3.167 \(\int \frac {(a+b x)^{10} (A+B x)}{x^{20}} \, dx\)

Optimal. Leaf size=227 \[ -\frac {a^{10} A}{19 x^{19}}-\frac {a^9 (a B+10 A b)}{18 x^{18}}-\frac {5 a^8 b (2 a B+9 A b)}{17 x^{17}}-\frac {15 a^7 b^2 (3 a B+8 A b)}{16 x^{16}}-\frac {2 a^6 b^3 (4 a B+7 A b)}{x^{15}}-\frac {3 a^5 b^4 (5 a B+6 A b)}{x^{14}}-\frac {42 a^4 b^5 (6 a B+5 A b)}{13 x^{13}}-\frac {5 a^3 b^6 (7 a B+4 A b)}{2 x^{12}}-\frac {15 a^2 b^7 (8 a B+3 A b)}{11 x^{11}}-\frac {b^9 (10 a B+A b)}{9 x^9}-\frac {a b^8 (9 a B+2 A b)}{2 x^{10}}-\frac {b^{10} B}{8 x^8} \]

[Out]

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

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Rubi [A]  time = 0.13, antiderivative size = 227, 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} \[ -\frac {15 a^7 b^2 (3 a B+8 A b)}{16 x^{16}}-\frac {2 a^6 b^3 (4 a B+7 A b)}{x^{15}}-\frac {3 a^5 b^4 (5 a B+6 A b)}{x^{14}}-\frac {42 a^4 b^5 (6 a B+5 A b)}{13 x^{13}}-\frac {5 a^3 b^6 (7 a B+4 A b)}{2 x^{12}}-\frac {15 a^2 b^7 (8 a B+3 A b)}{11 x^{11}}-\frac {a^9 (a B+10 A b)}{18 x^{18}}-\frac {5 a^8 b (2 a B+9 A b)}{17 x^{17}}-\frac {a^{10} A}{19 x^{19}}-\frac {a b^8 (9 a B+2 A b)}{2 x^{10}}-\frac {b^9 (10 a B+A b)}{9 x^9}-\frac {b^{10} B}{8 x^8} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^10*(A + B*x))/x^20,x]

[Out]

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

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

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Mathematica [A]  time = 0.09, size = 220, normalized size = 0.97 \[ -\frac {a^{10} (18 A+19 B x)}{342 x^{19}}-\frac {5 a^9 b (17 A+18 B x)}{153 x^{18}}-\frac {45 a^8 b^2 (16 A+17 B x)}{272 x^{17}}-\frac {a^7 b^3 (15 A+16 B x)}{2 x^{16}}-\frac {a^6 b^4 (14 A+15 B x)}{x^{15}}-\frac {18 a^5 b^5 (13 A+14 B x)}{13 x^{14}}-\frac {35 a^4 b^6 (12 A+13 B x)}{26 x^{13}}-\frac {10 a^3 b^7 (11 A+12 B x)}{11 x^{12}}-\frac {9 a^2 b^8 (10 A+11 B x)}{22 x^{11}}-\frac {a b^9 (9 A+10 B x)}{9 x^{10}}-\frac {b^{10} (8 A+9 B x)}{72 x^9} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^10*(A + B*x))/x^20,x]

[Out]

-1/72*(b^10*(8*A + 9*B*x))/x^9 - (a*b^9*(9*A + 10*B*x))/(9*x^10) - (9*a^2*b^8*(10*A + 11*B*x))/(22*x^11) - (10
*a^3*b^7*(11*A + 12*B*x))/(11*x^12) - (35*a^4*b^6*(12*A + 13*B*x))/(26*x^13) - (18*a^5*b^5*(13*A + 14*B*x))/(1
3*x^14) - (a^6*b^4*(14*A + 15*B*x))/x^15 - (a^7*b^3*(15*A + 16*B*x))/(2*x^16) - (45*a^8*b^2*(16*A + 17*B*x))/(
272*x^17) - (5*a^9*b*(17*A + 18*B*x))/(153*x^18) - (a^10*(18*A + 19*B*x))/(342*x^19)

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fricas [A]  time = 0.83, size = 243, normalized size = 1.07 \[ -\frac {831402 \, B b^{10} x^{11} + 350064 \, A a^{10} + 739024 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 3325608 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 9069840 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 16628040 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 21488544 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 19953648 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 13302432 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 6235515 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 1956240 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 369512 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{6651216 \, x^{19}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^20,x, algorithm="fricas")

[Out]

-1/6651216*(831402*B*b^10*x^11 + 350064*A*a^10 + 739024*(10*B*a*b^9 + A*b^10)*x^10 + 3325608*(9*B*a^2*b^8 + 2*
A*a*b^9)*x^9 + 9069840*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 16628040*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 21488544*(
6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 19953648*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 13302432*(4*B*a^7*b^3 + 7*A*a^6*b^
4)*x^4 + 6235515*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 1956240*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 369512*(B*a^10 + 10
*A*a^9*b)*x)/x^19

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giac [A]  time = 0.96, size = 243, normalized size = 1.07 \[ -\frac {831402 \, B b^{10} x^{11} + 7390240 \, B a b^{9} x^{10} + 739024 \, A b^{10} x^{10} + 29930472 \, B a^{2} b^{8} x^{9} + 6651216 \, A a b^{9} x^{9} + 72558720 \, B a^{3} b^{7} x^{8} + 27209520 \, A a^{2} b^{8} x^{8} + 116396280 \, B a^{4} b^{6} x^{7} + 66512160 \, A a^{3} b^{7} x^{7} + 128931264 \, B a^{5} b^{5} x^{6} + 107442720 \, A a^{4} b^{6} x^{6} + 99768240 \, B a^{6} b^{4} x^{5} + 119721888 \, A a^{5} b^{5} x^{5} + 53209728 \, B a^{7} b^{3} x^{4} + 93117024 \, A a^{6} b^{4} x^{4} + 18706545 \, B a^{8} b^{2} x^{3} + 49884120 \, A a^{7} b^{3} x^{3} + 3912480 \, B a^{9} b x^{2} + 17606160 \, A a^{8} b^{2} x^{2} + 369512 \, B a^{10} x + 3695120 \, A a^{9} b x + 350064 \, A a^{10}}{6651216 \, x^{19}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^20,x, algorithm="giac")

[Out]

-1/6651216*(831402*B*b^10*x^11 + 7390240*B*a*b^9*x^10 + 739024*A*b^10*x^10 + 29930472*B*a^2*b^8*x^9 + 6651216*
A*a*b^9*x^9 + 72558720*B*a^3*b^7*x^8 + 27209520*A*a^2*b^8*x^8 + 116396280*B*a^4*b^6*x^7 + 66512160*A*a^3*b^7*x
^7 + 128931264*B*a^5*b^5*x^6 + 107442720*A*a^4*b^6*x^6 + 99768240*B*a^6*b^4*x^5 + 119721888*A*a^5*b^5*x^5 + 53
209728*B*a^7*b^3*x^4 + 93117024*A*a^6*b^4*x^4 + 18706545*B*a^8*b^2*x^3 + 49884120*A*a^7*b^3*x^3 + 3912480*B*a^
9*b*x^2 + 17606160*A*a^8*b^2*x^2 + 369512*B*a^10*x + 3695120*A*a^9*b*x + 350064*A*a^10)/x^19

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maple [A]  time = 0.01, size = 208, normalized size = 0.92 \[ -\frac {B \,b^{10}}{8 x^{8}}-\frac {\left (A b +10 B a \right ) b^{9}}{9 x^{9}}-\frac {\left (2 A b +9 B a \right ) a \,b^{8}}{2 x^{10}}-\frac {15 \left (3 A b +8 B a \right ) a^{2} b^{7}}{11 x^{11}}-\frac {5 \left (4 A b +7 B a \right ) a^{3} b^{6}}{2 x^{12}}-\frac {42 \left (5 A b +6 B a \right ) a^{4} b^{5}}{13 x^{13}}-\frac {3 \left (6 A b +5 B a \right ) a^{5} b^{4}}{x^{14}}-\frac {2 \left (7 A b +4 B a \right ) a^{6} b^{3}}{x^{15}}-\frac {15 \left (8 A b +3 B a \right ) a^{7} b^{2}}{16 x^{16}}-\frac {A \,a^{10}}{19 x^{19}}-\frac {5 \left (9 A b +2 B a \right ) a^{8} b}{17 x^{17}}-\frac {\left (10 A b +B a \right ) a^{9}}{18 x^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10*(B*x+A)/x^20,x)

[Out]

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

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maxima [A]  time = 1.05, size = 243, normalized size = 1.07 \[ -\frac {831402 \, B b^{10} x^{11} + 350064 \, A a^{10} + 739024 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 3325608 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 9069840 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 16628040 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 21488544 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 19953648 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 13302432 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 6235515 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 1956240 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 369512 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{6651216 \, x^{19}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/x^20,x, algorithm="maxima")

[Out]

-1/6651216*(831402*B*b^10*x^11 + 350064*A*a^10 + 739024*(10*B*a*b^9 + A*b^10)*x^10 + 3325608*(9*B*a^2*b^8 + 2*
A*a*b^9)*x^9 + 9069840*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 16628040*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 21488544*(
6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 19953648*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 13302432*(4*B*a^7*b^3 + 7*A*a^6*b^
4)*x^4 + 6235515*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 1956240*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 369512*(B*a^10 + 10
*A*a^9*b)*x)/x^19

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mupad [B]  time = 0.14, size = 234, normalized size = 1.03 \[ -\frac {x\,\left (\frac {B\,a^{10}}{18}+\frac {5\,A\,b\,a^9}{9}\right )+\frac {A\,a^{10}}{19}+x^9\,\left (\frac {9\,B\,a^2\,b^8}{2}+A\,a\,b^9\right )+x^2\,\left (\frac {10\,B\,a^9\,b}{17}+\frac {45\,A\,a^8\,b^2}{17}\right )+x^{10}\,\left (\frac {A\,b^{10}}{9}+\frac {10\,B\,a\,b^9}{9}\right )+x^4\,\left (8\,B\,a^7\,b^3+14\,A\,a^6\,b^4\right )+x^5\,\left (15\,B\,a^6\,b^4+18\,A\,a^5\,b^5\right )+x^7\,\left (\frac {35\,B\,a^4\,b^6}{2}+10\,A\,a^3\,b^7\right )+x^3\,\left (\frac {45\,B\,a^8\,b^2}{16}+\frac {15\,A\,a^7\,b^3}{2}\right )+x^8\,\left (\frac {120\,B\,a^3\,b^7}{11}+\frac {45\,A\,a^2\,b^8}{11}\right )+x^6\,\left (\frac {252\,B\,a^5\,b^5}{13}+\frac {210\,A\,a^4\,b^6}{13}\right )+\frac {B\,b^{10}\,x^{11}}{8}}{x^{19}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^10)/x^20,x)

[Out]

-(x*((B*a^10)/18 + (5*A*a^9*b)/9) + (A*a^10)/19 + x^9*((9*B*a^2*b^8)/2 + A*a*b^9) + x^2*((45*A*a^8*b^2)/17 + (
10*B*a^9*b)/17) + x^10*((A*b^10)/9 + (10*B*a*b^9)/9) + x^4*(14*A*a^6*b^4 + 8*B*a^7*b^3) + x^5*(18*A*a^5*b^5 +
15*B*a^6*b^4) + x^7*(10*A*a^3*b^7 + (35*B*a^4*b^6)/2) + x^3*((15*A*a^7*b^3)/2 + (45*B*a^8*b^2)/16) + x^8*((45*
A*a^2*b^8)/11 + (120*B*a^3*b^7)/11) + x^6*((210*A*a^4*b^6)/13 + (252*B*a^5*b^5)/13) + (B*b^10*x^11)/8)/x^19

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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**20,x)

[Out]

Timed out

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