3.153 \(\int \frac{(a+b x)^{10}}{x^{19}} \, dx\)

Optimal. Leaf size=130 \[ -\frac{a^{10}}{18 x^{18}}-\frac{10 a^9 b}{17 x^{17}}-\frac{45 a^8 b^2}{16 x^{16}}-\frac{8 a^7 b^3}{x^{15}}-\frac{15 a^6 b^4}{x^{14}}-\frac{252 a^5 b^5}{13 x^{13}}-\frac{35 a^4 b^6}{2 x^{12}}-\frac{120 a^3 b^7}{11 x^{11}}-\frac{9 a^2 b^8}{2 x^{10}}-\frac{10 a b^9}{9 x^9}-\frac{b^{10}}{8 x^8} \]

[Out]

-a^10/(18*x^18) - (10*a^9*b)/(17*x^17) - (45*a^8*b^2)/(16*x^16) - (8*a^7*b^3)/x^
15 - (15*a^6*b^4)/x^14 - (252*a^5*b^5)/(13*x^13) - (35*a^4*b^6)/(2*x^12) - (120*
a^3*b^7)/(11*x^11) - (9*a^2*b^8)/(2*x^10) - (10*a*b^9)/(9*x^9) - b^10/(8*x^8)

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Rubi [A]  time = 0.125308, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{a^{10}}{18 x^{18}}-\frac{10 a^9 b}{17 x^{17}}-\frac{45 a^8 b^2}{16 x^{16}}-\frac{8 a^7 b^3}{x^{15}}-\frac{15 a^6 b^4}{x^{14}}-\frac{252 a^5 b^5}{13 x^{13}}-\frac{35 a^4 b^6}{2 x^{12}}-\frac{120 a^3 b^7}{11 x^{11}}-\frac{9 a^2 b^8}{2 x^{10}}-\frac{10 a b^9}{9 x^9}-\frac{b^{10}}{8 x^8} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^10/x^19,x]

[Out]

-a^10/(18*x^18) - (10*a^9*b)/(17*x^17) - (45*a^8*b^2)/(16*x^16) - (8*a^7*b^3)/x^
15 - (15*a^6*b^4)/x^14 - (252*a^5*b^5)/(13*x^13) - (35*a^4*b^6)/(2*x^12) - (120*
a^3*b^7)/(11*x^11) - (9*a^2*b^8)/(2*x^10) - (10*a*b^9)/(9*x^9) - b^10/(8*x^8)

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Rubi in Sympy [A]  time = 23.8417, size = 133, normalized size = 1.02 \[ - \frac{a^{10}}{18 x^{18}} - \frac{10 a^{9} b}{17 x^{17}} - \frac{45 a^{8} b^{2}}{16 x^{16}} - \frac{8 a^{7} b^{3}}{x^{15}} - \frac{15 a^{6} b^{4}}{x^{14}} - \frac{252 a^{5} b^{5}}{13 x^{13}} - \frac{35 a^{4} b^{6}}{2 x^{12}} - \frac{120 a^{3} b^{7}}{11 x^{11}} - \frac{9 a^{2} b^{8}}{2 x^{10}} - \frac{10 a b^{9}}{9 x^{9}} - \frac{b^{10}}{8 x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**10/x**19,x)

[Out]

-a**10/(18*x**18) - 10*a**9*b/(17*x**17) - 45*a**8*b**2/(16*x**16) - 8*a**7*b**3
/x**15 - 15*a**6*b**4/x**14 - 252*a**5*b**5/(13*x**13) - 35*a**4*b**6/(2*x**12)
- 120*a**3*b**7/(11*x**11) - 9*a**2*b**8/(2*x**10) - 10*a*b**9/(9*x**9) - b**10/
(8*x**8)

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Mathematica [A]  time = 0.00650269, size = 130, normalized size = 1. \[ -\frac{a^{10}}{18 x^{18}}-\frac{10 a^9 b}{17 x^{17}}-\frac{45 a^8 b^2}{16 x^{16}}-\frac{8 a^7 b^3}{x^{15}}-\frac{15 a^6 b^4}{x^{14}}-\frac{252 a^5 b^5}{13 x^{13}}-\frac{35 a^4 b^6}{2 x^{12}}-\frac{120 a^3 b^7}{11 x^{11}}-\frac{9 a^2 b^8}{2 x^{10}}-\frac{10 a b^9}{9 x^9}-\frac{b^{10}}{8 x^8} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^10/x^19,x]

[Out]

-a^10/(18*x^18) - (10*a^9*b)/(17*x^17) - (45*a^8*b^2)/(16*x^16) - (8*a^7*b^3)/x^
15 - (15*a^6*b^4)/x^14 - (252*a^5*b^5)/(13*x^13) - (35*a^4*b^6)/(2*x^12) - (120*
a^3*b^7)/(11*x^11) - (9*a^2*b^8)/(2*x^10) - (10*a*b^9)/(9*x^9) - b^10/(8*x^8)

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Maple [A]  time = 0.01, size = 113, normalized size = 0.9 \[ -{\frac{{a}^{10}}{18\,{x}^{18}}}-{\frac{10\,{a}^{9}b}{17\,{x}^{17}}}-{\frac{45\,{a}^{8}{b}^{2}}{16\,{x}^{16}}}-8\,{\frac{{a}^{7}{b}^{3}}{{x}^{15}}}-15\,{\frac{{a}^{6}{b}^{4}}{{x}^{14}}}-{\frac{252\,{a}^{5}{b}^{5}}{13\,{x}^{13}}}-{\frac{35\,{a}^{4}{b}^{6}}{2\,{x}^{12}}}-{\frac{120\,{a}^{3}{b}^{7}}{11\,{x}^{11}}}-{\frac{9\,{a}^{2}{b}^{8}}{2\,{x}^{10}}}-{\frac{10\,a{b}^{9}}{9\,{x}^{9}}}-{\frac{{b}^{10}}{8\,{x}^{8}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^10/x^19,x)

[Out]

-1/18*a^10/x^18-10/17*a^9*b/x^17-45/16*a^8*b^2/x^16-8*a^7*b^3/x^15-15*a^6*b^4/x^
14-252/13*a^5*b^5/x^13-35/2*a^4*b^6/x^12-120/11*a^3*b^7/x^11-9/2*a^2*b^8/x^10-10
/9*a*b^9/x^9-1/8*b^10/x^8

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Maxima [A]  time = 1.34401, size = 151, normalized size = 1.16 \[ -\frac{43758 \, b^{10} x^{10} + 388960 \, a b^{9} x^{9} + 1575288 \, a^{2} b^{8} x^{8} + 3818880 \, a^{3} b^{7} x^{7} + 6126120 \, a^{4} b^{6} x^{6} + 6785856 \, a^{5} b^{5} x^{5} + 5250960 \, a^{6} b^{4} x^{4} + 2800512 \, a^{7} b^{3} x^{3} + 984555 \, a^{8} b^{2} x^{2} + 205920 \, a^{9} b x + 19448 \, a^{10}}{350064 \, x^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^10/x^19,x, algorithm="maxima")

[Out]

-1/350064*(43758*b^10*x^10 + 388960*a*b^9*x^9 + 1575288*a^2*b^8*x^8 + 3818880*a^
3*b^7*x^7 + 6126120*a^4*b^6*x^6 + 6785856*a^5*b^5*x^5 + 5250960*a^6*b^4*x^4 + 28
00512*a^7*b^3*x^3 + 984555*a^8*b^2*x^2 + 205920*a^9*b*x + 19448*a^10)/x^18

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Fricas [A]  time = 0.186625, size = 151, normalized size = 1.16 \[ -\frac{43758 \, b^{10} x^{10} + 388960 \, a b^{9} x^{9} + 1575288 \, a^{2} b^{8} x^{8} + 3818880 \, a^{3} b^{7} x^{7} + 6126120 \, a^{4} b^{6} x^{6} + 6785856 \, a^{5} b^{5} x^{5} + 5250960 \, a^{6} b^{4} x^{4} + 2800512 \, a^{7} b^{3} x^{3} + 984555 \, a^{8} b^{2} x^{2} + 205920 \, a^{9} b x + 19448 \, a^{10}}{350064 \, x^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^10/x^19,x, algorithm="fricas")

[Out]

-1/350064*(43758*b^10*x^10 + 388960*a*b^9*x^9 + 1575288*a^2*b^8*x^8 + 3818880*a^
3*b^7*x^7 + 6126120*a^4*b^6*x^6 + 6785856*a^5*b^5*x^5 + 5250960*a^6*b^4*x^4 + 28
00512*a^7*b^3*x^3 + 984555*a^8*b^2*x^2 + 205920*a^9*b*x + 19448*a^10)/x^18

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Sympy [A]  time = 5.21072, size = 121, normalized size = 0.93 \[ - \frac{19448 a^{10} + 205920 a^{9} b x + 984555 a^{8} b^{2} x^{2} + 2800512 a^{7} b^{3} x^{3} + 5250960 a^{6} b^{4} x^{4} + 6785856 a^{5} b^{5} x^{5} + 6126120 a^{4} b^{6} x^{6} + 3818880 a^{3} b^{7} x^{7} + 1575288 a^{2} b^{8} x^{8} + 388960 a b^{9} x^{9} + 43758 b^{10} x^{10}}{350064 x^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**10/x**19,x)

[Out]

-(19448*a**10 + 205920*a**9*b*x + 984555*a**8*b**2*x**2 + 2800512*a**7*b**3*x**3
+ 5250960*a**6*b**4*x**4 + 6785856*a**5*b**5*x**5 + 6126120*a**4*b**6*x**6 + 38
18880*a**3*b**7*x**7 + 1575288*a**2*b**8*x**8 + 388960*a*b**9*x**9 + 43758*b**10
*x**10)/(350064*x**18)

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GIAC/XCAS [A]  time = 0.210345, size = 151, normalized size = 1.16 \[ -\frac{43758 \, b^{10} x^{10} + 388960 \, a b^{9} x^{9} + 1575288 \, a^{2} b^{8} x^{8} + 3818880 \, a^{3} b^{7} x^{7} + 6126120 \, a^{4} b^{6} x^{6} + 6785856 \, a^{5} b^{5} x^{5} + 5250960 \, a^{6} b^{4} x^{4} + 2800512 \, a^{7} b^{3} x^{3} + 984555 \, a^{8} b^{2} x^{2} + 205920 \, a^{9} b x + 19448 \, a^{10}}{350064 \, x^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^10/x^19,x, algorithm="giac")

[Out]

-1/350064*(43758*b^10*x^10 + 388960*a*b^9*x^9 + 1575288*a^2*b^8*x^8 + 3818880*a^
3*b^7*x^7 + 6126120*a^4*b^6*x^6 + 6785856*a^5*b^5*x^5 + 5250960*a^6*b^4*x^4 + 28
00512*a^7*b^3*x^3 + 984555*a^8*b^2*x^2 + 205920*a^9*b*x + 19448*a^10)/x^18