3.140 \(\int \frac{(a+b x)^{10}}{x^6} \, dx\)

Optimal. Leaf size=117 \[ -\frac{a^{10}}{5 x^5}-\frac{5 a^9 b}{2 x^4}-\frac{15 a^8 b^2}{x^3}-\frac{60 a^7 b^3}{x^2}-\frac{210 a^6 b^4}{x}+252 a^5 b^5 \log (x)+210 a^4 b^6 x+60 a^3 b^7 x^2+15 a^2 b^8 x^3+\frac{5}{2} a b^9 x^4+\frac{b^{10} x^5}{5} \]

[Out]

-a^10/(5*x^5) - (5*a^9*b)/(2*x^4) - (15*a^8*b^2)/x^3 - (60*a^7*b^3)/x^2 - (210*a
^6*b^4)/x + 210*a^4*b^6*x + 60*a^3*b^7*x^2 + 15*a^2*b^8*x^3 + (5*a*b^9*x^4)/2 +
(b^10*x^5)/5 + 252*a^5*b^5*Log[x]

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Rubi [A]  time = 0.11832, antiderivative size = 117, 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 \[ -\frac{a^{10}}{5 x^5}-\frac{5 a^9 b}{2 x^4}-\frac{15 a^8 b^2}{x^3}-\frac{60 a^7 b^3}{x^2}-\frac{210 a^6 b^4}{x}+252 a^5 b^5 \log (x)+210 a^4 b^6 x+60 a^3 b^7 x^2+15 a^2 b^8 x^3+\frac{5}{2} a b^9 x^4+\frac{b^{10} x^5}{5} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^10/x^6,x]

[Out]

-a^10/(5*x^5) - (5*a^9*b)/(2*x^4) - (15*a^8*b^2)/x^3 - (60*a^7*b^3)/x^2 - (210*a
^6*b^4)/x + 210*a^4*b^6*x + 60*a^3*b^7*x^2 + 15*a^2*b^8*x^3 + (5*a*b^9*x^4)/2 +
(b^10*x^5)/5 + 252*a^5*b^5*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{10}}{5 x^{5}} - \frac{5 a^{9} b}{2 x^{4}} - \frac{15 a^{8} b^{2}}{x^{3}} - \frac{60 a^{7} b^{3}}{x^{2}} - \frac{210 a^{6} b^{4}}{x} + 252 a^{5} b^{5} \log{\left (x \right )} + 210 a^{4} b^{6} x + 120 a^{3} b^{7} \int x\, dx + 15 a^{2} b^{8} x^{3} + \frac{5 a b^{9} x^{4}}{2} + \frac{b^{10} x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**10/x**6,x)

[Out]

-a**10/(5*x**5) - 5*a**9*b/(2*x**4) - 15*a**8*b**2/x**3 - 60*a**7*b**3/x**2 - 21
0*a**6*b**4/x + 252*a**5*b**5*log(x) + 210*a**4*b**6*x + 120*a**3*b**7*Integral(
x, x) + 15*a**2*b**8*x**3 + 5*a*b**9*x**4/2 + b**10*x**5/5

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Mathematica [A]  time = 0.0139001, size = 117, normalized size = 1. \[ -\frac{a^{10}}{5 x^5}-\frac{5 a^9 b}{2 x^4}-\frac{15 a^8 b^2}{x^3}-\frac{60 a^7 b^3}{x^2}-\frac{210 a^6 b^4}{x}+252 a^5 b^5 \log (x)+210 a^4 b^6 x+60 a^3 b^7 x^2+15 a^2 b^8 x^3+\frac{5}{2} a b^9 x^4+\frac{b^{10} x^5}{5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^10/x^6,x]

[Out]

-a^10/(5*x^5) - (5*a^9*b)/(2*x^4) - (15*a^8*b^2)/x^3 - (60*a^7*b^3)/x^2 - (210*a
^6*b^4)/x + 210*a^4*b^6*x + 60*a^3*b^7*x^2 + 15*a^2*b^8*x^3 + (5*a*b^9*x^4)/2 +
(b^10*x^5)/5 + 252*a^5*b^5*Log[x]

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Maple [A]  time = 0.011, size = 110, normalized size = 0.9 \[ -{\frac{{a}^{10}}{5\,{x}^{5}}}-{\frac{5\,{a}^{9}b}{2\,{x}^{4}}}-15\,{\frac{{a}^{8}{b}^{2}}{{x}^{3}}}-60\,{\frac{{a}^{7}{b}^{3}}{{x}^{2}}}-210\,{\frac{{a}^{6}{b}^{4}}{x}}+210\,{a}^{4}{b}^{6}x+60\,{a}^{3}{b}^{7}{x}^{2}+15\,{a}^{2}{b}^{8}{x}^{3}+{\frac{5\,a{b}^{9}{x}^{4}}{2}}+{\frac{{b}^{10}{x}^{5}}{5}}+252\,{a}^{5}{b}^{5}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^10/x^6,x)

[Out]

-1/5*a^10/x^5-5/2*a^9*b/x^4-15*a^8*b^2/x^3-60*a^7*b^3/x^2-210*a^6*b^4/x+210*a^4*
b^6*x+60*a^3*b^7*x^2+15*a^2*b^8*x^3+5/2*a*b^9*x^4+1/5*b^10*x^5+252*a^5*b^5*ln(x)

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Maxima [A]  time = 1.33163, size = 149, normalized size = 1.27 \[ \frac{1}{5} \, b^{10} x^{5} + \frac{5}{2} \, a b^{9} x^{4} + 15 \, a^{2} b^{8} x^{3} + 60 \, a^{3} b^{7} x^{2} + 210 \, a^{4} b^{6} x + 252 \, a^{5} b^{5} \log \left (x\right ) - \frac{2100 \, a^{6} b^{4} x^{4} + 600 \, a^{7} b^{3} x^{3} + 150 \, a^{8} b^{2} x^{2} + 25 \, a^{9} b x + 2 \, a^{10}}{10 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^10/x^6,x, algorithm="maxima")

[Out]

1/5*b^10*x^5 + 5/2*a*b^9*x^4 + 15*a^2*b^8*x^3 + 60*a^3*b^7*x^2 + 210*a^4*b^6*x +
252*a^5*b^5*log(x) - 1/10*(2100*a^6*b^4*x^4 + 600*a^7*b^3*x^3 + 150*a^8*b^2*x^2
+ 25*a^9*b*x + 2*a^10)/x^5

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Fricas [A]  time = 0.195875, size = 154, normalized size = 1.32 \[ \frac{2 \, b^{10} x^{10} + 25 \, a b^{9} x^{9} + 150 \, a^{2} b^{8} x^{8} + 600 \, a^{3} b^{7} x^{7} + 2100 \, a^{4} b^{6} x^{6} + 2520 \, a^{5} b^{5} x^{5} \log \left (x\right ) - 2100 \, a^{6} b^{4} x^{4} - 600 \, a^{7} b^{3} x^{3} - 150 \, a^{8} b^{2} x^{2} - 25 \, a^{9} b x - 2 \, a^{10}}{10 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^10/x^6,x, algorithm="fricas")

[Out]

1/10*(2*b^10*x^10 + 25*a*b^9*x^9 + 150*a^2*b^8*x^8 + 600*a^3*b^7*x^7 + 2100*a^4*
b^6*x^6 + 2520*a^5*b^5*x^5*log(x) - 2100*a^6*b^4*x^4 - 600*a^7*b^3*x^3 - 150*a^8
*b^2*x^2 - 25*a^9*b*x - 2*a^10)/x^5

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Sympy [A]  time = 2.23886, size = 119, normalized size = 1.02 \[ 252 a^{5} b^{5} \log{\left (x \right )} + 210 a^{4} b^{6} x + 60 a^{3} b^{7} x^{2} + 15 a^{2} b^{8} x^{3} + \frac{5 a b^{9} x^{4}}{2} + \frac{b^{10} x^{5}}{5} - \frac{2 a^{10} + 25 a^{9} b x + 150 a^{8} b^{2} x^{2} + 600 a^{7} b^{3} x^{3} + 2100 a^{6} b^{4} x^{4}}{10 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**10/x**6,x)

[Out]

252*a**5*b**5*log(x) + 210*a**4*b**6*x + 60*a**3*b**7*x**2 + 15*a**2*b**8*x**3 +
5*a*b**9*x**4/2 + b**10*x**5/5 - (2*a**10 + 25*a**9*b*x + 150*a**8*b**2*x**2 +
600*a**7*b**3*x**3 + 2100*a**6*b**4*x**4)/(10*x**5)

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GIAC/XCAS [A]  time = 0.204963, size = 150, normalized size = 1.28 \[ \frac{1}{5} \, b^{10} x^{5} + \frac{5}{2} \, a b^{9} x^{4} + 15 \, a^{2} b^{8} x^{3} + 60 \, a^{3} b^{7} x^{2} + 210 \, a^{4} b^{6} x + 252 \, a^{5} b^{5}{\rm ln}\left ({\left | x \right |}\right ) - \frac{2100 \, a^{6} b^{4} x^{4} + 600 \, a^{7} b^{3} x^{3} + 150 \, a^{8} b^{2} x^{2} + 25 \, a^{9} b x + 2 \, a^{10}}{10 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^10/x^6,x, algorithm="giac")

[Out]

1/5*b^10*x^5 + 5/2*a*b^9*x^4 + 15*a^2*b^8*x^3 + 60*a^3*b^7*x^2 + 210*a^4*b^6*x +
252*a^5*b^5*ln(abs(x)) - 1/10*(2100*a^6*b^4*x^4 + 600*a^7*b^3*x^3 + 150*a^8*b^2
*x^2 + 25*a^9*b*x + 2*a^10)/x^5