3.168 \(\int \frac {(a+b x)^{10} (A+B x)}{x^{21}} \, dx\)

Optimal. Leaf size=229 \[ -\frac {a^{10} A}{20 x^{20}}-\frac {a^9 (a B+10 A b)}{19 x^{19}}-\frac {5 a^8 b (2 a B+9 A b)}{18 x^{18}}-\frac {15 a^7 b^2 (3 a B+8 A b)}{17 x^{17}}-\frac {15 a^6 b^3 (4 a B+7 A b)}{8 x^{16}}-\frac {14 a^5 b^4 (5 a B+6 A b)}{5 x^{15}}-\frac {3 a^4 b^5 (6 a B+5 A b)}{x^{14}}-\frac {30 a^3 b^6 (7 a B+4 A b)}{13 x^{13}}-\frac {5 a^2 b^7 (8 a B+3 A b)}{4 x^{12}}-\frac {b^9 (10 a B+A b)}{10 x^{10}}-\frac {5 a b^8 (9 a B+2 A b)}{11 x^{11}}-\frac {b^{10} B}{9 x^9} \]

[Out]

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

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Rubi [A]  time = 0.14, 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} \[ -\frac {15 a^7 b^2 (3 a B+8 A b)}{17 x^{17}}-\frac {15 a^6 b^3 (4 a B+7 A b)}{8 x^{16}}-\frac {14 a^5 b^4 (5 a B+6 A b)}{5 x^{15}}-\frac {3 a^4 b^5 (6 a B+5 A b)}{x^{14}}-\frac {30 a^3 b^6 (7 a B+4 A b)}{13 x^{13}}-\frac {5 a^2 b^7 (8 a B+3 A b)}{4 x^{12}}-\frac {a^9 (a B+10 A b)}{19 x^{19}}-\frac {5 a^8 b (2 a B+9 A b)}{18 x^{18}}-\frac {a^{10} A}{20 x^{20}}-\frac {5 a b^8 (9 a B+2 A b)}{11 x^{11}}-\frac {b^9 (10 a B+A b)}{10 x^{10}}-\frac {b^{10} B}{9 x^9} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^10*(A + B*x))/x^21,x]

[Out]

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

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

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Mathematica [A]  time = 0.10, size = 222, normalized size = 0.97 \[ -\frac {a^{10} (19 A+20 B x)}{380 x^{20}}-\frac {5 a^9 b (18 A+19 B x)}{171 x^{19}}-\frac {5 a^8 b^2 (17 A+18 B x)}{34 x^{18}}-\frac {15 a^7 b^3 (16 A+17 B x)}{34 x^{17}}-\frac {7 a^6 b^4 (15 A+16 B x)}{8 x^{16}}-\frac {6 a^5 b^5 (14 A+15 B x)}{5 x^{15}}-\frac {15 a^4 b^6 (13 A+14 B x)}{13 x^{14}}-\frac {10 a^3 b^7 (12 A+13 B x)}{13 x^{13}}-\frac {15 a^2 b^8 (11 A+12 B x)}{44 x^{12}}-\frac {a b^9 (10 A+11 B x)}{11 x^{11}}-\frac {b^{10} (9 A+10 B x)}{90 x^{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^10*(A + B*x))/x^21,x]

[Out]

-1/90*(b^10*(9*A + 10*B*x))/x^10 - (a*b^9*(10*A + 11*B*x))/(11*x^11) - (15*a^2*b^8*(11*A + 12*B*x))/(44*x^12)
- (10*a^3*b^7*(12*A + 13*B*x))/(13*x^13) - (15*a^4*b^6*(13*A + 14*B*x))/(13*x^14) - (6*a^5*b^5*(14*A + 15*B*x)
)/(5*x^15) - (7*a^6*b^4*(15*A + 16*B*x))/(8*x^16) - (15*a^7*b^3*(16*A + 17*B*x))/(34*x^17) - (5*a^8*b^2*(17*A
+ 18*B*x))/(34*x^18) - (5*a^9*b*(18*A + 19*B*x))/(171*x^19) - (a^10*(19*A + 20*B*x))/(380*x^20)

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fricas [A]  time = 1.00, size = 243, normalized size = 1.06 \[ -\frac {1847560 \, B b^{10} x^{11} + 831402 \, A a^{10} + 1662804 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 7558200 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 20785050 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 38372400 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 49884120 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 46558512 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 31177575 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 14671800 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 4618900 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 875160 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{16628040 \, x^{20}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^21,x, algorithm="fricas")

[Out]

-1/16628040*(1847560*B*b^10*x^11 + 831402*A*a^10 + 1662804*(10*B*a*b^9 + A*b^10)*x^10 + 7558200*(9*B*a^2*b^8 +
2*A*a*b^9)*x^9 + 20785050*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 38372400*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 498841
20*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 46558512*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 31177575*(4*B*a^7*b^3 + 7*A*a^
6*b^4)*x^4 + 14671800*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 4618900*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 875160*(B*a^10
+ 10*A*a^9*b)*x)/x^20

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giac [A]  time = 0.96, size = 243, normalized size = 1.06 \[ -\frac {1847560 \, B b^{10} x^{11} + 16628040 \, B a b^{9} x^{10} + 1662804 \, A b^{10} x^{10} + 68023800 \, B a^{2} b^{8} x^{9} + 15116400 \, A a b^{9} x^{9} + 166280400 \, B a^{3} b^{7} x^{8} + 62355150 \, A a^{2} b^{8} x^{8} + 268606800 \, B a^{4} b^{6} x^{7} + 153489600 \, A a^{3} b^{7} x^{7} + 299304720 \, B a^{5} b^{5} x^{6} + 249420600 \, A a^{4} b^{6} x^{6} + 232792560 \, B a^{6} b^{4} x^{5} + 279351072 \, A a^{5} b^{5} x^{5} + 124710300 \, B a^{7} b^{3} x^{4} + 218243025 \, A a^{6} b^{4} x^{4} + 44015400 \, B a^{8} b^{2} x^{3} + 117374400 \, A a^{7} b^{3} x^{3} + 9237800 \, B a^{9} b x^{2} + 41570100 \, A a^{8} b^{2} x^{2} + 875160 \, B a^{10} x + 8751600 \, A a^{9} b x + 831402 \, A a^{10}}{16628040 \, x^{20}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^21,x, algorithm="giac")

[Out]

-1/16628040*(1847560*B*b^10*x^11 + 16628040*B*a*b^9*x^10 + 1662804*A*b^10*x^10 + 68023800*B*a^2*b^8*x^9 + 1511
6400*A*a*b^9*x^9 + 166280400*B*a^3*b^7*x^8 + 62355150*A*a^2*b^8*x^8 + 268606800*B*a^4*b^6*x^7 + 153489600*A*a^
3*b^7*x^7 + 299304720*B*a^5*b^5*x^6 + 249420600*A*a^4*b^6*x^6 + 232792560*B*a^6*b^4*x^5 + 279351072*A*a^5*b^5*
x^5 + 124710300*B*a^7*b^3*x^4 + 218243025*A*a^6*b^4*x^4 + 44015400*B*a^8*b^2*x^3 + 117374400*A*a^7*b^3*x^3 + 9
237800*B*a^9*b*x^2 + 41570100*A*a^8*b^2*x^2 + 875160*B*a^10*x + 8751600*A*a^9*b*x + 831402*A*a^10)/x^20

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maple [A]  time = 0.01, size = 208, normalized size = 0.91 \[ -\frac {B \,b^{10}}{9 x^{9}}-\frac {\left (A b +10 B a \right ) b^{9}}{10 x^{10}}-\frac {5 \left (2 A b +9 B a \right ) a \,b^{8}}{11 x^{11}}-\frac {5 \left (3 A b +8 B a \right ) a^{2} b^{7}}{4 x^{12}}-\frac {30 \left (4 A b +7 B a \right ) a^{3} b^{6}}{13 x^{13}}-\frac {3 \left (5 A b +6 B a \right ) a^{4} b^{5}}{x^{14}}-\frac {14 \left (6 A b +5 B a \right ) a^{5} b^{4}}{5 x^{15}}-\frac {15 \left (7 A b +4 B a \right ) a^{6} b^{3}}{8 x^{16}}-\frac {15 \left (8 A b +3 B a \right ) a^{7} b^{2}}{17 x^{17}}-\frac {A \,a^{10}}{20 x^{20}}-\frac {5 \left (9 A b +2 B a \right ) a^{8} b}{18 x^{18}}-\frac {\left (10 A b +B a \right ) a^{9}}{19 x^{19}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10*(B*x+A)/x^21,x)

[Out]

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

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maxima [A]  time = 1.03, size = 243, normalized size = 1.06 \[ -\frac {1847560 \, B b^{10} x^{11} + 831402 \, A a^{10} + 1662804 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 7558200 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 20785050 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 38372400 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 49884120 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 46558512 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 31177575 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 14671800 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 4618900 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 875160 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{16628040 \, x^{20}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^21,x, algorithm="maxima")

[Out]

-1/16628040*(1847560*B*b^10*x^11 + 831402*A*a^10 + 1662804*(10*B*a*b^9 + A*b^10)*x^10 + 7558200*(9*B*a^2*b^8 +
2*A*a*b^9)*x^9 + 20785050*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 38372400*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 498841
20*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 46558512*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 31177575*(4*B*a^7*b^3 + 7*A*a^
6*b^4)*x^4 + 14671800*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 4618900*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 875160*(B*a^10
+ 10*A*a^9*b)*x)/x^20

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mupad [B]  time = 0.13, size = 234, normalized size = 1.02 \[ -\frac {x\,\left (\frac {B\,a^{10}}{19}+\frac {10\,A\,b\,a^9}{19}\right )+\frac {A\,a^{10}}{20}+x^2\,\left (\frac {5\,B\,a^9\,b}{9}+\frac {5\,A\,a^8\,b^2}{2}\right )+x^9\,\left (\frac {45\,B\,a^2\,b^8}{11}+\frac {10\,A\,a\,b^9}{11}\right )+x^{10}\,\left (\frac {A\,b^{10}}{10}+B\,a\,b^9\right )+x^8\,\left (10\,B\,a^3\,b^7+\frac {15\,A\,a^2\,b^8}{4}\right )+x^6\,\left (18\,B\,a^5\,b^5+15\,A\,a^4\,b^6\right )+x^5\,\left (14\,B\,a^6\,b^4+\frac {84\,A\,a^5\,b^5}{5}\right )+x^4\,\left (\frac {15\,B\,a^7\,b^3}{2}+\frac {105\,A\,a^6\,b^4}{8}\right )+x^3\,\left (\frac {45\,B\,a^8\,b^2}{17}+\frac {120\,A\,a^7\,b^3}{17}\right )+x^7\,\left (\frac {210\,B\,a^4\,b^6}{13}+\frac {120\,A\,a^3\,b^7}{13}\right )+\frac {B\,b^{10}\,x^{11}}{9}}{x^{20}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^10)/x^21,x)

[Out]

-(x*((B*a^10)/19 + (10*A*a^9*b)/19) + (A*a^10)/20 + x^2*((5*A*a^8*b^2)/2 + (5*B*a^9*b)/9) + x^9*((45*B*a^2*b^8
)/11 + (10*A*a*b^9)/11) + x^10*((A*b^10)/10 + B*a*b^9) + x^8*((15*A*a^2*b^8)/4 + 10*B*a^3*b^7) + x^6*(15*A*a^4
*b^6 + 18*B*a^5*b^5) + x^5*((84*A*a^5*b^5)/5 + 14*B*a^6*b^4) + x^4*((105*A*a^6*b^4)/8 + (15*B*a^7*b^3)/2) + x^
3*((120*A*a^7*b^3)/17 + (45*B*a^8*b^2)/17) + x^7*((120*A*a^3*b^7)/13 + (210*B*a^4*b^6)/13) + (B*b^10*x^11)/9)/
x^20

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**10*(B*x+A)/x**21,x)

[Out]

Timed out

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