3.115 \(\int \frac{(a+b x)^7}{x^9} \, dx\)

Optimal. Leaf size=17 \[ -\frac{(a+b x)^8}{8 a x^8} \]

[Out]

-(a + b*x)^8/(8*a*x^8)

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Rubi [A]  time = 0.012129, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{(a+b x)^8}{8 a x^8} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^7/x^9,x]

[Out]

-(a + b*x)^8/(8*a*x^8)

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Rubi in Sympy [A]  time = 2.24652, size = 14, normalized size = 0.82 \[ - \frac{\left (a + b x\right )^{8}}{8 a x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**7/x**9,x)

[Out]

-(a + b*x)**8/(8*a*x**8)

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Mathematica [B]  time = 0.00552547, size = 87, normalized size = 5.12 \[ -\frac{a^7}{8 x^8}-\frac{a^6 b}{x^7}-\frac{7 a^5 b^2}{2 x^6}-\frac{7 a^4 b^3}{x^5}-\frac{35 a^3 b^4}{4 x^4}-\frac{7 a^2 b^5}{x^3}-\frac{7 a b^6}{2 x^2}-\frac{b^7}{x} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^7/x^9,x]

[Out]

-a^7/(8*x^8) - (a^6*b)/x^7 - (7*a^5*b^2)/(2*x^6) - (7*a^4*b^3)/x^5 - (35*a^3*b^4
)/(4*x^4) - (7*a^2*b^5)/x^3 - (7*a*b^6)/(2*x^2) - b^7/x

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Maple [B]  time = 0.007, size = 80, normalized size = 4.7 \[ -{\frac{7\,a{b}^{6}}{2\,{x}^{2}}}-7\,{\frac{{a}^{4}{b}^{3}}{{x}^{5}}}-{\frac{{b}^{7}}{x}}-{\frac{{a}^{6}b}{{x}^{7}}}-7\,{\frac{{a}^{2}{b}^{5}}{{x}^{3}}}-{\frac{{a}^{7}}{8\,{x}^{8}}}-{\frac{35\,{a}^{3}{b}^{4}}{4\,{x}^{4}}}-{\frac{7\,{a}^{5}{b}^{2}}{2\,{x}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^7/x^9,x)

[Out]

-7/2*a*b^6/x^2-7*a^4*b^3/x^5-b^7/x-a^6*b/x^7-7*a^2*b^5/x^3-1/8*a^7/x^8-35/4*a^3*
b^4/x^4-7/2*a^5*b^2/x^6

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Maxima [A]  time = 1.33515, size = 104, normalized size = 6.12 \[ -\frac{8 \, b^{7} x^{7} + 28 \, a b^{6} x^{6} + 56 \, a^{2} b^{5} x^{5} + 70 \, a^{3} b^{4} x^{4} + 56 \, a^{4} b^{3} x^{3} + 28 \, a^{5} b^{2} x^{2} + 8 \, a^{6} b x + a^{7}}{8 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^7/x^9,x, algorithm="maxima")

[Out]

-1/8*(8*b^7*x^7 + 28*a*b^6*x^6 + 56*a^2*b^5*x^5 + 70*a^3*b^4*x^4 + 56*a^4*b^3*x^
3 + 28*a^5*b^2*x^2 + 8*a^6*b*x + a^7)/x^8

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Fricas [A]  time = 0.192436, size = 104, normalized size = 6.12 \[ -\frac{8 \, b^{7} x^{7} + 28 \, a b^{6} x^{6} + 56 \, a^{2} b^{5} x^{5} + 70 \, a^{3} b^{4} x^{4} + 56 \, a^{4} b^{3} x^{3} + 28 \, a^{5} b^{2} x^{2} + 8 \, a^{6} b x + a^{7}}{8 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^7/x^9,x, algorithm="fricas")

[Out]

-1/8*(8*b^7*x^7 + 28*a*b^6*x^6 + 56*a^2*b^5*x^5 + 70*a^3*b^4*x^4 + 56*a^4*b^3*x^
3 + 28*a^5*b^2*x^2 + 8*a^6*b*x + a^7)/x^8

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Sympy [A]  time = 2.54978, size = 83, normalized size = 4.88 \[ - \frac{a^{7} + 8 a^{6} b x + 28 a^{5} b^{2} x^{2} + 56 a^{4} b^{3} x^{3} + 70 a^{3} b^{4} x^{4} + 56 a^{2} b^{5} x^{5} + 28 a b^{6} x^{6} + 8 b^{7} x^{7}}{8 x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**7/x**9,x)

[Out]

-(a**7 + 8*a**6*b*x + 28*a**5*b**2*x**2 + 56*a**4*b**3*x**3 + 70*a**3*b**4*x**4
+ 56*a**2*b**5*x**5 + 28*a*b**6*x**6 + 8*b**7*x**7)/(8*x**8)

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GIAC/XCAS [A]  time = 0.218573, size = 104, normalized size = 6.12 \[ -\frac{8 \, b^{7} x^{7} + 28 \, a b^{6} x^{6} + 56 \, a^{2} b^{5} x^{5} + 70 \, a^{3} b^{4} x^{4} + 56 \, a^{4} b^{3} x^{3} + 28 \, a^{5} b^{2} x^{2} + 8 \, a^{6} b x + a^{7}}{8 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^7/x^9,x, algorithm="giac")

[Out]

-1/8*(8*b^7*x^7 + 28*a*b^6*x^6 + 56*a^2*b^5*x^5 + 70*a^3*b^4*x^4 + 56*a^4*b^3*x^
3 + 28*a^5*b^2*x^2 + 8*a^6*b*x + a^7)/x^8