3.1069 \(\int (a+b x)^{10} (A+B x) (d+e x)^2 \, dx\)

Optimal. Leaf size=118 \[ \frac{e (a+b x)^{13} (-3 a B e+A b e+2 b B d)}{13 b^4}+\frac{(a+b x)^{12} (b d-a e) (-3 a B e+2 A b e+b B d)}{12 b^4}+\frac{(a+b x)^{11} (A b-a B) (b d-a e)^2}{11 b^4}+\frac{B e^2 (a+b x)^{14}}{14 b^4} \]

[Out]

((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)^1
3)/(13*b^4) + (B*e^2*(a + b*x)^14)/(14*b^4)

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Rubi [A]  time = 1.88172, antiderivative size = 118, 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 \[ \frac{e (a+b x)^{13} (-3 a B e+A b e+2 b B d)}{13 b^4}+\frac{(a+b x)^{12} (b d-a e) (-3 a B e+2 A b e+b B d)}{12 b^4}+\frac{(a+b x)^{11} (A b-a B) (b d-a e)^2}{11 b^4}+\frac{B e^2 (a+b x)^{14}}{14 b^4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^10*(A + B*x)*(d + e*x)^2,x]

[Out]

((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)^1
3)/(13*b^4) + (B*e^2*(a + b*x)^14)/(14*b^4)

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Rubi in Sympy [A]  time = 92.747, size = 112, normalized size = 0.95 \[ \frac{B e^{2} \left (a + b x\right )^{14}}{14 b^{4}} + \frac{e \left (a + b x\right )^{13} \left (A b e - 3 B a e + 2 B b d\right )}{13 b^{4}} - \frac{\left (a + b x\right )^{12} \left (a e - b d\right ) \left (2 A b e - 3 B a e + B b d\right )}{12 b^{4}} + \frac{\left (a + b x\right )^{11} \left (A b - B a\right ) \left (a e - b d\right )^{2}}{11 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**10*(B*x+A)*(e*x+d)**2,x)

[Out]

B*e**2*(a + b*x)**14/(14*b**4) + e*(a + b*x)**13*(A*b*e - 3*B*a*e + 2*B*b*d)/(13
*b**4) - (a + b*x)**12*(a*e - b*d)*(2*A*b*e - 3*B*a*e + B*b*d)/(12*b**4) + (a +
b*x)**11*(A*b - B*a)*(a*e - b*d)**2/(11*b**4)

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Mathematica [B]  time = 1.18388, size = 614, normalized size = 5.2 \[ \frac{x \left (1001 a^{10} \left (4 A \left (3 d^2+3 d e x+e^2 x^2\right )+B x \left (6 d^2+8 d e x+3 e^2 x^2\right )\right )+2002 a^9 b x \left (5 A \left (6 d^2+8 d e x+3 e^2 x^2\right )+2 B x \left (10 d^2+15 d e x+6 e^2 x^2\right )\right )+9009 a^8 b^2 x^2 \left (2 A \left (10 d^2+15 d e x+6 e^2 x^2\right )+B x \left (15 d^2+24 d e x+10 e^2 x^2\right )\right )+3432 a^7 b^3 x^3 \left (7 A \left (15 d^2+24 d e x+10 e^2 x^2\right )+4 B x \left (21 d^2+35 d e x+15 e^2 x^2\right )\right )+3003 a^6 b^4 x^4 \left (8 A \left (21 d^2+35 d e x+15 e^2 x^2\right )+5 B x \left (28 d^2+48 d e x+21 e^2 x^2\right )\right )+6006 a^5 b^5 x^5 \left (3 A \left (28 d^2+48 d e x+21 e^2 x^2\right )+2 B x \left (36 d^2+63 d e x+28 e^2 x^2\right )\right )+1001 a^4 b^6 x^6 \left (10 A \left (36 d^2+63 d e x+28 e^2 x^2\right )+7 B x \left (45 d^2+80 d e x+36 e^2 x^2\right )\right )+364 a^3 b^7 x^7 \left (11 A \left (45 d^2+80 d e x+36 e^2 x^2\right )+8 B x \left (55 d^2+99 d e x+45 e^2 x^2\right )\right )+273 a^2 b^8 x^8 \left (4 A \left (55 d^2+99 d e x+45 e^2 x^2\right )+3 B x \left (66 d^2+120 d e x+55 e^2 x^2\right )\right )+14 a b^9 x^9 \left (13 A \left (66 d^2+120 d e x+55 e^2 x^2\right )+10 B x \left (78 d^2+143 d e x+66 e^2 x^2\right )\right )+b^{10} x^{10} \left (14 A \left (78 d^2+143 d e x+66 e^2 x^2\right )+11 B x \left (91 d^2+168 d e x+78 e^2 x^2\right )\right )\right )}{12012} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^10*(A + B*x)*(d + e*x)^2,x]

[Out]

(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

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Maple [B]  time = 0.003, size = 769, normalized size = 6.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^10*(B*x+A)*(e*x+d)^2,x)

[Out]

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*(210*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+(210*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+(210*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^10*A*d*e+(1
0*A*a^9*b+B*a^10)*d^2)*x^2+a^10*A*d^2*x

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Maxima [A]  time = 1.3706, size = 1054, normalized size = 8.93 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10*(e*x + d)^2,x, algorithm="maxima")

[Out]

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 + 1
0*(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

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Fricas [A]  time = 0.196975, 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*x + A)*(b*x + a)^10*(e*x + d)^2,x, algorithm="fricas")

[Out]

1/14*x^14*e^2*b^10*B + 2/13*x^13*e*d*b^10*B + 10/13*x^13*e^2*b^9*a*B + 1/13*x^13
*e^2*b^10*A + 1/12*x^12*d^2*b^10*B + 5/3*x^12*e*d*b^9*a*B + 15/4*x^12*e^2*b^8*a^
2*B + 1/6*x^12*e*d*b^10*A + 5/6*x^12*e^2*b^9*a*A + 10/11*x^11*d^2*b^9*a*B + 90/1
1*x^11*e*d*b^8*a^2*B + 120/11*x^11*e^2*b^7*a^3*B + 1/11*x^11*d^2*b^10*A + 20/11*
x^11*e*d*b^9*a*A + 45/11*x^11*e^2*b^8*a^2*A + 9/2*x^10*d^2*b^8*a^2*B + 24*x^10*e
*d*b^7*a^3*B + 21*x^10*e^2*b^6*a^4*B + x^10*d^2*b^9*a*A + 9*x^10*e*d*b^8*a^2*A +
12*x^10*e^2*b^7*a^3*A + 40/3*x^9*d^2*b^7*a^3*B + 140/3*x^9*e*d*b^6*a^4*B + 28*x
^9*e^2*b^5*a^5*B + 5*x^9*d^2*b^8*a^2*A + 80/3*x^9*e*d*b^7*a^3*A + 70/3*x^9*e^2*b
^6*a^4*A + 105/4*x^8*d^2*b^6*a^4*B + 63*x^8*e*d*b^5*a^5*B + 105/4*x^8*e^2*b^4*a^
6*B + 15*x^8*d^2*b^7*a^3*A + 105/2*x^8*e*d*b^6*a^4*A + 63/2*x^8*e^2*b^5*a^5*A +
36*x^7*d^2*b^5*a^5*B + 60*x^7*e*d*b^4*a^6*B + 120/7*x^7*e^2*b^3*a^7*B + 30*x^7*d
^2*b^6*a^4*A + 72*x^7*e*d*b^5*a^5*A + 30*x^7*e^2*b^4*a^6*A + 35*x^6*d^2*b^4*a^6*
B + 40*x^6*e*d*b^3*a^7*B + 15/2*x^6*e^2*b^2*a^8*B + 42*x^6*d^2*b^5*a^5*A + 70*x^
6*e*d*b^4*a^6*A + 20*x^6*e^2*b^3*a^7*A + 24*x^5*d^2*b^3*a^7*B + 18*x^5*e*d*b^2*a
^8*B + 2*x^5*e^2*b*a^9*B + 42*x^5*d^2*b^4*a^6*A + 48*x^5*e*d*b^3*a^7*A + 9*x^5*e
^2*b^2*a^8*A + 45/4*x^4*d^2*b^2*a^8*B + 5*x^4*e*d*b*a^9*B + 1/4*x^4*e^2*a^10*B +
30*x^4*d^2*b^3*a^7*A + 45/2*x^4*e*d*b^2*a^8*A + 5/2*x^4*e^2*b*a^9*A + 10/3*x^3*
d^2*b*a^9*B + 2/3*x^3*e*d*a^10*B + 15*x^3*d^2*b^2*a^8*A + 20/3*x^3*e*d*b*a^9*A +
1/3*x^3*e^2*a^10*A + 1/2*x^2*d^2*a^10*B + 5*x^2*d^2*b*a^9*A + x^2*e*d*a^10*A +
x*d^2*a^10*A

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Sympy [A]  time = 0.488616, size = 921, normalized size = 7.81 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**10*(B*x+A)*(e*x+d)**2,x)

[Out]

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 + 105*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 + 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) + 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**1
0*d*e + 5*A*a**9*b*d**2 + B*a**10*d**2/2)

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GIAC/XCAS [A]  time = 0.210818, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10*(e*x + d)^2,x, algorithm="giac")

[Out]

Done