3.108 \(\int x^8 (a+b x)^{10} (A+B x) \, dx\)

Optimal. Leaf size=240 \[ \frac{a^8 (a+b x)^{11} (A b-a B)}{11 b^{10}}-\frac{a^7 (a+b x)^{12} (8 A b-9 a B)}{12 b^{10}}+\frac{4 a^6 (a+b x)^{13} (7 A b-9 a B)}{13 b^{10}}-\frac{2 a^5 (a+b x)^{14} (2 A b-3 a B)}{b^{10}}+\frac{14 a^4 (a+b x)^{15} (5 A b-9 a B)}{15 b^{10}}-\frac{7 a^3 (a+b x)^{16} (4 A b-9 a B)}{8 b^{10}}+\frac{28 a^2 (a+b x)^{17} (A b-3 a B)}{17 b^{10}}+\frac{(a+b x)^{19} (A b-9 a B)}{19 b^{10}}-\frac{2 a (a+b x)^{18} (2 A b-9 a B)}{9 b^{10}}+\frac{B (a+b x)^{20}}{20 b^{10}} \]

[Out]

(a^8*(A*b - a*B)*(a + b*x)^11)/(11*b^10) - (a^7*(8*A*b - 9*a*B)*(a + b*x)^12)/(1
2*b^10) + (4*a^6*(7*A*b - 9*a*B)*(a + b*x)^13)/(13*b^10) - (2*a^5*(2*A*b - 3*a*B
)*(a + b*x)^14)/b^10 + (14*a^4*(5*A*b - 9*a*B)*(a + b*x)^15)/(15*b^10) - (7*a^3*
(4*A*b - 9*a*B)*(a + b*x)^16)/(8*b^10) + (28*a^2*(A*b - 3*a*B)*(a + b*x)^17)/(17
*b^10) - (2*a*(2*A*b - 9*a*B)*(a + b*x)^18)/(9*b^10) + ((A*b - 9*a*B)*(a + b*x)^
19)/(19*b^10) + (B*(a + b*x)^20)/(20*b^10)

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Rubi [A]  time = 0.606041, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{a^8 (a+b x)^{11} (A b-a B)}{11 b^{10}}-\frac{a^7 (a+b x)^{12} (8 A b-9 a B)}{12 b^{10}}+\frac{4 a^6 (a+b x)^{13} (7 A b-9 a B)}{13 b^{10}}-\frac{2 a^5 (a+b x)^{14} (2 A b-3 a B)}{b^{10}}+\frac{14 a^4 (a+b x)^{15} (5 A b-9 a B)}{15 b^{10}}-\frac{7 a^3 (a+b x)^{16} (4 A b-9 a B)}{8 b^{10}}+\frac{28 a^2 (a+b x)^{17} (A b-3 a B)}{17 b^{10}}+\frac{(a+b x)^{19} (A b-9 a B)}{19 b^{10}}-\frac{2 a (a+b x)^{18} (2 A b-9 a B)}{9 b^{10}}+\frac{B (a+b x)^{20}}{20 b^{10}} \]

Antiderivative was successfully verified.

[In]  Int[x^8*(a + b*x)^10*(A + B*x),x]

[Out]

(a^8*(A*b - a*B)*(a + b*x)^11)/(11*b^10) - (a^7*(8*A*b - 9*a*B)*(a + b*x)^12)/(1
2*b^10) + (4*a^6*(7*A*b - 9*a*B)*(a + b*x)^13)/(13*b^10) - (2*a^5*(2*A*b - 3*a*B
)*(a + b*x)^14)/b^10 + (14*a^4*(5*A*b - 9*a*B)*(a + b*x)^15)/(15*b^10) - (7*a^3*
(4*A*b - 9*a*B)*(a + b*x)^16)/(8*b^10) + (28*a^2*(A*b - 3*a*B)*(a + b*x)^17)/(17
*b^10) - (2*a*(2*A*b - 9*a*B)*(a + b*x)^18)/(9*b^10) + ((A*b - 9*a*B)*(a + b*x)^
19)/(19*b^10) + (B*(a + b*x)^20)/(20*b^10)

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Rubi in Sympy [A]  time = 102.554, size = 238, normalized size = 0.99 \[ \frac{A a^{10} x^{9}}{9} + \frac{B b^{10} x^{20}}{20} + \frac{a^{9} x^{10} \left (10 A b + B a\right )}{10} + \frac{5 a^{8} b x^{11} \left (9 A b + 2 B a\right )}{11} + \frac{5 a^{7} b^{2} x^{12} \left (8 A b + 3 B a\right )}{4} + \frac{30 a^{6} b^{3} x^{13} \left (7 A b + 4 B a\right )}{13} + 3 a^{5} b^{4} x^{14} \left (6 A b + 5 B a\right ) + \frac{14 a^{4} b^{5} x^{15} \left (5 A b + 6 B a\right )}{5} + \frac{15 a^{3} b^{6} x^{16} \left (4 A b + 7 B a\right )}{8} + \frac{15 a^{2} b^{7} x^{17} \left (3 A b + 8 B a\right )}{17} + \frac{5 a b^{8} x^{18} \left (2 A b + 9 B a\right )}{18} + \frac{b^{9} x^{19} \left (A b + 10 B a\right )}{19} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8*(b*x+a)**10*(B*x+A),x)

[Out]

A*a**10*x**9/9 + B*b**10*x**20/20 + a**9*x**10*(10*A*b + B*a)/10 + 5*a**8*b*x**1
1*(9*A*b + 2*B*a)/11 + 5*a**7*b**2*x**12*(8*A*b + 3*B*a)/4 + 30*a**6*b**3*x**13*
(7*A*b + 4*B*a)/13 + 3*a**5*b**4*x**14*(6*A*b + 5*B*a) + 14*a**4*b**5*x**15*(5*A
*b + 6*B*a)/5 + 15*a**3*b**6*x**16*(4*A*b + 7*B*a)/8 + 15*a**2*b**7*x**17*(3*A*b
+ 8*B*a)/17 + 5*a*b**8*x**18*(2*A*b + 9*B*a)/18 + b**9*x**19*(A*b + 10*B*a)/19

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Mathematica [A]  time = 0.0544877, size = 229, normalized size = 0.95 \[ \frac{1}{9} a^{10} A x^9+\frac{1}{10} a^9 x^{10} (a B+10 A b)+\frac{5}{11} a^8 b x^{11} (2 a B+9 A b)+\frac{5}{4} a^7 b^2 x^{12} (3 a B+8 A b)+\frac{30}{13} a^6 b^3 x^{13} (4 a B+7 A b)+3 a^5 b^4 x^{14} (5 a B+6 A b)+\frac{14}{5} a^4 b^5 x^{15} (6 a B+5 A b)+\frac{15}{8} a^3 b^6 x^{16} (7 a B+4 A b)+\frac{15}{17} a^2 b^7 x^{17} (8 a B+3 A b)+\frac{1}{19} b^9 x^{19} (10 a B+A b)+\frac{5}{18} a b^8 x^{18} (9 a B+2 A b)+\frac{1}{20} b^{10} B x^{20} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8*(a + b*x)^10*(A + B*x),x]

[Out]

(a^10*A*x^9)/9 + (a^9*(10*A*b + a*B)*x^10)/10 + (5*a^8*b*(9*A*b + 2*a*B)*x^11)/1
1 + (5*a^7*b^2*(8*A*b + 3*a*B)*x^12)/4 + (30*a^6*b^3*(7*A*b + 4*a*B)*x^13)/13 +
3*a^5*b^4*(6*A*b + 5*a*B)*x^14 + (14*a^4*b^5*(5*A*b + 6*a*B)*x^15)/5 + (15*a^3*b
^6*(4*A*b + 7*a*B)*x^16)/8 + (15*a^2*b^7*(3*A*b + 8*a*B)*x^17)/17 + (5*a*b^8*(2*
A*b + 9*a*B)*x^18)/18 + (b^9*(A*b + 10*a*B)*x^19)/19 + (b^10*B*x^20)/20

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Maple [A]  time = 0.002, size = 244, normalized size = 1. \[{\frac{{b}^{10}B{x}^{20}}{20}}+{\frac{ \left ({b}^{10}A+10\,a{b}^{9}B \right ){x}^{19}}{19}}+{\frac{ \left ( 10\,a{b}^{9}A+45\,{a}^{2}{b}^{8}B \right ){x}^{18}}{18}}+{\frac{ \left ( 45\,{a}^{2}{b}^{8}A+120\,{a}^{3}{b}^{7}B \right ){x}^{17}}{17}}+{\frac{ \left ( 120\,{a}^{3}{b}^{7}A+210\,{a}^{4}{b}^{6}B \right ){x}^{16}}{16}}+{\frac{ \left ( 210\,{a}^{4}{b}^{6}A+252\,{a}^{5}{b}^{5}B \right ){x}^{15}}{15}}+{\frac{ \left ( 252\,{a}^{5}{b}^{5}A+210\,{a}^{6}{b}^{4}B \right ){x}^{14}}{14}}+{\frac{ \left ( 210\,{a}^{6}{b}^{4}A+120\,{a}^{7}{b}^{3}B \right ){x}^{13}}{13}}+{\frac{ \left ( 120\,{a}^{7}{b}^{3}A+45\,{a}^{8}{b}^{2}B \right ){x}^{12}}{12}}+{\frac{ \left ( 45\,{a}^{8}{b}^{2}A+10\,{a}^{9}bB \right ){x}^{11}}{11}}+{\frac{ \left ( 10\,{a}^{9}bA+{a}^{10}B \right ){x}^{10}}{10}}+{\frac{{a}^{10}A{x}^{9}}{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8*(b*x+a)^10*(B*x+A),x)

[Out]

1/20*b^10*B*x^20+1/19*(A*b^10+10*B*a*b^9)*x^19+1/18*(10*A*a*b^9+45*B*a^2*b^8)*x^
18+1/17*(45*A*a^2*b^8+120*B*a^3*b^7)*x^17+1/16*(120*A*a^3*b^7+210*B*a^4*b^6)*x^1
6+1/15*(210*A*a^4*b^6+252*B*a^5*b^5)*x^15+1/14*(252*A*a^5*b^5+210*B*a^6*b^4)*x^1
4+1/13*(210*A*a^6*b^4+120*B*a^7*b^3)*x^13+1/12*(120*A*a^7*b^3+45*B*a^8*b^2)*x^12
+1/11*(45*A*a^8*b^2+10*B*a^9*b)*x^11+1/10*(10*A*a^9*b+B*a^10)*x^10+1/9*a^10*A*x^
9

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Maxima [A]  time = 1.33996, size = 328, normalized size = 1.37 \[ \frac{1}{20} \, B b^{10} x^{20} + \frac{1}{9} \, A a^{10} x^{9} + \frac{1}{19} \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{19} + \frac{5}{18} \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{18} + \frac{15}{17} \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{17} + \frac{15}{8} \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{16} + \frac{14}{5} \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{15} + 3 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{14} + \frac{30}{13} \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{13} + \frac{5}{4} \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{12} + \frac{5}{11} \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{11} + \frac{1}{10} \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x^{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10*x^8,x, algorithm="maxima")

[Out]

1/20*B*b^10*x^20 + 1/9*A*a^10*x^9 + 1/19*(10*B*a*b^9 + A*b^10)*x^19 + 5/18*(9*B*
a^2*b^8 + 2*A*a*b^9)*x^18 + 15/17*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^17 + 15/8*(7*B*a
^4*b^6 + 4*A*a^3*b^7)*x^16 + 14/5*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^15 + 3*(5*B*a^6*
b^4 + 6*A*a^5*b^5)*x^14 + 30/13*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^13 + 5/4*(3*B*a^8*
b^2 + 8*A*a^7*b^3)*x^12 + 5/11*(2*B*a^9*b + 9*A*a^8*b^2)*x^11 + 1/10*(B*a^10 + 1
0*A*a^9*b)*x^10

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Fricas [A]  time = 0.181452, size = 1, normalized size = 0. \[ \frac{1}{20} x^{20} b^{10} B + \frac{10}{19} x^{19} b^{9} a B + \frac{1}{19} x^{19} b^{10} A + \frac{5}{2} x^{18} b^{8} a^{2} B + \frac{5}{9} x^{18} b^{9} a A + \frac{120}{17} x^{17} b^{7} a^{3} B + \frac{45}{17} x^{17} b^{8} a^{2} A + \frac{105}{8} x^{16} b^{6} a^{4} B + \frac{15}{2} x^{16} b^{7} a^{3} A + \frac{84}{5} x^{15} b^{5} a^{5} B + 14 x^{15} b^{6} a^{4} A + 15 x^{14} b^{4} a^{6} B + 18 x^{14} b^{5} a^{5} A + \frac{120}{13} x^{13} b^{3} a^{7} B + \frac{210}{13} x^{13} b^{4} a^{6} A + \frac{15}{4} x^{12} b^{2} a^{8} B + 10 x^{12} b^{3} a^{7} A + \frac{10}{11} x^{11} b a^{9} B + \frac{45}{11} x^{11} b^{2} a^{8} A + \frac{1}{10} x^{10} a^{10} B + x^{10} b a^{9} A + \frac{1}{9} x^{9} a^{10} A \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10*x^8,x, algorithm="fricas")

[Out]

1/20*x^20*b^10*B + 10/19*x^19*b^9*a*B + 1/19*x^19*b^10*A + 5/2*x^18*b^8*a^2*B +
5/9*x^18*b^9*a*A + 120/17*x^17*b^7*a^3*B + 45/17*x^17*b^8*a^2*A + 105/8*x^16*b^6
*a^4*B + 15/2*x^16*b^7*a^3*A + 84/5*x^15*b^5*a^5*B + 14*x^15*b^6*a^4*A + 15*x^14
*b^4*a^6*B + 18*x^14*b^5*a^5*A + 120/13*x^13*b^3*a^7*B + 210/13*x^13*b^4*a^6*A +
15/4*x^12*b^2*a^8*B + 10*x^12*b^3*a^7*A + 10/11*x^11*b*a^9*B + 45/11*x^11*b^2*a
^8*A + 1/10*x^10*a^10*B + x^10*b*a^9*A + 1/9*x^9*a^10*A

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Sympy [A]  time = 0.261811, size = 264, normalized size = 1.1 \[ \frac{A a^{10} x^{9}}{9} + \frac{B b^{10} x^{20}}{20} + x^{19} \left (\frac{A b^{10}}{19} + \frac{10 B a b^{9}}{19}\right ) + x^{18} \left (\frac{5 A a b^{9}}{9} + \frac{5 B a^{2} b^{8}}{2}\right ) + x^{17} \left (\frac{45 A a^{2} b^{8}}{17} + \frac{120 B a^{3} b^{7}}{17}\right ) + x^{16} \left (\frac{15 A a^{3} b^{7}}{2} + \frac{105 B a^{4} b^{6}}{8}\right ) + x^{15} \left (14 A a^{4} b^{6} + \frac{84 B a^{5} b^{5}}{5}\right ) + x^{14} \left (18 A a^{5} b^{5} + 15 B a^{6} b^{4}\right ) + x^{13} \left (\frac{210 A a^{6} b^{4}}{13} + \frac{120 B a^{7} b^{3}}{13}\right ) + x^{12} \left (10 A a^{7} b^{3} + \frac{15 B a^{8} b^{2}}{4}\right ) + x^{11} \left (\frac{45 A a^{8} b^{2}}{11} + \frac{10 B a^{9} b}{11}\right ) + x^{10} \left (A a^{9} b + \frac{B a^{10}}{10}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8*(b*x+a)**10*(B*x+A),x)

[Out]

A*a**10*x**9/9 + B*b**10*x**20/20 + x**19*(A*b**10/19 + 10*B*a*b**9/19) + x**18*
(5*A*a*b**9/9 + 5*B*a**2*b**8/2) + x**17*(45*A*a**2*b**8/17 + 120*B*a**3*b**7/17
) + x**16*(15*A*a**3*b**7/2 + 105*B*a**4*b**6/8) + x**15*(14*A*a**4*b**6 + 84*B*
a**5*b**5/5) + x**14*(18*A*a**5*b**5 + 15*B*a**6*b**4) + x**13*(210*A*a**6*b**4/
13 + 120*B*a**7*b**3/13) + x**12*(10*A*a**7*b**3 + 15*B*a**8*b**2/4) + x**11*(45
*A*a**8*b**2/11 + 10*B*a**9*b/11) + x**10*(A*a**9*b + B*a**10/10)

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GIAC/XCAS [A]  time = 0.235297, size = 329, normalized size = 1.37 \[ \frac{1}{20} \, B b^{10} x^{20} + \frac{10}{19} \, B a b^{9} x^{19} + \frac{1}{19} \, A b^{10} x^{19} + \frac{5}{2} \, B a^{2} b^{8} x^{18} + \frac{5}{9} \, A a b^{9} x^{18} + \frac{120}{17} \, B a^{3} b^{7} x^{17} + \frac{45}{17} \, A a^{2} b^{8} x^{17} + \frac{105}{8} \, B a^{4} b^{6} x^{16} + \frac{15}{2} \, A a^{3} b^{7} x^{16} + \frac{84}{5} \, B a^{5} b^{5} x^{15} + 14 \, A a^{4} b^{6} x^{15} + 15 \, B a^{6} b^{4} x^{14} + 18 \, A a^{5} b^{5} x^{14} + \frac{120}{13} \, B a^{7} b^{3} x^{13} + \frac{210}{13} \, A a^{6} b^{4} x^{13} + \frac{15}{4} \, B a^{8} b^{2} x^{12} + 10 \, A a^{7} b^{3} x^{12} + \frac{10}{11} \, B a^{9} b x^{11} + \frac{45}{11} \, A a^{8} b^{2} x^{11} + \frac{1}{10} \, B a^{10} x^{10} + A a^{9} b x^{10} + \frac{1}{9} \, A a^{10} x^{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10*x^8,x, algorithm="giac")

[Out]

1/20*B*b^10*x^20 + 10/19*B*a*b^9*x^19 + 1/19*A*b^10*x^19 + 5/2*B*a^2*b^8*x^18 +
5/9*A*a*b^9*x^18 + 120/17*B*a^3*b^7*x^17 + 45/17*A*a^2*b^8*x^17 + 105/8*B*a^4*b^
6*x^16 + 15/2*A*a^3*b^7*x^16 + 84/5*B*a^5*b^5*x^15 + 14*A*a^4*b^6*x^15 + 15*B*a^
6*b^4*x^14 + 18*A*a^5*b^5*x^14 + 120/13*B*a^7*b^3*x^13 + 210/13*A*a^6*b^4*x^13 +
15/4*B*a^8*b^2*x^12 + 10*A*a^7*b^3*x^12 + 10/11*B*a^9*b*x^11 + 45/11*A*a^8*b^2*
x^11 + 1/10*B*a^10*x^10 + A*a^9*b*x^10 + 1/9*A*a^10*x^9