3.96 \(\int \frac{(a+b x)^5}{x^{13}} \, dx\)

Optimal. Leaf size=67 \[ -\frac{a^5}{12 x^{12}}-\frac{5 a^4 b}{11 x^{11}}-\frac{a^3 b^2}{x^{10}}-\frac{10 a^2 b^3}{9 x^9}-\frac{5 a b^4}{8 x^8}-\frac{b^5}{7 x^7} \]

[Out]

-a^5/(12*x^12) - (5*a^4*b)/(11*x^11) - (a^3*b^2)/x^10 - (10*a^2*b^3)/(9*x^9) - (
5*a*b^4)/(8*x^8) - b^5/(7*x^7)

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Rubi [A]  time = 0.0526455, antiderivative size = 67, 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 \[ -\frac{a^5}{12 x^{12}}-\frac{5 a^4 b}{11 x^{11}}-\frac{a^3 b^2}{x^{10}}-\frac{10 a^2 b^3}{9 x^9}-\frac{5 a b^4}{8 x^8}-\frac{b^5}{7 x^7} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^5/x^13,x]

[Out]

-a^5/(12*x^12) - (5*a^4*b)/(11*x^11) - (a^3*b^2)/x^10 - (10*a^2*b^3)/(9*x^9) - (
5*a*b^4)/(8*x^8) - b^5/(7*x^7)

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Rubi in Sympy [A]  time = 10.5955, size = 65, normalized size = 0.97 \[ - \frac{a^{5}}{12 x^{12}} - \frac{5 a^{4} b}{11 x^{11}} - \frac{a^{3} b^{2}}{x^{10}} - \frac{10 a^{2} b^{3}}{9 x^{9}} - \frac{5 a b^{4}}{8 x^{8}} - \frac{b^{5}}{7 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**5/x**13,x)

[Out]

-a**5/(12*x**12) - 5*a**4*b/(11*x**11) - a**3*b**2/x**10 - 10*a**2*b**3/(9*x**9)
- 5*a*b**4/(8*x**8) - b**5/(7*x**7)

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Mathematica [A]  time = 0.00546371, size = 67, normalized size = 1. \[ -\frac{a^5}{12 x^{12}}-\frac{5 a^4 b}{11 x^{11}}-\frac{a^3 b^2}{x^{10}}-\frac{10 a^2 b^3}{9 x^9}-\frac{5 a b^4}{8 x^8}-\frac{b^5}{7 x^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^5/x^13,x]

[Out]

-a^5/(12*x^12) - (5*a^4*b)/(11*x^11) - (a^3*b^2)/x^10 - (10*a^2*b^3)/(9*x^9) - (
5*a*b^4)/(8*x^8) - b^5/(7*x^7)

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Maple [A]  time = 0.007, size = 58, normalized size = 0.9 \[ -{\frac{{a}^{5}}{12\,{x}^{12}}}-{\frac{5\,{a}^{4}b}{11\,{x}^{11}}}-{\frac{{a}^{3}{b}^{2}}{{x}^{10}}}-{\frac{10\,{a}^{2}{b}^{3}}{9\,{x}^{9}}}-{\frac{5\,a{b}^{4}}{8\,{x}^{8}}}-{\frac{{b}^{5}}{7\,{x}^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^5/x^13,x)

[Out]

-1/12*a^5/x^12-5/11*a^4*b/x^11-a^3*b^2/x^10-10/9*a^2*b^3/x^9-5/8*a*b^4/x^8-1/7*b
^5/x^7

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Maxima [A]  time = 1.48474, size = 77, normalized size = 1.15 \[ -\frac{792 \, b^{5} x^{5} + 3465 \, a b^{4} x^{4} + 6160 \, a^{2} b^{3} x^{3} + 5544 \, a^{3} b^{2} x^{2} + 2520 \, a^{4} b x + 462 \, a^{5}}{5544 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/x^13,x, algorithm="maxima")

[Out]

-1/5544*(792*b^5*x^5 + 3465*a*b^4*x^4 + 6160*a^2*b^3*x^3 + 5544*a^3*b^2*x^2 + 25
20*a^4*b*x + 462*a^5)/x^12

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Fricas [A]  time = 0.190063, size = 77, normalized size = 1.15 \[ -\frac{792 \, b^{5} x^{5} + 3465 \, a b^{4} x^{4} + 6160 \, a^{2} b^{3} x^{3} + 5544 \, a^{3} b^{2} x^{2} + 2520 \, a^{4} b x + 462 \, a^{5}}{5544 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/x^13,x, algorithm="fricas")

[Out]

-1/5544*(792*b^5*x^5 + 3465*a*b^4*x^4 + 6160*a^2*b^3*x^3 + 5544*a^3*b^2*x^2 + 25
20*a^4*b*x + 462*a^5)/x^12

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Sympy [A]  time = 2.51644, size = 61, normalized size = 0.91 \[ - \frac{462 a^{5} + 2520 a^{4} b x + 5544 a^{3} b^{2} x^{2} + 6160 a^{2} b^{3} x^{3} + 3465 a b^{4} x^{4} + 792 b^{5} x^{5}}{5544 x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**5/x**13,x)

[Out]

-(462*a**5 + 2520*a**4*b*x + 5544*a**3*b**2*x**2 + 6160*a**2*b**3*x**3 + 3465*a*
b**4*x**4 + 792*b**5*x**5)/(5544*x**12)

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GIAC/XCAS [A]  time = 0.207071, size = 77, normalized size = 1.15 \[ -\frac{792 \, b^{5} x^{5} + 3465 \, a b^{4} x^{4} + 6160 \, a^{2} b^{3} x^{3} + 5544 \, a^{3} b^{2} x^{2} + 2520 \, a^{4} b x + 462 \, a^{5}}{5544 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/x^13,x, algorithm="giac")

[Out]

-1/5544*(792*b^5*x^5 + 3465*a*b^4*x^4 + 6160*a^2*b^3*x^3 + 5544*a^3*b^2*x^2 + 25
20*a^4*b*x + 462*a^5)/x^12