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

Optimal. Leaf size=38 \[ \frac{(a+b x)^{11} (A b-a B)}{11 b^2}+\frac{B (a+b x)^{12}}{12 b^2} \]

[Out]

((A*b - a*B)*(a + b*x)^11)/(11*b^2) + (B*(a + b*x)^12)/(12*b^2)

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Rubi [A]  time = 0.0527079, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{(a+b x)^{11} (A b-a B)}{11 b^2}+\frac{B (a+b x)^{12}}{12 b^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^10*(A + B*x),x]

[Out]

((A*b - a*B)*(a + b*x)^11)/(11*b^2) + (B*(a + b*x)^12)/(12*b^2)

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Rubi in Sympy [A]  time = 32.7378, size = 31, normalized size = 0.82 \[ \frac{B \left (a + b x\right )^{12}}{12 b^{2}} + \frac{\left (a + b x\right )^{11} \left (A b - B a\right )}{11 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*(a + b*x)**12/(12*b**2) + (a + b*x)**11*(A*b - B*a)/(11*b**2)

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Mathematica [B]  time = 0.102354, size = 198, normalized size = 5.21 \[ \frac{1}{132} x \left (66 a^{10} (2 A+B x)+220 a^9 b x (3 A+2 B x)+495 a^8 b^2 x^2 (4 A+3 B x)+792 a^7 b^3 x^3 (5 A+4 B x)+924 a^6 b^4 x^4 (6 A+5 B x)+792 a^5 b^5 x^5 (7 A+6 B x)+495 a^4 b^6 x^6 (8 A+7 B x)+220 a^3 b^7 x^7 (9 A+8 B x)+66 a^2 b^8 x^8 (10 A+9 B x)+12 a b^9 x^9 (11 A+10 B x)+b^{10} x^{10} (12 A+11 B x)\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x*(66*a^10*(2*A + B*x) + 220*a^9*b*x*(3*A + 2*B*x) + 495*a^8*b^2*x^2*(4*A + 3*B
*x) + 792*a^7*b^3*x^3*(5*A + 4*B*x) + 924*a^6*b^4*x^4*(6*A + 5*B*x) + 792*a^5*b^
5*x^5*(7*A + 6*B*x) + 495*a^4*b^6*x^6*(8*A + 7*B*x) + 220*a^3*b^7*x^7*(9*A + 8*B
*x) + 66*a^2*b^8*x^8*(10*A + 9*B*x) + 12*a*b^9*x^9*(11*A + 10*B*x) + b^10*x^10*(
12*A + 11*B*x)))/132

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Maple [B]  time = 0.003, size = 241, normalized size = 6.3 \[{\frac{{b}^{10}B{x}^{12}}{12}}+{\frac{ \left ({b}^{10}A+10\,a{b}^{9}B \right ){x}^{11}}{11}}+{\frac{ \left ( 10\,a{b}^{9}A+45\,{a}^{2}{b}^{8}B \right ){x}^{10}}{10}}+{\frac{ \left ( 45\,{a}^{2}{b}^{8}A+120\,{a}^{3}{b}^{7}B \right ){x}^{9}}{9}}+{\frac{ \left ( 120\,{a}^{3}{b}^{7}A+210\,{a}^{4}{b}^{6}B \right ){x}^{8}}{8}}+{\frac{ \left ( 210\,{a}^{4}{b}^{6}A+252\,{a}^{5}{b}^{5}B \right ){x}^{7}}{7}}+{\frac{ \left ( 252\,{a}^{5}{b}^{5}A+210\,{a}^{6}{b}^{4}B \right ){x}^{6}}{6}}+{\frac{ \left ( 210\,{a}^{6}{b}^{4}A+120\,{a}^{7}{b}^{3}B \right ){x}^{5}}{5}}+{\frac{ \left ( 120\,{a}^{7}{b}^{3}A+45\,{a}^{8}{b}^{2}B \right ){x}^{4}}{4}}+{\frac{ \left ( 45\,{a}^{8}{b}^{2}A+10\,{a}^{9}bB \right ){x}^{3}}{3}}+{\frac{ \left ( 10\,{a}^{9}bA+{a}^{10}B \right ){x}^{2}}{2}}+{a}^{10}Ax \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^10*(B*x+A),x)

[Out]

1/12*b^10*B*x^12+1/11*(A*b^10+10*B*a*b^9)*x^11+1/10*(10*A*a*b^9+45*B*a^2*b^8)*x^
10+1/9*(45*A*a^2*b^8+120*B*a^3*b^7)*x^9+1/8*(120*A*a^3*b^7+210*B*a^4*b^6)*x^8+1/
7*(210*A*a^4*b^6+252*B*a^5*b^5)*x^7+1/6*(252*A*a^5*b^5+210*B*a^6*b^4)*x^6+1/5*(2
10*A*a^6*b^4+120*B*a^7*b^3)*x^5+1/4*(120*A*a^7*b^3+45*B*a^8*b^2)*x^4+1/3*(45*A*a
^8*b^2+10*B*a^9*b)*x^3+1/2*(10*A*a^9*b+B*a^10)*x^2+a^10*A*x

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Maxima [A]  time = 1.34855, size = 324, normalized size = 8.53 \[ \frac{1}{12} \, B b^{10} x^{12} + A a^{10} x + \frac{1}{11} \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{11} + \frac{1}{2} \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{10} + \frac{5}{3} \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{9} + \frac{15}{4} \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{8} + 6 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{7} + 7 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{6} + 6 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{5} + \frac{15}{4} \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{4} + \frac{5}{3} \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{3} + \frac{1}{2} \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*B*b^10*x^12 + A*a^10*x + 1/11*(10*B*a*b^9 + A*b^10)*x^11 + 1/2*(9*B*a^2*b^8
+ 2*A*a*b^9)*x^10 + 5/3*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^9 + 15/4*(7*B*a^4*b^6 + 4
*A*a^3*b^7)*x^8 + 6*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^7 + 7*(5*B*a^6*b^4 + 6*A*a^5*b
^5)*x^6 + 6*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^5 + 15/4*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x
^4 + 5/3*(2*B*a^9*b + 9*A*a^8*b^2)*x^3 + 1/2*(B*a^10 + 10*A*a^9*b)*x^2

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Fricas [A]  time = 0.180989, size = 1, normalized size = 0.03 \[ \frac{1}{12} x^{12} b^{10} B + \frac{10}{11} x^{11} b^{9} a B + \frac{1}{11} x^{11} b^{10} A + \frac{9}{2} x^{10} b^{8} a^{2} B + x^{10} b^{9} a A + \frac{40}{3} x^{9} b^{7} a^{3} B + 5 x^{9} b^{8} a^{2} A + \frac{105}{4} x^{8} b^{6} a^{4} B + 15 x^{8} b^{7} a^{3} A + 36 x^{7} b^{5} a^{5} B + 30 x^{7} b^{6} a^{4} A + 35 x^{6} b^{4} a^{6} B + 42 x^{6} b^{5} a^{5} A + 24 x^{5} b^{3} a^{7} B + 42 x^{5} b^{4} a^{6} A + \frac{45}{4} x^{4} b^{2} a^{8} B + 30 x^{4} b^{3} a^{7} A + \frac{10}{3} x^{3} b a^{9} B + 15 x^{3} b^{2} a^{8} A + \frac{1}{2} x^{2} a^{10} B + 5 x^{2} b a^{9} A + x a^{10} A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*x^12*b^10*B + 10/11*x^11*b^9*a*B + 1/11*x^11*b^10*A + 9/2*x^10*b^8*a^2*B +
x^10*b^9*a*A + 40/3*x^9*b^7*a^3*B + 5*x^9*b^8*a^2*A + 105/4*x^8*b^6*a^4*B + 15*x
^8*b^7*a^3*A + 36*x^7*b^5*a^5*B + 30*x^7*b^6*a^4*A + 35*x^6*b^4*a^6*B + 42*x^6*b
^5*a^5*A + 24*x^5*b^3*a^7*B + 42*x^5*b^4*a^6*A + 45/4*x^4*b^2*a^8*B + 30*x^4*b^3
*a^7*A + 10/3*x^3*b*a^9*B + 15*x^3*b^2*a^8*A + 1/2*x^2*a^10*B + 5*x^2*b*a^9*A +
x*a^10*A

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Sympy [A]  time = 0.256985, size = 248, normalized size = 6.53 \[ A a^{10} x + \frac{B b^{10} x^{12}}{12} + x^{11} \left (\frac{A b^{10}}{11} + \frac{10 B a b^{9}}{11}\right ) + x^{10} \left (A a b^{9} + \frac{9 B a^{2} b^{8}}{2}\right ) + x^{9} \left (5 A a^{2} b^{8} + \frac{40 B a^{3} b^{7}}{3}\right ) + x^{8} \left (15 A a^{3} b^{7} + \frac{105 B a^{4} b^{6}}{4}\right ) + x^{7} \left (30 A a^{4} b^{6} + 36 B a^{5} b^{5}\right ) + x^{6} \left (42 A a^{5} b^{5} + 35 B a^{6} b^{4}\right ) + x^{5} \left (42 A a^{6} b^{4} + 24 B a^{7} b^{3}\right ) + x^{4} \left (30 A a^{7} b^{3} + \frac{45 B a^{8} b^{2}}{4}\right ) + x^{3} \left (15 A a^{8} b^{2} + \frac{10 B a^{9} b}{3}\right ) + x^{2} \left (5 A a^{9} b + \frac{B a^{10}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a**10*x + B*b**10*x**12/12 + x**11*(A*b**10/11 + 10*B*a*b**9/11) + x**10*(A*a*
b**9 + 9*B*a**2*b**8/2) + x**9*(5*A*a**2*b**8 + 40*B*a**3*b**7/3) + x**8*(15*A*a
**3*b**7 + 105*B*a**4*b**6/4) + x**7*(30*A*a**4*b**6 + 36*B*a**5*b**5) + x**6*(4
2*A*a**5*b**5 + 35*B*a**6*b**4) + x**5*(42*A*a**6*b**4 + 24*B*a**7*b**3) + x**4*
(30*A*a**7*b**3 + 45*B*a**8*b**2/4) + x**3*(15*A*a**8*b**2 + 10*B*a**9*b/3) + x*
*2*(5*A*a**9*b + B*a**10/2)

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GIAC/XCAS [A]  time = 0.294595, size = 325, normalized size = 8.55 \[ \frac{1}{12} \, B b^{10} x^{12} + \frac{10}{11} \, B a b^{9} x^{11} + \frac{1}{11} \, A b^{10} x^{11} + \frac{9}{2} \, B a^{2} b^{8} x^{10} + A a b^{9} x^{10} + \frac{40}{3} \, B a^{3} b^{7} x^{9} + 5 \, A a^{2} b^{8} x^{9} + \frac{105}{4} \, B a^{4} b^{6} x^{8} + 15 \, A a^{3} b^{7} x^{8} + 36 \, B a^{5} b^{5} x^{7} + 30 \, A a^{4} b^{6} x^{7} + 35 \, B a^{6} b^{4} x^{6} + 42 \, A a^{5} b^{5} x^{6} + 24 \, B a^{7} b^{3} x^{5} + 42 \, A a^{6} b^{4} x^{5} + \frac{45}{4} \, B a^{8} b^{2} x^{4} + 30 \, A a^{7} b^{3} x^{4} + \frac{10}{3} \, B a^{9} b x^{3} + 15 \, A a^{8} b^{2} x^{3} + \frac{1}{2} \, B a^{10} x^{2} + 5 \, A a^{9} b x^{2} + A a^{10} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*B*b^10*x^12 + 10/11*B*a*b^9*x^11 + 1/11*A*b^10*x^11 + 9/2*B*a^2*b^8*x^10 +
A*a*b^9*x^10 + 40/3*B*a^3*b^7*x^9 + 5*A*a^2*b^8*x^9 + 105/4*B*a^4*b^6*x^8 + 15*A
*a^3*b^7*x^8 + 36*B*a^5*b^5*x^7 + 30*A*a^4*b^6*x^7 + 35*B*a^6*b^4*x^6 + 42*A*a^5
*b^5*x^6 + 24*B*a^7*b^3*x^5 + 42*A*a^6*b^4*x^5 + 45/4*B*a^8*b^2*x^4 + 30*A*a^7*b
^3*x^4 + 10/3*B*a^9*b*x^3 + 15*A*a^8*b^2*x^3 + 1/2*B*a^10*x^2 + 5*A*a^9*b*x^2 +
A*a^10*x