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

Optimal. Leaf size=229 \[ \frac {1}{11} a^{10} A x^{11}+\frac {1}{12} a^9 x^{12} (a B+10 A b)+\frac {5}{13} a^8 b x^{13} (2 a B+9 A b)+\frac {15}{14} a^7 b^2 x^{14} (3 a B+8 A b)+2 a^6 b^3 x^{15} (4 a B+7 A b)+\frac {21}{8} a^5 b^4 x^{16} (5 a B+6 A b)+\frac {42}{17} a^4 b^5 x^{17} (6 a B+5 A b)+\frac {5}{3} a^3 b^6 x^{18} (7 a B+4 A b)+\frac {15}{19} a^2 b^7 x^{19} (8 a B+3 A b)+\frac {1}{21} b^9 x^{21} (10 a B+A b)+\frac {1}{4} a b^8 x^{20} (9 a B+2 A b)+\frac {1}{22} b^{10} B x^{22} \]

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Rubi [A]  time = 0.29, antiderivative size = 229, 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 {15}{19} a^2 b^7 x^{19} (8 a B+3 A b)+\frac {5}{3} a^3 b^6 x^{18} (7 a B+4 A b)+\frac {42}{17} a^4 b^5 x^{17} (6 a B+5 A b)+\frac {21}{8} a^5 b^4 x^{16} (5 a B+6 A b)+2 a^6 b^3 x^{15} (4 a B+7 A b)+\frac {15}{14} a^7 b^2 x^{14} (3 a B+8 A b)+\frac {5}{13} a^8 b x^{13} (2 a B+9 A b)+\frac {1}{12} a^9 x^{12} (a B+10 A b)+\frac {1}{11} a^{10} A x^{11}+\frac {1}{21} b^9 x^{21} (10 a B+A b)+\frac {1}{4} a b^8 x^{20} (9 a B+2 A b)+\frac {1}{22} b^{10} B x^{22} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^10*(a + b*x)^10*(A + B*x),x]

[Out]

(a^10*A*x^11)/11 + (a^9*(10*A*b + a*B)*x^12)/12 + (5*a^8*b*(9*A*b + 2*a*B)*x^13)/13 + (15*a^7*b^2*(8*A*b + 3*a
*B)*x^14)/14 + 2*a^6*b^3*(7*A*b + 4*a*B)*x^15 + (21*a^5*b^4*(6*A*b + 5*a*B)*x^16)/8 + (42*a^4*b^5*(5*A*b + 6*a
*B)*x^17)/17 + (5*a^3*b^6*(4*A*b + 7*a*B)*x^18)/3 + (15*a^2*b^7*(3*A*b + 8*a*B)*x^19)/19 + (a*b^8*(2*A*b + 9*a
*B)*x^20)/4 + (b^9*(A*b + 10*a*B)*x^21)/21 + (b^10*B*x^22)/22

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^{10} (a+b x)^{10} (A+B x) \, dx &=\int \left (a^{10} A x^{10}+a^9 (10 A b+a B) x^{11}+5 a^8 b (9 A b+2 a B) x^{12}+15 a^7 b^2 (8 A b+3 a B) x^{13}+30 a^6 b^3 (7 A b+4 a B) x^{14}+42 a^5 b^4 (6 A b+5 a B) x^{15}+42 a^4 b^5 (5 A b+6 a B) x^{16}+30 a^3 b^6 (4 A b+7 a B) x^{17}+15 a^2 b^7 (3 A b+8 a B) x^{18}+5 a b^8 (2 A b+9 a B) x^{19}+b^9 (A b+10 a B) x^{20}+b^{10} B x^{21}\right ) \, dx\\ &=\frac {1}{11} a^{10} A x^{11}+\frac {1}{12} a^9 (10 A b+a B) x^{12}+\frac {5}{13} a^8 b (9 A b+2 a B) x^{13}+\frac {15}{14} a^7 b^2 (8 A b+3 a B) x^{14}+2 a^6 b^3 (7 A b+4 a B) x^{15}+\frac {21}{8} a^5 b^4 (6 A b+5 a B) x^{16}+\frac {42}{17} a^4 b^5 (5 A b+6 a B) x^{17}+\frac {5}{3} a^3 b^6 (4 A b+7 a B) x^{18}+\frac {15}{19} a^2 b^7 (3 A b+8 a B) x^{19}+\frac {1}{4} a b^8 (2 A b+9 a B) x^{20}+\frac {1}{21} b^9 (A b+10 a B) x^{21}+\frac {1}{22} b^{10} B x^{22}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 229, normalized size = 1.00 \begin {gather*} \frac {1}{11} a^{10} A x^{11}+\frac {1}{12} a^9 x^{12} (a B+10 A b)+\frac {5}{13} a^8 b x^{13} (2 a B+9 A b)+\frac {15}{14} a^7 b^2 x^{14} (3 a B+8 A b)+2 a^6 b^3 x^{15} (4 a B+7 A b)+\frac {21}{8} a^5 b^4 x^{16} (5 a B+6 A b)+\frac {42}{17} a^4 b^5 x^{17} (6 a B+5 A b)+\frac {5}{3} a^3 b^6 x^{18} (7 a B+4 A b)+\frac {15}{19} a^2 b^7 x^{19} (8 a B+3 A b)+\frac {1}{21} b^9 x^{21} (10 a B+A b)+\frac {1}{4} a b^8 x^{20} (9 a B+2 A b)+\frac {1}{22} b^{10} B x^{22} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^10*A*x^11)/11 + (a^9*(10*A*b + a*B)*x^12)/12 + (5*a^8*b*(9*A*b + 2*a*B)*x^13)/13 + (15*a^7*b^2*(8*A*b + 3*a
*B)*x^14)/14 + 2*a^6*b^3*(7*A*b + 4*a*B)*x^15 + (21*a^5*b^4*(6*A*b + 5*a*B)*x^16)/8 + (42*a^4*b^5*(5*A*b + 6*a
*B)*x^17)/17 + (5*a^3*b^6*(4*A*b + 7*a*B)*x^18)/3 + (15*a^2*b^7*(3*A*b + 8*a*B)*x^19)/19 + (a*b^8*(2*A*b + 9*a
*B)*x^20)/4 + (b^9*(A*b + 10*a*B)*x^21)/21 + (b^10*B*x^22)/22

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.12, size = 245, normalized size = 1.07 \begin {gather*} \frac {1}{22} x^{22} b^{10} B + \frac {10}{21} x^{21} b^{9} a B + \frac {1}{21} x^{21} b^{10} A + \frac {9}{4} x^{20} b^{8} a^{2} B + \frac {1}{2} x^{20} b^{9} a A + \frac {120}{19} x^{19} b^{7} a^{3} B + \frac {45}{19} x^{19} b^{8} a^{2} A + \frac {35}{3} x^{18} b^{6} a^{4} B + \frac {20}{3} x^{18} b^{7} a^{3} A + \frac {252}{17} x^{17} b^{5} a^{5} B + \frac {210}{17} x^{17} b^{6} a^{4} A + \frac {105}{8} x^{16} b^{4} a^{6} B + \frac {63}{4} x^{16} b^{5} a^{5} A + 8 x^{15} b^{3} a^{7} B + 14 x^{15} b^{4} a^{6} A + \frac {45}{14} x^{14} b^{2} a^{8} B + \frac {60}{7} x^{14} b^{3} a^{7} A + \frac {10}{13} x^{13} b a^{9} B + \frac {45}{13} x^{13} b^{2} a^{8} A + \frac {1}{12} x^{12} a^{10} B + \frac {5}{6} x^{12} b a^{9} A + \frac {1}{11} x^{11} a^{10} A \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/22*x^22*b^10*B + 10/21*x^21*b^9*a*B + 1/21*x^21*b^10*A + 9/4*x^20*b^8*a^2*B + 1/2*x^20*b^9*a*A + 120/19*x^19
*b^7*a^3*B + 45/19*x^19*b^8*a^2*A + 35/3*x^18*b^6*a^4*B + 20/3*x^18*b^7*a^3*A + 252/17*x^17*b^5*a^5*B + 210/17
*x^17*b^6*a^4*A + 105/8*x^16*b^4*a^6*B + 63/4*x^16*b^5*a^5*A + 8*x^15*b^3*a^7*B + 14*x^15*b^4*a^6*A + 45/14*x^
14*b^2*a^8*B + 60/7*x^14*b^3*a^7*A + 10/13*x^13*b*a^9*B + 45/13*x^13*b^2*a^8*A + 1/12*x^12*a^10*B + 5/6*x^12*b
*a^9*A + 1/11*x^11*a^10*A

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giac [A]  time = 1.25, size = 245, normalized size = 1.07 \begin {gather*} \frac {1}{22} \, B b^{10} x^{22} + \frac {10}{21} \, B a b^{9} x^{21} + \frac {1}{21} \, A b^{10} x^{21} + \frac {9}{4} \, B a^{2} b^{8} x^{20} + \frac {1}{2} \, A a b^{9} x^{20} + \frac {120}{19} \, B a^{3} b^{7} x^{19} + \frac {45}{19} \, A a^{2} b^{8} x^{19} + \frac {35}{3} \, B a^{4} b^{6} x^{18} + \frac {20}{3} \, A a^{3} b^{7} x^{18} + \frac {252}{17} \, B a^{5} b^{5} x^{17} + \frac {210}{17} \, A a^{4} b^{6} x^{17} + \frac {105}{8} \, B a^{6} b^{4} x^{16} + \frac {63}{4} \, A a^{5} b^{5} x^{16} + 8 \, B a^{7} b^{3} x^{15} + 14 \, A a^{6} b^{4} x^{15} + \frac {45}{14} \, B a^{8} b^{2} x^{14} + \frac {60}{7} \, A a^{7} b^{3} x^{14} + \frac {10}{13} \, B a^{9} b x^{13} + \frac {45}{13} \, A a^{8} b^{2} x^{13} + \frac {1}{12} \, B a^{10} x^{12} + \frac {5}{6} \, A a^{9} b x^{12} + \frac {1}{11} \, A a^{10} x^{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/22*B*b^10*x^22 + 10/21*B*a*b^9*x^21 + 1/21*A*b^10*x^21 + 9/4*B*a^2*b^8*x^20 + 1/2*A*a*b^9*x^20 + 120/19*B*a^
3*b^7*x^19 + 45/19*A*a^2*b^8*x^19 + 35/3*B*a^4*b^6*x^18 + 20/3*A*a^3*b^7*x^18 + 252/17*B*a^5*b^5*x^17 + 210/17
*A*a^4*b^6*x^17 + 105/8*B*a^6*b^4*x^16 + 63/4*A*a^5*b^5*x^16 + 8*B*a^7*b^3*x^15 + 14*A*a^6*b^4*x^15 + 45/14*B*
a^8*b^2*x^14 + 60/7*A*a^7*b^3*x^14 + 10/13*B*a^9*b*x^13 + 45/13*A*a^8*b^2*x^13 + 1/12*B*a^10*x^12 + 5/6*A*a^9*
b*x^12 + 1/11*A*a^10*x^11

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maple [A]  time = 0.00, size = 244, normalized size = 1.07 \begin {gather*} \frac {B \,b^{10} x^{22}}{22}+\frac {A \,a^{10} x^{11}}{11}+\frac {\left (b^{10} A +10 a \,b^{9} B \right ) x^{21}}{21}+\frac {\left (10 a \,b^{9} A +45 a^{2} b^{8} B \right ) x^{20}}{20}+\frac {\left (45 a^{2} b^{8} A +120 a^{3} b^{7} B \right ) x^{19}}{19}+\frac {\left (120 a^{3} b^{7} A +210 a^{4} b^{6} B \right ) x^{18}}{18}+\frac {\left (210 a^{4} b^{6} A +252 a^{5} b^{5} B \right ) x^{17}}{17}+\frac {\left (252 a^{5} b^{5} A +210 a^{6} b^{4} B \right ) x^{16}}{16}+\frac {\left (210 a^{6} b^{4} A +120 a^{7} b^{3} B \right ) x^{15}}{15}+\frac {\left (120 a^{7} b^{3} A +45 a^{8} b^{2} B \right ) x^{14}}{14}+\frac {\left (45 a^{8} b^{2} A +10 a^{9} b B \right ) x^{13}}{13}+\frac {\left (10 a^{9} b A +a^{10} B \right ) x^{12}}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/22*b^10*B*x^22+1/21*(A*b^10+10*B*a*b^9)*x^21+1/20*(10*A*a*b^9+45*B*a^2*b^8)*x^20+1/19*(45*A*a^2*b^8+120*B*a^
3*b^7)*x^19+1/18*(120*A*a^3*b^7+210*B*a^4*b^6)*x^18+1/17*(210*A*a^4*b^6+252*B*a^5*b^5)*x^17+1/16*(252*A*a^5*b^
5+210*B*a^6*b^4)*x^16+1/15*(210*A*a^6*b^4+120*B*a^7*b^3)*x^15+1/14*(120*A*a^7*b^3+45*B*a^8*b^2)*x^14+1/13*(45*
A*a^8*b^2+10*B*a^9*b)*x^13+1/12*(10*A*a^9*b+B*a^10)*x^12+1/11*a^10*A*x^11

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maxima [A]  time = 1.11, size = 243, normalized size = 1.06 \begin {gather*} \frac {1}{22} \, B b^{10} x^{22} + \frac {1}{11} \, A a^{10} x^{11} + \frac {1}{21} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{21} + \frac {1}{4} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{20} + \frac {15}{19} \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{19} + \frac {5}{3} \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{18} + \frac {42}{17} \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{17} + \frac {21}{8} \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{16} + 2 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{15} + \frac {15}{14} \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{14} + \frac {5}{13} \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{13} + \frac {1}{12} \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x^{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/22*B*b^10*x^22 + 1/11*A*a^10*x^11 + 1/21*(10*B*a*b^9 + A*b^10)*x^21 + 1/4*(9*B*a^2*b^8 + 2*A*a*b^9)*x^20 + 1
5/19*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^19 + 5/3*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^18 + 42/17*(6*B*a^5*b^5 + 5*A*a^4*b^
6)*x^17 + 21/8*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^16 + 2*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^15 + 15/14*(3*B*a^8*b^2 + 8*
A*a^7*b^3)*x^14 + 5/13*(2*B*a^9*b + 9*A*a^8*b^2)*x^13 + 1/12*(B*a^10 + 10*A*a^9*b)*x^12

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mupad [B]  time = 0.14, size = 211, normalized size = 0.92 \begin {gather*} x^{12}\,\left (\frac {B\,a^{10}}{12}+\frac {5\,A\,b\,a^9}{6}\right )+x^{21}\,\left (\frac {A\,b^{10}}{21}+\frac {10\,B\,a\,b^9}{21}\right )+\frac {A\,a^{10}\,x^{11}}{11}+\frac {B\,b^{10}\,x^{22}}{22}+\frac {15\,a^7\,b^2\,x^{14}\,\left (8\,A\,b+3\,B\,a\right )}{14}+2\,a^6\,b^3\,x^{15}\,\left (7\,A\,b+4\,B\,a\right )+\frac {21\,a^5\,b^4\,x^{16}\,\left (6\,A\,b+5\,B\,a\right )}{8}+\frac {42\,a^4\,b^5\,x^{17}\,\left (5\,A\,b+6\,B\,a\right )}{17}+\frac {5\,a^3\,b^6\,x^{18}\,\left (4\,A\,b+7\,B\,a\right )}{3}+\frac {15\,a^2\,b^7\,x^{19}\,\left (3\,A\,b+8\,B\,a\right )}{19}+\frac {5\,a^8\,b\,x^{13}\,\left (9\,A\,b+2\,B\,a\right )}{13}+\frac {a\,b^8\,x^{20}\,\left (2\,A\,b+9\,B\,a\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^10*(A + B*x)*(a + b*x)^10,x)

[Out]

x^12*((B*a^10)/12 + (5*A*a^9*b)/6) + x^21*((A*b^10)/21 + (10*B*a*b^9)/21) + (A*a^10*x^11)/11 + (B*b^10*x^22)/2
2 + (15*a^7*b^2*x^14*(8*A*b + 3*B*a))/14 + 2*a^6*b^3*x^15*(7*A*b + 4*B*a) + (21*a^5*b^4*x^16*(6*A*b + 5*B*a))/
8 + (42*a^4*b^5*x^17*(5*A*b + 6*B*a))/17 + (5*a^3*b^6*x^18*(4*A*b + 7*B*a))/3 + (15*a^2*b^7*x^19*(3*A*b + 8*B*
a))/19 + (5*a^8*b*x^13*(9*A*b + 2*B*a))/13 + (a*b^8*x^20*(2*A*b + 9*B*a))/4

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sympy [A]  time = 0.15, size = 269, normalized size = 1.17 \begin {gather*} \frac {A a^{10} x^{11}}{11} + \frac {B b^{10} x^{22}}{22} + x^{21} \left (\frac {A b^{10}}{21} + \frac {10 B a b^{9}}{21}\right ) + x^{20} \left (\frac {A a b^{9}}{2} + \frac {9 B a^{2} b^{8}}{4}\right ) + x^{19} \left (\frac {45 A a^{2} b^{8}}{19} + \frac {120 B a^{3} b^{7}}{19}\right ) + x^{18} \left (\frac {20 A a^{3} b^{7}}{3} + \frac {35 B a^{4} b^{6}}{3}\right ) + x^{17} \left (\frac {210 A a^{4} b^{6}}{17} + \frac {252 B a^{5} b^{5}}{17}\right ) + x^{16} \left (\frac {63 A a^{5} b^{5}}{4} + \frac {105 B a^{6} b^{4}}{8}\right ) + x^{15} \left (14 A a^{6} b^{4} + 8 B a^{7} b^{3}\right ) + x^{14} \left (\frac {60 A a^{7} b^{3}}{7} + \frac {45 B a^{8} b^{2}}{14}\right ) + x^{13} \left (\frac {45 A a^{8} b^{2}}{13} + \frac {10 B a^{9} b}{13}\right ) + x^{12} \left (\frac {5 A a^{9} b}{6} + \frac {B a^{10}}{12}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a**10*x**11/11 + B*b**10*x**22/22 + x**21*(A*b**10/21 + 10*B*a*b**9/21) + x**20*(A*a*b**9/2 + 9*B*a**2*b**8/
4) + x**19*(45*A*a**2*b**8/19 + 120*B*a**3*b**7/19) + x**18*(20*A*a**3*b**7/3 + 35*B*a**4*b**6/3) + x**17*(210
*A*a**4*b**6/17 + 252*B*a**5*b**5/17) + x**16*(63*A*a**5*b**5/4 + 105*B*a**6*b**4/8) + x**15*(14*A*a**6*b**4 +
8*B*a**7*b**3) + x**14*(60*A*a**7*b**3/7 + 45*B*a**8*b**2/14) + x**13*(45*A*a**8*b**2/13 + 10*B*a**9*b/13) +
x**12*(5*A*a**9*b/6 + B*a**10/12)

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