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

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

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Rubi [A]  time = 0.19, antiderivative size = 231, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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*}

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Mathematica [A]  time = 0.05, size = 231, normalized size = 1.00 \begin {gather*} \frac {1}{10} a^{10} A x^{10}+\frac {1}{11} a^9 x^{11} (a B+10 A b)+\frac {5}{12} a^8 b x^{12} (2 a B+9 A b)+\frac {15}{13} a^7 b^2 x^{13} (3 a B+8 A b)+\frac {15}{7} a^6 b^3 x^{14} (4 a B+7 A b)+\frac {14}{5} a^5 b^4 x^{15} (5 a B+6 A b)+\frac {21}{8} a^4 b^5 x^{16} (6 a B+5 A b)+\frac {30}{17} a^3 b^6 x^{17} (7 a B+4 A b)+\frac {5}{6} a^2 b^7 x^{18} (8 a B+3 A b)+\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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^9 (a+b x)^{10} (A+B x) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.84, size = 245, normalized size = 1.06 \begin {gather*} \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 {gather*}

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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giac [A]  time = 1.22, size = 245, normalized size = 1.06 \begin {gather*} \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 {gather*}

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

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maple [A]  time = 0.00, size = 244, normalized size = 1.06 \begin {gather*} \frac {B \,b^{10} x^{21}}{21}+\frac {A \,a^{10} x^{10}}{10}+\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} b B \right ) x^{12}}{12}+\frac {\left (10 a^{9} b A +a^{10} B \right ) x^{11}}{11} \end {gather*}

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 = 1.07, size = 243, normalized size = 1.05 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.38, size = 211, normalized size = 0.91 \begin {gather*} x^{11}\,\left (\frac {B\,a^{10}}{11}+\frac {10\,A\,b\,a^9}{11}\right )+x^{20}\,\left (\frac {A\,b^{10}}{20}+\frac {B\,a\,b^9}{2}\right )+\frac {A\,a^{10}\,x^{10}}{10}+\frac {B\,b^{10}\,x^{21}}{21}+\frac {15\,a^7\,b^2\,x^{13}\,\left (8\,A\,b+3\,B\,a\right )}{13}+\frac {15\,a^6\,b^3\,x^{14}\,\left (7\,A\,b+4\,B\,a\right )}{7}+\frac {14\,a^5\,b^4\,x^{15}\,\left (6\,A\,b+5\,B\,a\right )}{5}+\frac {21\,a^4\,b^5\,x^{16}\,\left (5\,A\,b+6\,B\,a\right )}{8}+\frac {30\,a^3\,b^6\,x^{17}\,\left (4\,A\,b+7\,B\,a\right )}{17}+\frac {5\,a^2\,b^7\,x^{18}\,\left (3\,A\,b+8\,B\,a\right )}{6}+\frac {5\,a^8\,b\,x^{12}\,\left (9\,A\,b+2\,B\,a\right )}{12}+\frac {5\,a\,b^8\,x^{19}\,\left (2\,A\,b+9\,B\,a\right )}{19} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9*(A + B*x)*(a + b*x)^10,x)

[Out]

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

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sympy [A]  time = 0.23, size = 269, normalized size = 1.16 \begin {gather*} \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 {gather*}

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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