3.242 \(\int \frac{(a+b x)^9}{x^{10}} \, dx\)

Optimal. Leaf size=109 \[ -\frac{a^9}{9 x^9}-\frac{9 a^8 b}{8 x^8}-\frac{36 a^7 b^2}{7 x^7}-\frac{14 a^6 b^3}{x^6}-\frac{126 a^5 b^4}{5 x^5}-\frac{63 a^4 b^5}{2 x^4}-\frac{28 a^3 b^6}{x^3}-\frac{18 a^2 b^7}{x^2}-\frac{9 a b^8}{x}+b^9 \log (x) \]

[Out]

-a^9/(9*x^9) - (9*a^8*b)/(8*x^8) - (36*a^7*b^2)/(7*x^7) - (14*a^6*b^3)/x^6 - (12
6*a^5*b^4)/(5*x^5) - (63*a^4*b^5)/(2*x^4) - (28*a^3*b^6)/x^3 - (18*a^2*b^7)/x^2
- (9*a*b^8)/x + b^9*Log[x]

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Rubi [A]  time = 0.111354, antiderivative size = 109, 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 \[ -\frac{a^9}{9 x^9}-\frac{9 a^8 b}{8 x^8}-\frac{36 a^7 b^2}{7 x^7}-\frac{14 a^6 b^3}{x^6}-\frac{126 a^5 b^4}{5 x^5}-\frac{63 a^4 b^5}{2 x^4}-\frac{28 a^3 b^6}{x^3}-\frac{18 a^2 b^7}{x^2}-\frac{9 a b^8}{x}+b^9 \log (x) \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^9/x^10,x]

[Out]

-a^9/(9*x^9) - (9*a^8*b)/(8*x^8) - (36*a^7*b^2)/(7*x^7) - (14*a^6*b^3)/x^6 - (12
6*a^5*b^4)/(5*x^5) - (63*a^4*b^5)/(2*x^4) - (28*a^3*b^6)/x^3 - (18*a^2*b^7)/x^2
- (9*a*b^8)/x + b^9*Log[x]

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Rubi in Sympy [A]  time = 20.3246, size = 110, normalized size = 1.01 \[ - \frac{a^{9}}{9 x^{9}} - \frac{9 a^{8} b}{8 x^{8}} - \frac{36 a^{7} b^{2}}{7 x^{7}} - \frac{14 a^{6} b^{3}}{x^{6}} - \frac{126 a^{5} b^{4}}{5 x^{5}} - \frac{63 a^{4} b^{5}}{2 x^{4}} - \frac{28 a^{3} b^{6}}{x^{3}} - \frac{18 a^{2} b^{7}}{x^{2}} - \frac{9 a b^{8}}{x} + b^{9} \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**9/x**10,x)

[Out]

-a**9/(9*x**9) - 9*a**8*b/(8*x**8) - 36*a**7*b**2/(7*x**7) - 14*a**6*b**3/x**6 -
126*a**5*b**4/(5*x**5) - 63*a**4*b**5/(2*x**4) - 28*a**3*b**6/x**3 - 18*a**2*b*
*7/x**2 - 9*a*b**8/x + b**9*log(x)

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Mathematica [A]  time = 0.00698235, size = 109, normalized size = 1. \[ -\frac{a^9}{9 x^9}-\frac{9 a^8 b}{8 x^8}-\frac{36 a^7 b^2}{7 x^7}-\frac{14 a^6 b^3}{x^6}-\frac{126 a^5 b^4}{5 x^5}-\frac{63 a^4 b^5}{2 x^4}-\frac{28 a^3 b^6}{x^3}-\frac{18 a^2 b^7}{x^2}-\frac{9 a b^8}{x}+b^9 \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^9/x^10,x]

[Out]

-a^9/(9*x^9) - (9*a^8*b)/(8*x^8) - (36*a^7*b^2)/(7*x^7) - (14*a^6*b^3)/x^6 - (12
6*a^5*b^4)/(5*x^5) - (63*a^4*b^5)/(2*x^4) - (28*a^3*b^6)/x^3 - (18*a^2*b^7)/x^2
- (9*a*b^8)/x + b^9*Log[x]

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Maple [A]  time = 0.012, size = 100, normalized size = 0.9 \[ -{\frac{{a}^{9}}{9\,{x}^{9}}}-{\frac{9\,{a}^{8}b}{8\,{x}^{8}}}-{\frac{36\,{a}^{7}{b}^{2}}{7\,{x}^{7}}}-14\,{\frac{{a}^{6}{b}^{3}}{{x}^{6}}}-{\frac{126\,{a}^{5}{b}^{4}}{5\,{x}^{5}}}-{\frac{63\,{a}^{4}{b}^{5}}{2\,{x}^{4}}}-28\,{\frac{{a}^{3}{b}^{6}}{{x}^{3}}}-18\,{\frac{{a}^{2}{b}^{7}}{{x}^{2}}}-9\,{\frac{a{b}^{8}}{x}}+{b}^{9}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^9/x^10,x)

[Out]

-1/9*a^9/x^9-9/8*a^8*b/x^8-36/7*a^7*b^2/x^7-14*a^6*b^3/x^6-126/5*a^5*b^4/x^5-63/
2*a^4*b^5/x^4-28*a^3*b^6/x^3-18*a^2*b^7/x^2-9*a*b^8/x+b^9*ln(x)

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Maxima [A]  time = 1.34518, size = 135, normalized size = 1.24 \[ b^{9} \log \left (x\right ) - \frac{22680 \, a b^{8} x^{8} + 45360 \, a^{2} b^{7} x^{7} + 70560 \, a^{3} b^{6} x^{6} + 79380 \, a^{4} b^{5} x^{5} + 63504 \, a^{5} b^{4} x^{4} + 35280 \, a^{6} b^{3} x^{3} + 12960 \, a^{7} b^{2} x^{2} + 2835 \, a^{8} b x + 280 \, a^{9}}{2520 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^9/x^10,x, algorithm="maxima")

[Out]

b^9*log(x) - 1/2520*(22680*a*b^8*x^8 + 45360*a^2*b^7*x^7 + 70560*a^3*b^6*x^6 + 7
9380*a^4*b^5*x^5 + 63504*a^5*b^4*x^4 + 35280*a^6*b^3*x^3 + 12960*a^7*b^2*x^2 + 2
835*a^8*b*x + 280*a^9)/x^9

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Fricas [A]  time = 0.210099, size = 139, normalized size = 1.28 \[ \frac{2520 \, b^{9} x^{9} \log \left (x\right ) - 22680 \, a b^{8} x^{8} - 45360 \, a^{2} b^{7} x^{7} - 70560 \, a^{3} b^{6} x^{6} - 79380 \, a^{4} b^{5} x^{5} - 63504 \, a^{5} b^{4} x^{4} - 35280 \, a^{6} b^{3} x^{3} - 12960 \, a^{7} b^{2} x^{2} - 2835 \, a^{8} b x - 280 \, a^{9}}{2520 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^9/x^10,x, algorithm="fricas")

[Out]

1/2520*(2520*b^9*x^9*log(x) - 22680*a*b^8*x^8 - 45360*a^2*b^7*x^7 - 70560*a^3*b^
6*x^6 - 79380*a^4*b^5*x^5 - 63504*a^5*b^4*x^4 - 35280*a^6*b^3*x^3 - 12960*a^7*b^
2*x^2 - 2835*a^8*b*x - 280*a^9)/x^9

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Sympy [A]  time = 3.16019, size = 105, normalized size = 0.96 \[ b^{9} \log{\left (x \right )} - \frac{280 a^{9} + 2835 a^{8} b x + 12960 a^{7} b^{2} x^{2} + 35280 a^{6} b^{3} x^{3} + 63504 a^{5} b^{4} x^{4} + 79380 a^{4} b^{5} x^{5} + 70560 a^{3} b^{6} x^{6} + 45360 a^{2} b^{7} x^{7} + 22680 a b^{8} x^{8}}{2520 x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**9/x**10,x)

[Out]

b**9*log(x) - (280*a**9 + 2835*a**8*b*x + 12960*a**7*b**2*x**2 + 35280*a**6*b**3
*x**3 + 63504*a**5*b**4*x**4 + 79380*a**4*b**5*x**5 + 70560*a**3*b**6*x**6 + 453
60*a**2*b**7*x**7 + 22680*a*b**8*x**8)/(2520*x**9)

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GIAC/XCAS [A]  time = 0.204813, size = 136, normalized size = 1.25 \[ b^{9}{\rm ln}\left ({\left | x \right |}\right ) - \frac{22680 \, a b^{8} x^{8} + 45360 \, a^{2} b^{7} x^{7} + 70560 \, a^{3} b^{6} x^{6} + 79380 \, a^{4} b^{5} x^{5} + 63504 \, a^{5} b^{4} x^{4} + 35280 \, a^{6} b^{3} x^{3} + 12960 \, a^{7} b^{2} x^{2} + 2835 \, a^{8} b x + 280 \, a^{9}}{2520 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^9/x^10,x, algorithm="giac")

[Out]

b^9*ln(abs(x)) - 1/2520*(22680*a*b^8*x^8 + 45360*a^2*b^7*x^7 + 70560*a^3*b^6*x^6
+ 79380*a^4*b^5*x^5 + 63504*a^5*b^4*x^4 + 35280*a^6*b^3*x^3 + 12960*a^7*b^2*x^2
+ 2835*a^8*b*x + 280*a^9)/x^9