3.95 \(\int \frac{(a+b x)^5}{x^{12}} \, dx\)

Optimal. Leaf size=69 \[ -\frac{a^5}{11 x^{11}}-\frac{a^4 b}{2 x^{10}}-\frac{10 a^3 b^2}{9 x^9}-\frac{5 a^2 b^3}{4 x^8}-\frac{5 a b^4}{7 x^7}-\frac{b^5}{6 x^6} \]

[Out]

-a^5/(11*x^11) - (a^4*b)/(2*x^10) - (10*a^3*b^2)/(9*x^9) - (5*a^2*b^3)/(4*x^8) -
(5*a*b^4)/(7*x^7) - b^5/(6*x^6)

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Rubi [A]  time = 0.0501538, 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 \[ -\frac{a^5}{11 x^{11}}-\frac{a^4 b}{2 x^{10}}-\frac{10 a^3 b^2}{9 x^9}-\frac{5 a^2 b^3}{4 x^8}-\frac{5 a b^4}{7 x^7}-\frac{b^5}{6 x^6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^5/x^12,x]

[Out]

-a^5/(11*x^11) - (a^4*b)/(2*x^10) - (10*a^3*b^2)/(9*x^9) - (5*a^2*b^3)/(4*x^8) -
(5*a*b^4)/(7*x^7) - b^5/(6*x^6)

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Rubi in Sympy [A]  time = 11.1844, size = 66, normalized size = 0.96 \[ - \frac{a^{5}}{11 x^{11}} - \frac{a^{4} b}{2 x^{10}} - \frac{10 a^{3} b^{2}}{9 x^{9}} - \frac{5 a^{2} b^{3}}{4 x^{8}} - \frac{5 a b^{4}}{7 x^{7}} - \frac{b^{5}}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**5/x**12,x)

[Out]

-a**5/(11*x**11) - a**4*b/(2*x**10) - 10*a**3*b**2/(9*x**9) - 5*a**2*b**3/(4*x**
8) - 5*a*b**4/(7*x**7) - b**5/(6*x**6)

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Mathematica [A]  time = 0.00526788, size = 69, normalized size = 1. \[ -\frac{a^5}{11 x^{11}}-\frac{a^4 b}{2 x^{10}}-\frac{10 a^3 b^2}{9 x^9}-\frac{5 a^2 b^3}{4 x^8}-\frac{5 a b^4}{7 x^7}-\frac{b^5}{6 x^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^5/x^12,x]

[Out]

-a^5/(11*x^11) - (a^4*b)/(2*x^10) - (10*a^3*b^2)/(9*x^9) - (5*a^2*b^3)/(4*x^8) -
(5*a*b^4)/(7*x^7) - b^5/(6*x^6)

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Maple [A]  time = 0.009, size = 58, normalized size = 0.8 \[ -{\frac{{a}^{5}}{11\,{x}^{11}}}-{\frac{{a}^{4}b}{2\,{x}^{10}}}-{\frac{10\,{a}^{3}{b}^{2}}{9\,{x}^{9}}}-{\frac{5\,{a}^{2}{b}^{3}}{4\,{x}^{8}}}-{\frac{5\,a{b}^{4}}{7\,{x}^{7}}}-{\frac{{b}^{5}}{6\,{x}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^5/x^12,x)

[Out]

-1/11*a^5/x^11-1/2*a^4*b/x^10-10/9*a^3*b^2/x^9-5/4*a^2*b^3/x^8-5/7*a*b^4/x^7-1/6
*b^5/x^6

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Maxima [A]  time = 1.33255, size = 77, normalized size = 1.12 \[ -\frac{462 \, b^{5} x^{5} + 1980 \, a b^{4} x^{4} + 3465 \, a^{2} b^{3} x^{3} + 3080 \, a^{3} b^{2} x^{2} + 1386 \, a^{4} b x + 252 \, a^{5}}{2772 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/x^12,x, algorithm="maxima")

[Out]

-1/2772*(462*b^5*x^5 + 1980*a*b^4*x^4 + 3465*a^2*b^3*x^3 + 3080*a^3*b^2*x^2 + 13
86*a^4*b*x + 252*a^5)/x^11

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Fricas [A]  time = 0.189828, size = 77, normalized size = 1.12 \[ -\frac{462 \, b^{5} x^{5} + 1980 \, a b^{4} x^{4} + 3465 \, a^{2} b^{3} x^{3} + 3080 \, a^{3} b^{2} x^{2} + 1386 \, a^{4} b x + 252 \, a^{5}}{2772 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/x^12,x, algorithm="fricas")

[Out]

-1/2772*(462*b^5*x^5 + 1980*a*b^4*x^4 + 3465*a^2*b^3*x^3 + 3080*a^3*b^2*x^2 + 13
86*a^4*b*x + 252*a^5)/x^11

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Sympy [A]  time = 2.37898, size = 61, normalized size = 0.88 \[ - \frac{252 a^{5} + 1386 a^{4} b x + 3080 a^{3} b^{2} x^{2} + 3465 a^{2} b^{3} x^{3} + 1980 a b^{4} x^{4} + 462 b^{5} x^{5}}{2772 x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**5/x**12,x)

[Out]

-(252*a**5 + 1386*a**4*b*x + 3080*a**3*b**2*x**2 + 3465*a**2*b**3*x**3 + 1980*a*
b**4*x**4 + 462*b**5*x**5)/(2772*x**11)

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GIAC/XCAS [A]  time = 0.206446, size = 77, normalized size = 1.12 \[ -\frac{462 \, b^{5} x^{5} + 1980 \, a b^{4} x^{4} + 3465 \, a^{2} b^{3} x^{3} + 3080 \, a^{3} b^{2} x^{2} + 1386 \, a^{4} b x + 252 \, a^{5}}{2772 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/x^12,x, algorithm="giac")

[Out]

-1/2772*(462*b^5*x^5 + 1980*a*b^4*x^4 + 3465*a^2*b^3*x^3 + 3080*a^3*b^2*x^2 + 13
86*a^4*b*x + 252*a^5)/x^11