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

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

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Rubi [A]  time = 0.19, antiderivative size = 240, 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 {28 a^2 (a+b x)^{17} (A b-3 a B)}{17 b^{10}}-\frac {7 a^3 (a+b x)^{16} (4 A b-9 a B)}{8 b^{10}}+\frac {14 a^4 (a+b x)^{15} (5 A b-9 a B)}{15 b^{10}}-\frac {2 a^5 (a+b x)^{14} (2 A b-3 a B)}{b^{10}}+\frac {4 a^6 (a+b x)^{13} (7 A b-9 a B)}{13 b^{10}}-\frac {a^7 (a+b x)^{12} (8 A b-9 a B)}{12 b^{10}}+\frac {a^8 (a+b x)^{11} (A b-a B)}{11 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}} \end {gather*}

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)/(12*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^1
0)

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^8 (a+b x)^{10} (A+B x) \, dx &=\int \left (-\frac {a^8 (-A b+a B) (a+b x)^{10}}{b^9}+\frac {a^7 (-8 A b+9 a B) (a+b x)^{11}}{b^9}-\frac {4 a^6 (-7 A b+9 a B) (a+b x)^{12}}{b^9}+\frac {28 a^5 (-2 A b+3 a B) (a+b x)^{13}}{b^9}-\frac {14 a^4 (-5 A b+9 a B) (a+b x)^{14}}{b^9}+\frac {14 a^3 (-4 A b+9 a B) (a+b x)^{15}}{b^9}-\frac {28 a^2 (-A b+3 a B) (a+b x)^{16}}{b^9}+\frac {4 a (-2 A b+9 a B) (a+b x)^{17}}{b^9}+\frac {(A b-9 a B) (a+b x)^{18}}{b^9}+\frac {B (a+b x)^{19}}{b^9}\right ) \, dx\\ &=\frac {a^8 (A b-a B) (a+b x)^{11}}{11 b^{10}}-\frac {a^7 (8 A b-9 a B) (a+b x)^{12}}{12 b^{10}}+\frac {4 a^6 (7 A b-9 a B) (a+b x)^{13}}{13 b^{10}}-\frac {2 a^5 (2 A b-3 a B) (a+b x)^{14}}{b^{10}}+\frac {14 a^4 (5 A b-9 a B) (a+b x)^{15}}{15 b^{10}}-\frac {7 a^3 (4 A b-9 a B) (a+b x)^{16}}{8 b^{10}}+\frac {28 a^2 (A b-3 a B) (a+b x)^{17}}{17 b^{10}}-\frac {2 a (2 A b-9 a B) (a+b x)^{18}}{9 b^{10}}+\frac {(A b-9 a B) (a+b x)^{19}}{19 b^{10}}+\frac {B (a+b x)^{20}}{20 b^{10}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 229, normalized size = 0.95 \begin {gather*} \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} \end {gather*}

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)/11 + (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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^8 (a+b x)^{10} (A+B x) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.17, size = 244, normalized size = 1.02 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(b*x+a)^10*(B*x+A),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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giac [A]  time = 1.25, size = 244, normalized size = 1.02 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(b*x+a)^10*(B*x+A),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

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maple [A]  time = 0.00, size = 244, normalized size = 1.02 \begin {gather*} \frac {B \,b^{10} x^{20}}{20}+\frac {A \,a^{10} x^{9}}{9}+\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} b B \right ) x^{11}}{11}+\frac {\left (10 a^{9} b A +a^{10} B \right ) x^{10}}{10} \end {gather*}

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^16+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^14+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 = 1.17, size = 243, normalized size = 1.01 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(b*x+a)^10*(B*x+A),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 + 10*A*a^9*b)*x^10

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mupad [B]  time = 0.11, size = 210, normalized size = 0.88 \begin {gather*} x^{10}\,\left (\frac {B\,a^{10}}{10}+A\,b\,a^9\right )+x^{19}\,\left (\frac {A\,b^{10}}{19}+\frac {10\,B\,a\,b^9}{19}\right )+\frac {A\,a^{10}\,x^9}{9}+\frac {B\,b^{10}\,x^{20}}{20}+\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^8\,b\,x^{11}\,\left (9\,A\,b+2\,B\,a\right )}{11}+\frac {5\,a\,b^8\,x^{18}\,\left (2\,A\,b+9\,B\,a\right )}{18} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(A + B*x)*(a + b*x)^10,x)

[Out]

x^10*((B*a^10)/10 + A*a^9*b) + x^19*((A*b^10)/19 + (10*B*a*b^9)/19) + (A*a^10*x^9)/9 + (B*b^10*x^20)/20 + (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^8*b*x^11*(9*A*b + 2*B*a))/11 + (5*a*b^8*x^18*(2*A*b + 9*B*a))/18

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sympy [A]  time = 0.28, size = 264, normalized size = 1.10 \begin {gather*} \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 ) \end {gather*}

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**1
0*(A*a**9*b + B*a**10/10)

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