3.91 \(\int \frac{(a+b x)^5}{x^8} \, dx\)

Optimal. Leaf size=36 \[ \frac{b (a+b x)^6}{42 a^2 x^6}-\frac{(a+b x)^6}{7 a x^7} \]

[Out]

-(a + b*x)^6/(7*a*x^7) + (b*(a + b*x)^6)/(42*a^2*x^6)

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Rubi [A]  time = 0.0230804, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{b (a+b x)^6}{42 a^2 x^6}-\frac{(a+b x)^6}{7 a x^7} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^5/x^8,x]

[Out]

-(a + b*x)^6/(7*a*x^7) + (b*(a + b*x)^6)/(42*a^2*x^6)

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Rubi in Sympy [A]  time = 3.57116, size = 29, normalized size = 0.81 \[ - \frac{\left (a + b x\right )^{6}}{7 a x^{7}} + \frac{b \left (a + b x\right )^{6}}{42 a^{2} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**5/x**8,x)

[Out]

-(a + b*x)**6/(7*a*x**7) + b*(a + b*x)**6/(42*a**2*x**6)

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Mathematica [A]  time = 0.00519844, size = 67, normalized size = 1.86 \[ -\frac{a^5}{7 x^7}-\frac{5 a^4 b}{6 x^6}-\frac{2 a^3 b^2}{x^5}-\frac{5 a^2 b^3}{2 x^4}-\frac{5 a b^4}{3 x^3}-\frac{b^5}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^5/x^8,x]

[Out]

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

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Maple [A]  time = 0.009, size = 58, normalized size = 1.6 \[ -{\frac{{b}^{5}}{2\,{x}^{2}}}-{\frac{5\,a{b}^{4}}{3\,{x}^{3}}}-2\,{\frac{{a}^{3}{b}^{2}}{{x}^{5}}}-{\frac{5\,{a}^{4}b}{6\,{x}^{6}}}-{\frac{5\,{a}^{2}{b}^{3}}{2\,{x}^{4}}}-{\frac{{a}^{5}}{7\,{x}^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^5/x^8,x)

[Out]

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

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Maxima [A]  time = 1.34056, size = 77, normalized size = 2.14 \[ -\frac{21 \, b^{5} x^{5} + 70 \, a b^{4} x^{4} + 105 \, a^{2} b^{3} x^{3} + 84 \, a^{3} b^{2} x^{2} + 35 \, a^{4} b x + 6 \, a^{5}}{42 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/x^8,x, algorithm="maxima")

[Out]

-1/42*(21*b^5*x^5 + 70*a*b^4*x^4 + 105*a^2*b^3*x^3 + 84*a^3*b^2*x^2 + 35*a^4*b*x
+ 6*a^5)/x^7

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Fricas [A]  time = 0.18883, size = 77, normalized size = 2.14 \[ -\frac{21 \, b^{5} x^{5} + 70 \, a b^{4} x^{4} + 105 \, a^{2} b^{3} x^{3} + 84 \, a^{3} b^{2} x^{2} + 35 \, a^{4} b x + 6 \, a^{5}}{42 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/x^8,x, algorithm="fricas")

[Out]

-1/42*(21*b^5*x^5 + 70*a*b^4*x^4 + 105*a^2*b^3*x^3 + 84*a^3*b^2*x^2 + 35*a^4*b*x
+ 6*a^5)/x^7

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Sympy [A]  time = 1.99454, size = 61, normalized size = 1.69 \[ - \frac{6 a^{5} + 35 a^{4} b x + 84 a^{3} b^{2} x^{2} + 105 a^{2} b^{3} x^{3} + 70 a b^{4} x^{4} + 21 b^{5} x^{5}}{42 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**5/x**8,x)

[Out]

-(6*a**5 + 35*a**4*b*x + 84*a**3*b**2*x**2 + 105*a**2*b**3*x**3 + 70*a*b**4*x**4
+ 21*b**5*x**5)/(42*x**7)

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GIAC/XCAS [A]  time = 0.205518, size = 77, normalized size = 2.14 \[ -\frac{21 \, b^{5} x^{5} + 70 \, a b^{4} x^{4} + 105 \, a^{2} b^{3} x^{3} + 84 \, a^{3} b^{2} x^{2} + 35 \, a^{4} b x + 6 \, a^{5}}{42 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/x^8,x, algorithm="giac")

[Out]

-1/42*(21*b^5*x^5 + 70*a*b^4*x^4 + 105*a^2*b^3*x^3 + 84*a^3*b^2*x^2 + 35*a^4*b*x
+ 6*a^5)/x^7