3.94 \(\int \frac{(a+b x)^5}{x^{11}} \, dx\)

Optimal. Leaf size=69 \[ -\frac{a^5}{10 x^{10}}-\frac{5 a^4 b}{9 x^9}-\frac{5 a^3 b^2}{4 x^8}-\frac{10 a^2 b^3}{7 x^7}-\frac{5 a b^4}{6 x^6}-\frac{b^5}{5 x^5} \]

[Out]

-a^5/(10*x^10) - (5*a^4*b)/(9*x^9) - (5*a^3*b^2)/(4*x^8) - (10*a^2*b^3)/(7*x^7)
- (5*a*b^4)/(6*x^6) - b^5/(5*x^5)

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Rubi [A]  time = 0.0515521, antiderivative size = 69, 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 \[ -\frac{a^5}{10 x^{10}}-\frac{5 a^4 b}{9 x^9}-\frac{5 a^3 b^2}{4 x^8}-\frac{10 a^2 b^3}{7 x^7}-\frac{5 a b^4}{6 x^6}-\frac{b^5}{5 x^5} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^5/x^11,x]

[Out]

-a^5/(10*x^10) - (5*a^4*b)/(9*x^9) - (5*a^3*b^2)/(4*x^8) - (10*a^2*b^3)/(7*x^7)
- (5*a*b^4)/(6*x^6) - b^5/(5*x^5)

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Rubi in Sympy [A]  time = 10.7924, size = 68, normalized size = 0.99 \[ - \frac{a^{5}}{10 x^{10}} - \frac{5 a^{4} b}{9 x^{9}} - \frac{5 a^{3} b^{2}}{4 x^{8}} - \frac{10 a^{2} b^{3}}{7 x^{7}} - \frac{5 a b^{4}}{6 x^{6}} - \frac{b^{5}}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**5/x**11,x)

[Out]

-a**5/(10*x**10) - 5*a**4*b/(9*x**9) - 5*a**3*b**2/(4*x**8) - 10*a**2*b**3/(7*x*
*7) - 5*a*b**4/(6*x**6) - b**5/(5*x**5)

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Mathematica [A]  time = 0.00538659, size = 69, normalized size = 1. \[ -\frac{a^5}{10 x^{10}}-\frac{5 a^4 b}{9 x^9}-\frac{5 a^3 b^2}{4 x^8}-\frac{10 a^2 b^3}{7 x^7}-\frac{5 a b^4}{6 x^6}-\frac{b^5}{5 x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^5/x^11,x]

[Out]

-a^5/(10*x^10) - (5*a^4*b)/(9*x^9) - (5*a^3*b^2)/(4*x^8) - (10*a^2*b^3)/(7*x^7)
- (5*a*b^4)/(6*x^6) - b^5/(5*x^5)

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Maple [A]  time = 0.008, size = 58, normalized size = 0.8 \[ -{\frac{{a}^{5}}{10\,{x}^{10}}}-{\frac{5\,{a}^{4}b}{9\,{x}^{9}}}-{\frac{5\,{a}^{3}{b}^{2}}{4\,{x}^{8}}}-{\frac{10\,{a}^{2}{b}^{3}}{7\,{x}^{7}}}-{\frac{5\,a{b}^{4}}{6\,{x}^{6}}}-{\frac{{b}^{5}}{5\,{x}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^5/x^11,x)

[Out]

-1/10*a^5/x^10-5/9*a^4*b/x^9-5/4*a^3*b^2/x^8-10/7*a^2*b^3/x^7-5/6*a*b^4/x^6-1/5*
b^5/x^5

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Maxima [A]  time = 1.3437, size = 77, normalized size = 1.12 \[ -\frac{252 \, b^{5} x^{5} + 1050 \, a b^{4} x^{4} + 1800 \, a^{2} b^{3} x^{3} + 1575 \, a^{3} b^{2} x^{2} + 700 \, a^{4} b x + 126 \, a^{5}}{1260 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/x^11,x, algorithm="maxima")

[Out]

-1/1260*(252*b^5*x^5 + 1050*a*b^4*x^4 + 1800*a^2*b^3*x^3 + 1575*a^3*b^2*x^2 + 70
0*a^4*b*x + 126*a^5)/x^10

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Fricas [A]  time = 0.192151, size = 77, normalized size = 1.12 \[ -\frac{252 \, b^{5} x^{5} + 1050 \, a b^{4} x^{4} + 1800 \, a^{2} b^{3} x^{3} + 1575 \, a^{3} b^{2} x^{2} + 700 \, a^{4} b x + 126 \, a^{5}}{1260 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/x^11,x, algorithm="fricas")

[Out]

-1/1260*(252*b^5*x^5 + 1050*a*b^4*x^4 + 1800*a^2*b^3*x^3 + 1575*a^3*b^2*x^2 + 70
0*a^4*b*x + 126*a^5)/x^10

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Sympy [A]  time = 2.25446, size = 61, normalized size = 0.88 \[ - \frac{126 a^{5} + 700 a^{4} b x + 1575 a^{3} b^{2} x^{2} + 1800 a^{2} b^{3} x^{3} + 1050 a b^{4} x^{4} + 252 b^{5} x^{5}}{1260 x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**5/x**11,x)

[Out]

-(126*a**5 + 700*a**4*b*x + 1575*a**3*b**2*x**2 + 1800*a**2*b**3*x**3 + 1050*a*b
**4*x**4 + 252*b**5*x**5)/(1260*x**10)

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GIAC/XCAS [A]  time = 0.206667, size = 77, normalized size = 1.12 \[ -\frac{252 \, b^{5} x^{5} + 1050 \, a b^{4} x^{4} + 1800 \, a^{2} b^{3} x^{3} + 1575 \, a^{3} b^{2} x^{2} + 700 \, a^{4} b x + 126 \, a^{5}}{1260 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/x^11,x, algorithm="giac")

[Out]

-1/1260*(252*b^5*x^5 + 1050*a*b^4*x^4 + 1800*a^2*b^3*x^3 + 1575*a^3*b^2*x^2 + 70
0*a^4*b*x + 126*a^5)/x^10