (d+e x)^{15}}{15 e^7} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^8*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

((b*d - a*e)^6*(d + e*x)^9)/(9*e^7) - (3*b*(b*d - a*e)^5*(d + e*x)^10)/(5*e^7) +
(15*b^2*(b*d - a*e)^4*(d + e*x)^11)/(11*e^7) - (5*b^3*(b*d - a*e)^3*(d + e*x)^1
2)/(3*e^7) + (15*b^4*(b*d - a*e)^2*(d + e*x)^13)/(13*e^7) - (3*b^5*(b*d - a*e)*(
d + e*x)^14)/(7*e^7) + (b^6*(d + e*x)^15)/(15*e^7)

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Rubi in Sympy [A]  time = 135.12, size = 158, normalized size = 0.91 \[ \frac{b^{6} \left (d + e x\right )^{15}}{15 e^{7}} + \frac{3 b^{5} \left (d + e x\right )^{14} \left (a e - b d\right )}{7 e^{7}} + \frac{15 b^{4} \left (d + e x\right )^{13} \left (a e - b d\right )^{2}}{13 e^{7}} + \frac{5 b^{3} \left (d + e x\right )^{12} \left (a e - b d\right )^{3}}{3 e^{7}} + \frac{15 b^{2} \left (d + e x\right )^{11} \left (a e - b d\right )^{4}}{11 e^{7}} + \frac{3 b \left (d + e x\right )^{10} \left (a e - b d\right )^{5}}{5 e^{7}} + \frac{\left (d + e x\right )^{9} \left (a e - b d\right )^{6}}{9 e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**8*(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

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

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Mathematica [B]  time = 0.204363, size = 771, normalized size = 4.46 \[ a^6 d^8 x+a^5 d^7 x^2 (4 a e+3 b d)+\frac{1}{13} b^4 e^6 x^{13} \left (15 a^2 e^2+48 a b d e+28 b^2 d^2\right )+\frac{1}{3} a^4 d^6 x^3 \left (28 a^2 e^2+48 a b d e+15 b^2 d^2\right )+\frac{1}{3} b^3 e^5 x^{12} \left (5 a^3 e^3+30 a^2 b d e^2+42 a b^2 d^2 e+14 b^3 d^3\right )+a^3 d^5 x^4 \left (14 a^3 e^3+42 a^2 b d e^2+30 a b^2 d^2 e+5 b^3 d^3\right )+\frac{1}{11} b^2 e^4 x^{11} \left (15 a^4 e^4+160 a^3 b d e^3+420 a^2 b^2 d^2 e^2+336 a b^3 d^3 e+70 b^4 d^4\right )+\frac{1}{5} a^2 d^4 x^5 \left (70 a^4 e^4+336 a^3 b d e^3+420 a^2 b^2 d^2 e^2+160 a b^3 d^3 e+15 b^4 d^4\right )+\frac{1}{5} b e^3 x^{10} \left (3 a^5 e^5+60 a^4 b d e^4+280 a^3 b^2 d^2 e^3+420 a^2 b^3 d^3 e^2+210 a b^4 d^4 e+28 b^5 d^5\right )+\frac{1}{3} a d^3 x^6 \left (28 a^5 e^5+210 a^4 b d e^4+420 a^3 b^2 d^2 e^3+280 a^2 b^3 d^3 e^2+60 a b^4 d^4 e+3 b^5 d^5\right )+\frac{1}{9} e^2 x^9 \left (a^6 e^6+48 a^5 b d e^5+420 a^4 b^2 d^2 e^4+1120 a^3 b^3 d^3 e^3+1050 a^2 b^4 d^4 e^2+336 a b^5 d^5 e+28 b^6 d^6\right )+d e x^8 \left (a^6 e^6+21 a^5 b d e^5+105 a^4 b^2 d^2 e^4+175 a^3 b^3 d^3 e^3+105 a^2 b^4 d^4 e^2+21 a b^5 d^5 e+b^6 d^6\right )+\frac{1}{7} d^2 x^7 \left (28 a^6 e^6+336 a^5 b d e^5+1050 a^4 b^2 d^2 e^4+1120 a^3 b^3 d^3 e^3+420 a^2 b^4 d^4 e^2+48 a b^5 d^5 e+b^6 d^6\right )+\frac{1}{7} b^5 e^7 x^{14} (3 a e+4 b d)+\frac{1}{15} b^6 e^8 x^{15} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^8*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

a^6*d^8*x + a^5*d^7*(3*b*d + 4*a*e)*x^2 + (a^4*d^6*(15*b^2*d^2 + 48*a*b*d*e + 28
*a^2*e^2)*x^3)/3 + a^3*d^5*(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 + (a^2*d^4*(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*(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)/3 + (d^2*(
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*e*(b^6*d^6 + 21*a*b^5*d^
5*e + 105*a^2*b^4*d^4*e^2 + 175*a^3*b^3*d^3*e^3 + 105*a^4*b^2*d^2*e^4 + 21*a^5*b
*d*e^5 + a^6*e^6)*x^8 + (e^2*(28*b^6*d^6 + 336*a*b^5*d^5*e + 1050*a^2*b^4*d^4*e^
2 + 1120*a^3*b^3*d^3*e^3 + 420*a^4*b^2*d^2*e^4 + 48*a^5*b*d*e^5 + a^6*e^6)*x^9)/
9 + (b*e^3*(28*b^5*d^5 + 210*a*b^4*d^4*e + 420*a^2*b^3*d^3*e^2 + 280*a^3*b^2*d^2
*e^3 + 60*a^4*b*d*e^4 + 3*a^5*e^5)*x^10)/5 + (b^2*e^4*(70*b^4*d^4 + 336*a*b^3*d^
3*e + 420*a^2*b^2*d^2*e^2 + 160*a^3*b*d*e^3 + 15*a^4*e^4)*x^11)/11 + (b^3*e^5*(1
4*b^3*d^3 + 42*a*b^2*d^2*e + 30*a^2*b*d*e^2 + 5*a^3*e^3)*x^12)/3 + (b^4*e^6*(28*
b^2*d^2 + 48*a*b*d*e + 15*a^2*e^2)*x^13)/13 + (b^5*e^7*(4*b*d + 3*a*e)*x^14)/7 +
(b^6*e^8*x^15)/15

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Maple [B]  time = 0.003, size = 803, normalized size = 4.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^8*(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

1/15*e^8*b^6*x^15+1/14*(6*a*b^5*e^8+8*b^6*d*e^7)*x^14+1/13*(15*a^2*b^4*e^8+48*a*
b^5*d*e^7+28*b^6*d^2*e^6)*x^13+1/12*(20*a^3*b^3*e^8+120*a^2*b^4*d*e^7+168*a*b^5*
d^2*e^6+56*b^6*d^3*e^5)*x^12+1/11*(15*a^4*b^2*e^8+160*a^3*b^3*d*e^7+420*a^2*b^4*
d^2*e^6+336*a*b^5*d^3*e^5+70*b^6*d^4*e^4)*x^11+1/10*(6*a^5*b*e^8+120*a^4*b^2*d*e
^7+560*a^3*b^3*d^2*e^6+840*a^2*b^4*d^3*e^5+420*a*b^5*d^4*e^4+56*b^6*d^5*e^3)*x^1
0+1/9*(a^6*e^8+48*a^5*b*d*e^7+420*a^4*b^2*d^2*e^6+1120*a^3*b^3*d^3*e^5+1050*a^2*
b^4*d^4*e^4+336*a*b^5*d^5*e^3+28*b^6*d^6*e^2)*x^9+1/8*(8*a^6*d*e^7+168*a^5*b*d^2
*e^6+840*a^4*b^2*d^3*e^5+1400*a^3*b^3*d^4*e^4+840*a^2*b^4*d^5*e^3+168*a*b^5*d^6*
e^2+8*b^6*d^7*e)*x^8+1/7*(28*a^6*d^2*e^6+336*a^5*b*d^3*e^5+1050*a^4*b^2*d^4*e^4+
1120*a^3*b^3*d^5*e^3+420*a^2*b^4*d^6*e^2+48*a*b^5*d^7*e+b^6*d^8)*x^7+1/6*(56*a^6
*d^3*e^5+420*a^5*b*d^4*e^4+840*a^4*b^2*d^5*e^3+560*a^3*b^3*d^6*e^2+120*a^2*b^4*d
^7*e+6*a*b^5*d^8)*x^6+1/5*(70*a^6*d^4*e^4+336*a^5*b*d^5*e^3+420*a^4*b^2*d^6*e^2+
160*a^3*b^3*d^7*e+15*a^2*b^4*d^8)*x^5+1/4*(56*a^6*d^5*e^3+168*a^5*b*d^6*e^2+120*
a^4*b^2*d^7*e+20*a^3*b^3*d^8)*x^4+1/3*(28*a^6*d^6*e^2+48*a^5*b*d^7*e+15*a^4*b^2*
d^8)*x^3+1/2*(8*a^6*d^7*e+6*a^5*b*d^8)*x^2+d^8*a^6*x

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Maxima [A]  time = 0.699265, size = 1076, normalized size = 6.22 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(e*x + d)^8,x, algorithm="maxima")

[Out]

1/15*b^6*e^8*x^15 + a^6*d^8*x + 1/7*(4*b^6*d*e^7 + 3*a*b^5*e^8)*x^14 + 1/13*(28*
b^6*d^2*e^6 + 48*a*b^5*d*e^7 + 15*a^2*b^4*e^8)*x^13 + 1/3*(14*b^6*d^3*e^5 + 42*a
*b^5*d^2*e^6 + 30*a^2*b^4*d*e^7 + 5*a^3*b^3*e^8)*x^12 + 1/11*(70*b^6*d^4*e^4 + 3
36*a*b^5*d^3*e^5 + 420*a^2*b^4*d^2*e^6 + 160*a^3*b^3*d*e^7 + 15*a^4*b^2*e^8)*x^1
1 + 1/5*(28*b^6*d^5*e^3 + 210*a*b^5*d^4*e^4 + 420*a^2*b^4*d^3*e^5 + 280*a^3*b^3*
d^2*e^6 + 60*a^4*b^2*d*e^7 + 3*a^5*b*e^8)*x^10 + 1/9*(28*b^6*d^6*e^2 + 336*a*b^5
*d^5*e^3 + 1050*a^2*b^4*d^4*e^4 + 1120*a^3*b^3*d^3*e^5 + 420*a^4*b^2*d^2*e^6 + 4
8*a^5*b*d*e^7 + a^6*e^8)*x^9 + (b^6*d^7*e + 21*a*b^5*d^6*e^2 + 105*a^2*b^4*d^5*e
^3 + 175*a^3*b^3*d^4*e^4 + 105*a^4*b^2*d^3*e^5 + 21*a^5*b*d^2*e^6 + a^6*d*e^7)*x
^8 + 1/7*(b^6*d^8 + 48*a*b^5*d^7*e + 420*a^2*b^4*d^6*e^2 + 1120*a^3*b^3*d^5*e^3
+ 1050*a^4*b^2*d^4*e^4 + 336*a^5*b*d^3*e^5 + 28*a^6*d^2*e^6)*x^7 + 1/3*(3*a*b^5*
d^8 + 60*a^2*b^4*d^7*e + 280*a^3*b^3*d^6*e^2 + 420*a^4*b^2*d^5*e^3 + 210*a^5*b*d
^4*e^4 + 28*a^6*d^3*e^5)*x^6 + 1/5*(15*a^2*b^4*d^8 + 160*a^3*b^3*d^7*e + 420*a^4
*b^2*d^6*e^2 + 336*a^5*b*d^5*e^3 + 70*a^6*d^4*e^4)*x^5 + (5*a^3*b^3*d^8 + 30*a^4
*b^2*d^7*e + 42*a^5*b*d^6*e^2 + 14*a^6*d^5*e^3)*x^4 + 1/3*(15*a^4*b^2*d^8 + 48*a
^5*b*d^7*e + 28*a^6*d^6*e^2)*x^3 + (3*a^5*b*d^8 + 4*a^6*d^7*e)*x^2

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Fricas [A]  time = 0.182426, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(e*x + d)^8,x, algorithm="fricas")

[Out]

1/15*x^15*e^8*b^6 + 4/7*x^14*e^7*d*b^6 + 3/7*x^14*e^8*b^5*a + 28/13*x^13*e^6*d^2
*b^6 + 48/13*x^13*e^7*d*b^5*a + 15/13*x^13*e^8*b^4*a^2 + 14/3*x^12*e^5*d^3*b^6 +
14*x^12*e^6*d^2*b^5*a + 10*x^12*e^7*d*b^4*a^2 + 5/3*x^12*e^8*b^3*a^3 + 70/11*x^
11*e^4*d^4*b^6 + 336/11*x^11*e^5*d^3*b^5*a + 420/11*x^11*e^6*d^2*b^4*a^2 + 160/1
1*x^11*e^7*d*b^3*a^3 + 15/11*x^11*e^8*b^2*a^4 + 28/5*x^10*e^3*d^5*b^6 + 42*x^10*
e^4*d^4*b^5*a + 84*x^10*e^5*d^3*b^4*a^2 + 56*x^10*e^6*d^2*b^3*a^3 + 12*x^10*e^7*
d*b^2*a^4 + 3/5*x^10*e^8*b*a^5 + 28/9*x^9*e^2*d^6*b^6 + 112/3*x^9*e^3*d^5*b^5*a
+ 350/3*x^9*e^4*d^4*b^4*a^2 + 1120/9*x^9*e^5*d^3*b^3*a^3 + 140/3*x^9*e^6*d^2*b^2
*a^4 + 16/3*x^9*e^7*d*b*a^5 + 1/9*x^9*e^8*a^6 + x^8*e*d^7*b^6 + 21*x^8*e^2*d^6*b
^5*a + 105*x^8*e^3*d^5*b^4*a^2 + 175*x^8*e^4*d^4*b^3*a^3 + 105*x^8*e^5*d^3*b^2*a
^4 + 21*x^8*e^6*d^2*b*a^5 + x^8*e^7*d*a^6 + 1/7*x^7*d^8*b^6 + 48/7*x^7*e*d^7*b^5
*a + 60*x^7*e^2*d^6*b^4*a^2 + 160*x^7*e^3*d^5*b^3*a^3 + 150*x^7*e^4*d^4*b^2*a^4
+ 48*x^7*e^5*d^3*b*a^5 + 4*x^7*e^6*d^2*a^6 + x^6*d^8*b^5*a + 20*x^6*e*d^7*b^4*a^
2 + 280/3*x^6*e^2*d^6*b^3*a^3 + 140*x^6*e^3*d^5*b^2*a^4 + 70*x^6*e^4*d^4*b*a^5 +
28/3*x^6*e^5*d^3*a^6 + 3*x^5*d^8*b^4*a^2 + 32*x^5*e*d^7*b^3*a^3 + 84*x^5*e^2*d^
6*b^2*a^4 + 336/5*x^5*e^3*d^5*b*a^5 + 14*x^5*e^4*d^4*a^6 + 5*x^4*d^8*b^3*a^3 + 3
0*x^4*e*d^7*b^2*a^4 + 42*x^4*e^2*d^6*b*a^5 + 14*x^4*e^3*d^5*a^6 + 5*x^3*d^8*b^2*
a^4 + 16*x^3*e*d^7*b*a^5 + 28/3*x^3*e^2*d^6*a^6 + 3*x^2*d^8*b*a^5 + 4*x^2*e*d^7*
a^6 + x*d^8*a^6

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Sympy [A]  time = 0.490806, size = 884, normalized size = 5.11 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**8*(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

a**6*d**8*x + b**6*e**8*x**15/15 + x**14*(3*a*b**5*e**8/7 + 4*b**6*d*e**7/7) + x
**13*(15*a**2*b**4*e**8/13 + 48*a*b**5*d*e**7/13 + 28*b**6*d**2*e**6/13) + x**12
*(5*a**3*b**3*e**8/3 + 10*a**2*b**4*d*e**7 + 14*a*b**5*d**2*e**6 + 14*b**6*d**3*
e**5/3) + x**11*(15*a**4*b**2*e**8/11 + 160*a**3*b**3*d*e**7/11 + 420*a**2*b**4*
d**2*e**6/11 + 336*a*b**5*d**3*e**5/11 + 70*b**6*d**4*e**4/11) + x**10*(3*a**5*b
*e**8/5 + 12*a**4*b**2*d*e**7 + 56*a**3*b**3*d**2*e**6 + 84*a**2*b**4*d**3*e**5
+ 42*a*b**5*d**4*e**4 + 28*b**6*d**5*e**3/5) + x**9*(a**6*e**8/9 + 16*a**5*b*d*e
**7/3 + 140*a**4*b**2*d**2*e**6/3 + 1120*a**3*b**3*d**3*e**5/9 + 350*a**2*b**4*d
**4*e**4/3 + 112*a*b**5*d**5*e**3/3 + 28*b**6*d**6*e**2/9) + x**8*(a**6*d*e**7 +
21*a**5*b*d**2*e**6 + 105*a**4*b**2*d**3*e**5 + 175*a**3*b**3*d**4*e**4 + 105*a
**2*b**4*d**5*e**3 + 21*a*b**5*d**6*e**2 + b**6*d**7*e) + x**7*(4*a**6*d**2*e**6
+ 48*a**5*b*d**3*e**5 + 150*a**4*b**2*d**4*e**4 + 160*a**3*b**3*d**5*e**3 + 60*
a**2*b**4*d**6*e**2 + 48*a*b**5*d**7*e/7 + b**6*d**8/7) + x**6*(28*a**6*d**3*e**
5/3 + 70*a**5*b*d**4*e**4 + 140*a**4*b**2*d**5*e**3 + 280*a**3*b**3*d**6*e**2/3
+ 20*a**2*b**4*d**7*e + a*b**5*d**8) + x**5*(14*a**6*d**4*e**4 + 336*a**5*b*d**5
*e**3/5 + 84*a**4*b**2*d**6*e**2 + 32*a**3*b**3*d**7*e + 3*a**2*b**4*d**8) + x**
4*(14*a**6*d**5*e**3 + 42*a**5*b*d**6*e**2 + 30*a**4*b**2*d**7*e + 5*a**3*b**3*d
**8) + x**3*(28*a**6*d**6*e**2/3 + 16*a**5*b*d**7*e + 5*a**4*b**2*d**8) + x**2*(
4*a**6*d**7*e + 3*a**5*b*d**8)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.208745, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(e*x + d)^8,x, algorithm="giac")

[Out]

Done