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

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]

-(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^1
3) - (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^1
1) - (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)

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Rubi [A]  time = 0.495038, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ -\frac{a^{10} A}{19 x^{19}}-\frac{a^9 (a B+10 A b)}{18 x^{18}}-\frac{5 a^8 b (2 a B+9 A b)}{17 x^{17}}-\frac{15 a^7 b^2 (3 a B+8 A b)}{16 x^{16}}-\frac{2 a^6 b^3 (4 a B+7 A b)}{x^{15}}-\frac{3 a^5 b^4 (5 a B+6 A b)}{x^{14}}-\frac{42 a^4 b^5 (6 a B+5 A b)}{13 x^{13}}-\frac{5 a^3 b^6 (7 a B+4 A b)}{2 x^{12}}-\frac{15 a^2 b^7 (8 a B+3 A b)}{11 x^{11}}-\frac{b^9 (10 a B+A b)}{9 x^9}-\frac{a b^8 (9 a B+2 A b)}{2 x^{10}}-\frac{b^{10} B}{8 x^8} \]

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^1
3) - (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^1
1) - (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)

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Rubi in Sympy [A]  time = 77.5899, size = 236, normalized size = 1.04 \[ - \frac{A a^{10}}{19 x^{19}} - \frac{B b^{10}}{8 x^{8}} - \frac{a^{9} \left (10 A b + B a\right )}{18 x^{18}} - \frac{5 a^{8} b \left (9 A b + 2 B a\right )}{17 x^{17}} - \frac{15 a^{7} b^{2} \left (8 A b + 3 B a\right )}{16 x^{16}} - \frac{2 a^{6} b^{3} \left (7 A b + 4 B a\right )}{x^{15}} - \frac{3 a^{5} b^{4} \left (6 A b + 5 B a\right )}{x^{14}} - \frac{42 a^{4} b^{5} \left (5 A b + 6 B a\right )}{13 x^{13}} - \frac{5 a^{3} b^{6} \left (4 A b + 7 B a\right )}{2 x^{12}} - \frac{15 a^{2} b^{7} \left (3 A b + 8 B a\right )}{11 x^{11}} - \frac{a b^{8} \left (2 A b + 9 B a\right )}{2 x^{10}} - \frac{b^{9} \left (A b + 10 B a\right )}{9 x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**10*(B*x+A)/x**20,x)

[Out]

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

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Mathematica [A]  time = 0.128141, 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]

-(b^10*(8*A + 9*B*x))/(72*x^9) - (a*b^9*(9*A + 10*B*x))/(9*x^10) - (9*a^2*b^8*(1
0*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))/(13*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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Maple [A]  time = 0.01, size = 208, normalized size = 0.9 \[ -{\frac{A{a}^{10}}{19\,{x}^{19}}}-{\frac{{a}^{9} \left ( 10\,Ab+Ba \right ) }{18\,{x}^{18}}}-{\frac{5\,{a}^{8}b \left ( 9\,Ab+2\,Ba \right ) }{17\,{x}^{17}}}-{\frac{15\,{a}^{7}{b}^{2} \left ( 8\,Ab+3\,Ba \right ) }{16\,{x}^{16}}}-2\,{\frac{{a}^{6}{b}^{3} \left ( 7\,Ab+4\,Ba \right ) }{{x}^{15}}}-3\,{\frac{{a}^{5}{b}^{4} \left ( 6\,Ab+5\,Ba \right ) }{{x}^{14}}}-{\frac{42\,{a}^{4}{b}^{5} \left ( 5\,Ab+6\,Ba \right ) }{13\,{x}^{13}}}-{\frac{5\,{a}^{3}{b}^{6} \left ( 4\,Ab+7\,Ba \right ) }{2\,{x}^{12}}}-{\frac{15\,{a}^{2}{b}^{7} \left ( 3\,Ab+8\,Ba \right ) }{11\,{x}^{11}}}-{\frac{a{b}^{8} \left ( 2\,Ab+9\,Ba \right ) }{2\,{x}^{10}}}-{\frac{{b}^{9} \left ( Ab+10\,Ba \right ) }{9\,{x}^{9}}}-{\frac{B{b}^{10}}{8\,{x}^{8}}} \]

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 = 1.36153, size = 328, normalized size = 1.44 \[ -\frac{831402 \, B b^{10} x^{11} + 350064 \, A a^{10} + 739024 \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 3325608 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 9069840 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 16628040 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 21488544 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 19953648 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 13302432 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 6235515 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 1956240 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 369512 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{6651216 \, x^{19}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/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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Fricas [A]  time = 0.196571, size = 328, normalized size = 1.44 \[ -\frac{831402 \, B b^{10} x^{11} + 350064 \, A a^{10} + 739024 \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 3325608 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 9069840 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 16628040 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 21488544 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 19953648 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 13302432 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 6235515 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 1956240 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 369512 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{6651216 \, x^{19}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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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GIAC/XCAS [A]  time = 0.371221, size = 328, normalized size = 1.44 \[ -\frac{831402 \, B b^{10} x^{11} + 7390240 \, B a b^{9} x^{10} + 739024 \, A b^{10} x^{10} + 29930472 \, B a^{2} b^{8} x^{9} + 6651216 \, A a b^{9} x^{9} + 72558720 \, B a^{3} b^{7} x^{8} + 27209520 \, A a^{2} b^{8} x^{8} + 116396280 \, B a^{4} b^{6} x^{7} + 66512160 \, A a^{3} b^{7} x^{7} + 128931264 \, B a^{5} b^{5} x^{6} + 107442720 \, A a^{4} b^{6} x^{6} + 99768240 \, B a^{6} b^{4} x^{5} + 119721888 \, A a^{5} b^{5} x^{5} + 53209728 \, B a^{7} b^{3} x^{4} + 93117024 \, A a^{6} b^{4} x^{4} + 18706545 \, B a^{8} b^{2} x^{3} + 49884120 \, A a^{7} b^{3} x^{3} + 3912480 \, B a^{9} b x^{2} + 17606160 \, A a^{8} b^{2} x^{2} + 369512 \, B a^{10} x + 3695120 \, A a^{9} b x + 350064 \, A a^{10}}{6651216 \, x^{19}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/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 + 299
30472*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 + 36951
2*B*a^10*x + 3695120*A*a^9*b*x + 350064*A*a^10)/x^19