3.138 \(\int x^9 (a+b x)^{10} (A+B x) \, dx\)

Optimal. Leaf size=231 \[ \frac{5}{6} a^2 b^7 x^{18} (8 a B+3 A b)+\frac{30}{17} a^3 b^6 x^{17} (7 a B+4 A b)+\frac{21}{8} a^4 b^5 x^{16} (6 a B+5 A b)+\frac{14}{5} a^5 b^4 x^{15} (5 a B+6 A b)+\frac{15}{7} a^6 b^3 x^{14} (4 a B+7 A b)+\frac{15}{13} a^7 b^2 x^{13} (3 a B+8 A b)+\frac{5}{12} a^8 b x^{12} (2 a B+9 A b)+\frac{1}{11} a^9 x^{11} (a B+10 A b)+\frac{1}{10} a^{10} A x^{10}+\frac{1}{20} b^9 x^{20} (10 a B+A b)+\frac{5}{19} a b^8 x^{19} (9 a B+2 A b)+\frac{1}{21} b^{10} B x^{21} \]

[Out]

(a^10*A*x^10)/10 + (a^9*(10*A*b + a*B)*x^11)/11 + (5*a^8*b*(9*A*b + 2*a*B)*x^12)/12 + (15*a^7*b^2*(8*A*b + 3*a
*B)*x^13)/13 + (15*a^6*b^3*(7*A*b + 4*a*B)*x^14)/7 + (14*a^5*b^4*(6*A*b + 5*a*B)*x^15)/5 + (21*a^4*b^5*(5*A*b
+ 6*a*B)*x^16)/8 + (30*a^3*b^6*(4*A*b + 7*a*B)*x^17)/17 + (5*a^2*b^7*(3*A*b + 8*a*B)*x^18)/6 + (5*a*b^8*(2*A*b
+ 9*a*B)*x^19)/19 + (b^9*(A*b + 10*a*B)*x^20)/20 + (b^10*B*x^21)/21

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Rubi [A]  time = 0.187381, antiderivative size = 231, 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, Rules used = {76} \[ \frac{5}{6} a^2 b^7 x^{18} (8 a B+3 A b)+\frac{30}{17} a^3 b^6 x^{17} (7 a B+4 A b)+\frac{21}{8} a^4 b^5 x^{16} (6 a B+5 A b)+\frac{14}{5} a^5 b^4 x^{15} (5 a B+6 A b)+\frac{15}{7} a^6 b^3 x^{14} (4 a B+7 A b)+\frac{15}{13} a^7 b^2 x^{13} (3 a B+8 A b)+\frac{5}{12} a^8 b x^{12} (2 a B+9 A b)+\frac{1}{11} a^9 x^{11} (a B+10 A b)+\frac{1}{10} a^{10} A x^{10}+\frac{1}{20} b^9 x^{20} (10 a B+A b)+\frac{5}{19} a b^8 x^{19} (9 a B+2 A b)+\frac{1}{21} b^{10} B x^{21} \]

Antiderivative was successfully verified.

[In]

Int[x^9*(a + b*x)^10*(A + B*x),x]

[Out]

(a^10*A*x^10)/10 + (a^9*(10*A*b + a*B)*x^11)/11 + (5*a^8*b*(9*A*b + 2*a*B)*x^12)/12 + (15*a^7*b^2*(8*A*b + 3*a
*B)*x^13)/13 + (15*a^6*b^3*(7*A*b + 4*a*B)*x^14)/7 + (14*a^5*b^4*(6*A*b + 5*a*B)*x^15)/5 + (21*a^4*b^5*(5*A*b
+ 6*a*B)*x^16)/8 + (30*a^3*b^6*(4*A*b + 7*a*B)*x^17)/17 + (5*a^2*b^7*(3*A*b + 8*a*B)*x^18)/6 + (5*a*b^8*(2*A*b
+ 9*a*B)*x^19)/19 + (b^9*(A*b + 10*a*B)*x^20)/20 + (b^10*B*x^21)/21

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

\begin{align*} \int x^9 (a+b x)^{10} (A+B x) \, dx &=\int \left (a^{10} A x^9+a^9 (10 A b+a B) x^{10}+5 a^8 b (9 A b+2 a B) x^{11}+15 a^7 b^2 (8 A b+3 a B) x^{12}+30 a^6 b^3 (7 A b+4 a B) x^{13}+42 a^5 b^4 (6 A b+5 a B) x^{14}+42 a^4 b^5 (5 A b+6 a B) x^{15}+30 a^3 b^6 (4 A b+7 a B) x^{16}+15 a^2 b^7 (3 A b+8 a B) x^{17}+5 a b^8 (2 A b+9 a B) x^{18}+b^9 (A b+10 a B) x^{19}+b^{10} B x^{20}\right ) \, dx\\ &=\frac{1}{10} a^{10} A x^{10}+\frac{1}{11} a^9 (10 A b+a B) x^{11}+\frac{5}{12} a^8 b (9 A b+2 a B) x^{12}+\frac{15}{13} a^7 b^2 (8 A b+3 a B) x^{13}+\frac{15}{7} a^6 b^3 (7 A b+4 a B) x^{14}+\frac{14}{5} a^5 b^4 (6 A b+5 a B) x^{15}+\frac{21}{8} a^4 b^5 (5 A b+6 a B) x^{16}+\frac{30}{17} a^3 b^6 (4 A b+7 a B) x^{17}+\frac{5}{6} a^2 b^7 (3 A b+8 a B) x^{18}+\frac{5}{19} a b^8 (2 A b+9 a B) x^{19}+\frac{1}{20} b^9 (A b+10 a B) x^{20}+\frac{1}{21} b^{10} B x^{21}\\ \end{align*}

Mathematica [A]  time = 0.028302, size = 231, normalized size = 1. \[ \frac{5}{6} a^2 b^7 x^{18} (8 a B+3 A b)+\frac{30}{17} a^3 b^6 x^{17} (7 a B+4 A b)+\frac{21}{8} a^4 b^5 x^{16} (6 a B+5 A b)+\frac{14}{5} a^5 b^4 x^{15} (5 a B+6 A b)+\frac{15}{7} a^6 b^3 x^{14} (4 a B+7 A b)+\frac{15}{13} a^7 b^2 x^{13} (3 a B+8 A b)+\frac{5}{12} a^8 b x^{12} (2 a B+9 A b)+\frac{1}{11} a^9 x^{11} (a B+10 A b)+\frac{1}{10} a^{10} A x^{10}+\frac{1}{20} b^9 x^{20} (10 a B+A b)+\frac{5}{19} a b^8 x^{19} (9 a B+2 A b)+\frac{1}{21} b^{10} B x^{21} \]

Antiderivative was successfully verified.

[In]

Integrate[x^9*(a + b*x)^10*(A + B*x),x]

[Out]

(a^10*A*x^10)/10 + (a^9*(10*A*b + a*B)*x^11)/11 + (5*a^8*b*(9*A*b + 2*a*B)*x^12)/12 + (15*a^7*b^2*(8*A*b + 3*a
*B)*x^13)/13 + (15*a^6*b^3*(7*A*b + 4*a*B)*x^14)/7 + (14*a^5*b^4*(6*A*b + 5*a*B)*x^15)/5 + (21*a^4*b^5*(5*A*b
+ 6*a*B)*x^16)/8 + (30*a^3*b^6*(4*A*b + 7*a*B)*x^17)/17 + (5*a^2*b^7*(3*A*b + 8*a*B)*x^18)/6 + (5*a*b^8*(2*A*b
+ 9*a*B)*x^19)/19 + (b^9*(A*b + 10*a*B)*x^20)/20 + (b^10*B*x^21)/21

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Maple [A]  time = 0.002, size = 244, normalized size = 1.1 \begin{align*}{\frac{{b}^{10}B{x}^{21}}{21}}+{\frac{ \left ({b}^{10}A+10\,a{b}^{9}B \right ){x}^{20}}{20}}+{\frac{ \left ( 10\,a{b}^{9}A+45\,{a}^{2}{b}^{8}B \right ){x}^{19}}{19}}+{\frac{ \left ( 45\,{a}^{2}{b}^{8}A+120\,{a}^{3}{b}^{7}B \right ){x}^{18}}{18}}+{\frac{ \left ( 120\,{a}^{3}{b}^{7}A+210\,{a}^{4}{b}^{6}B \right ){x}^{17}}{17}}+{\frac{ \left ( 210\,{a}^{4}{b}^{6}A+252\,{a}^{5}{b}^{5}B \right ){x}^{16}}{16}}+{\frac{ \left ( 252\,{a}^{5}{b}^{5}A+210\,{a}^{6}{b}^{4}B \right ){x}^{15}}{15}}+{\frac{ \left ( 210\,{a}^{6}{b}^{4}A+120\,{a}^{7}{b}^{3}B \right ){x}^{14}}{14}}+{\frac{ \left ( 120\,{a}^{7}{b}^{3}A+45\,{a}^{8}{b}^{2}B \right ){x}^{13}}{13}}+{\frac{ \left ( 45\,{a}^{8}{b}^{2}A+10\,{a}^{9}bB \right ){x}^{12}}{12}}+{\frac{ \left ( 10\,{a}^{9}bA+{a}^{10}B \right ){x}^{11}}{11}}+{\frac{{a}^{10}A{x}^{10}}{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9*(b*x+a)^10*(B*x+A),x)

[Out]

1/21*b^10*B*x^21+1/20*(A*b^10+10*B*a*b^9)*x^20+1/19*(10*A*a*b^9+45*B*a^2*b^8)*x^19+1/18*(45*A*a^2*b^8+120*B*a^
3*b^7)*x^18+1/17*(120*A*a^3*b^7+210*B*a^4*b^6)*x^17+1/16*(210*A*a^4*b^6+252*B*a^5*b^5)*x^16+1/15*(252*A*a^5*b^
5+210*B*a^6*b^4)*x^15+1/14*(210*A*a^6*b^4+120*B*a^7*b^3)*x^14+1/13*(120*A*a^7*b^3+45*B*a^8*b^2)*x^13+1/12*(45*
A*a^8*b^2+10*B*a^9*b)*x^12+1/11*(10*A*a^9*b+B*a^10)*x^11+1/10*a^10*A*x^10

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Maxima [A]  time = 0.989657, size = 328, normalized size = 1.42 \begin{align*} \frac{1}{21} \, B b^{10} x^{21} + \frac{1}{10} \, A a^{10} x^{10} + \frac{1}{20} \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{20} + \frac{5}{19} \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{19} + \frac{5}{6} \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{18} + \frac{30}{17} \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{17} + \frac{21}{8} \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{16} + \frac{14}{5} \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{15} + \frac{15}{7} \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{14} + \frac{15}{13} \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{13} + \frac{5}{12} \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{12} + \frac{1}{11} \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x^{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*(b*x+a)^10*(B*x+A),x, algorithm="maxima")

[Out]

1/21*B*b^10*x^21 + 1/10*A*a^10*x^10 + 1/20*(10*B*a*b^9 + A*b^10)*x^20 + 5/19*(9*B*a^2*b^8 + 2*A*a*b^9)*x^19 +
5/6*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^18 + 30/17*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^17 + 21/8*(6*B*a^5*b^5 + 5*A*a^4*b^
6)*x^16 + 14/5*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^15 + 15/7*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^14 + 15/13*(3*B*a^8*b^2 +
8*A*a^7*b^3)*x^13 + 5/12*(2*B*a^9*b + 9*A*a^8*b^2)*x^12 + 1/11*(B*a^10 + 10*A*a^9*b)*x^11

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Fricas [A]  time = 1.28686, size = 630, normalized size = 2.73 \begin{align*} \frac{1}{21} x^{21} b^{10} B + \frac{1}{2} x^{20} b^{9} a B + \frac{1}{20} x^{20} b^{10} A + \frac{45}{19} x^{19} b^{8} a^{2} B + \frac{10}{19} x^{19} b^{9} a A + \frac{20}{3} x^{18} b^{7} a^{3} B + \frac{5}{2} x^{18} b^{8} a^{2} A + \frac{210}{17} x^{17} b^{6} a^{4} B + \frac{120}{17} x^{17} b^{7} a^{3} A + \frac{63}{4} x^{16} b^{5} a^{5} B + \frac{105}{8} x^{16} b^{6} a^{4} A + 14 x^{15} b^{4} a^{6} B + \frac{84}{5} x^{15} b^{5} a^{5} A + \frac{60}{7} x^{14} b^{3} a^{7} B + 15 x^{14} b^{4} a^{6} A + \frac{45}{13} x^{13} b^{2} a^{8} B + \frac{120}{13} x^{13} b^{3} a^{7} A + \frac{5}{6} x^{12} b a^{9} B + \frac{15}{4} x^{12} b^{2} a^{8} A + \frac{1}{11} x^{11} a^{10} B + \frac{10}{11} x^{11} b a^{9} A + \frac{1}{10} x^{10} a^{10} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*(b*x+a)^10*(B*x+A),x, algorithm="fricas")

[Out]

1/21*x^21*b^10*B + 1/2*x^20*b^9*a*B + 1/20*x^20*b^10*A + 45/19*x^19*b^8*a^2*B + 10/19*x^19*b^9*a*A + 20/3*x^18
*b^7*a^3*B + 5/2*x^18*b^8*a^2*A + 210/17*x^17*b^6*a^4*B + 120/17*x^17*b^7*a^3*A + 63/4*x^16*b^5*a^5*B + 105/8*
x^16*b^6*a^4*A + 14*x^15*b^4*a^6*B + 84/5*x^15*b^5*a^5*A + 60/7*x^14*b^3*a^7*B + 15*x^14*b^4*a^6*A + 45/13*x^1
3*b^2*a^8*B + 120/13*x^13*b^3*a^7*A + 5/6*x^12*b*a^9*B + 15/4*x^12*b^2*a^8*A + 1/11*x^11*a^10*B + 10/11*x^11*b
*a^9*A + 1/10*x^10*a^10*A

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Sympy [A]  time = 0.117116, size = 269, normalized size = 1.16 \begin{align*} \frac{A a^{10} x^{10}}{10} + \frac{B b^{10} x^{21}}{21} + x^{20} \left (\frac{A b^{10}}{20} + \frac{B a b^{9}}{2}\right ) + x^{19} \left (\frac{10 A a b^{9}}{19} + \frac{45 B a^{2} b^{8}}{19}\right ) + x^{18} \left (\frac{5 A a^{2} b^{8}}{2} + \frac{20 B a^{3} b^{7}}{3}\right ) + x^{17} \left (\frac{120 A a^{3} b^{7}}{17} + \frac{210 B a^{4} b^{6}}{17}\right ) + x^{16} \left (\frac{105 A a^{4} b^{6}}{8} + \frac{63 B a^{5} b^{5}}{4}\right ) + x^{15} \left (\frac{84 A a^{5} b^{5}}{5} + 14 B a^{6} b^{4}\right ) + x^{14} \left (15 A a^{6} b^{4} + \frac{60 B a^{7} b^{3}}{7}\right ) + x^{13} \left (\frac{120 A a^{7} b^{3}}{13} + \frac{45 B a^{8} b^{2}}{13}\right ) + x^{12} \left (\frac{15 A a^{8} b^{2}}{4} + \frac{5 B a^{9} b}{6}\right ) + x^{11} \left (\frac{10 A a^{9} b}{11} + \frac{B a^{10}}{11}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**9*(b*x+a)**10*(B*x+A),x)

[Out]

A*a**10*x**10/10 + B*b**10*x**21/21 + x**20*(A*b**10/20 + B*a*b**9/2) + x**19*(10*A*a*b**9/19 + 45*B*a**2*b**8
/19) + x**18*(5*A*a**2*b**8/2 + 20*B*a**3*b**7/3) + x**17*(120*A*a**3*b**7/17 + 210*B*a**4*b**6/17) + x**16*(1
05*A*a**4*b**6/8 + 63*B*a**5*b**5/4) + x**15*(84*A*a**5*b**5/5 + 14*B*a**6*b**4) + x**14*(15*A*a**6*b**4 + 60*
B*a**7*b**3/7) + x**13*(120*A*a**7*b**3/13 + 45*B*a**8*b**2/13) + x**12*(15*A*a**8*b**2/4 + 5*B*a**9*b/6) + x*
*11*(10*A*a**9*b/11 + B*a**10/11)

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Giac [A]  time = 1.16853, size = 331, normalized size = 1.43 \begin{align*} \frac{1}{21} \, B b^{10} x^{21} + \frac{1}{2} \, B a b^{9} x^{20} + \frac{1}{20} \, A b^{10} x^{20} + \frac{45}{19} \, B a^{2} b^{8} x^{19} + \frac{10}{19} \, A a b^{9} x^{19} + \frac{20}{3} \, B a^{3} b^{7} x^{18} + \frac{5}{2} \, A a^{2} b^{8} x^{18} + \frac{210}{17} \, B a^{4} b^{6} x^{17} + \frac{120}{17} \, A a^{3} b^{7} x^{17} + \frac{63}{4} \, B a^{5} b^{5} x^{16} + \frac{105}{8} \, A a^{4} b^{6} x^{16} + 14 \, B a^{6} b^{4} x^{15} + \frac{84}{5} \, A a^{5} b^{5} x^{15} + \frac{60}{7} \, B a^{7} b^{3} x^{14} + 15 \, A a^{6} b^{4} x^{14} + \frac{45}{13} \, B a^{8} b^{2} x^{13} + \frac{120}{13} \, A a^{7} b^{3} x^{13} + \frac{5}{6} \, B a^{9} b x^{12} + \frac{15}{4} \, A a^{8} b^{2} x^{12} + \frac{1}{11} \, B a^{10} x^{11} + \frac{10}{11} \, A a^{9} b x^{11} + \frac{1}{10} \, A a^{10} x^{10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*(b*x+a)^10*(B*x+A),x, algorithm="giac")

[Out]

1/21*B*b^10*x^21 + 1/2*B*a*b^9*x^20 + 1/20*A*b^10*x^20 + 45/19*B*a^2*b^8*x^19 + 10/19*A*a*b^9*x^19 + 20/3*B*a^
3*b^7*x^18 + 5/2*A*a^2*b^8*x^18 + 210/17*B*a^4*b^6*x^17 + 120/17*A*a^3*b^7*x^17 + 63/4*B*a^5*b^5*x^16 + 105/8*
A*a^4*b^6*x^16 + 14*B*a^6*b^4*x^15 + 84/5*A*a^5*b^5*x^15 + 60/7*B*a^7*b^3*x^14 + 15*A*a^6*b^4*x^14 + 45/13*B*a
^8*b^2*x^13 + 120/13*A*a^7*b^3*x^13 + 5/6*B*a^9*b*x^12 + 15/4*A*a^8*b^2*x^12 + 1/11*B*a^10*x^11 + 10/11*A*a^9*
b*x^11 + 1/10*A*a^10*x^10