3.1600 \(\int \frac{\left (a+\frac{b}{x}\right )^8}{x^4} \, dx\)

Optimal. Leaf size=56 \[ -\frac{a^2 (a x+b)^9}{495 b^3 x^9}+\frac{a (a x+b)^9}{55 b^2 x^{10}}-\frac{(a x+b)^9}{11 b x^{11}} \]

[Out]

-(b + a*x)^9/(11*b*x^11) + (a*(b + a*x)^9)/(55*b^2*x^10) - (a^2*(b + a*x)^9)/(49
5*b^3*x^9)

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Rubi [A]  time = 0.0548057, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{a^2 (a x+b)^9}{495 b^3 x^9}+\frac{a (a x+b)^9}{55 b^2 x^{10}}-\frac{(a x+b)^9}{11 b x^{11}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^8/x^4,x]

[Out]

-(b + a*x)^9/(11*b*x^11) + (a*(b + a*x)^9)/(55*b^2*x^10) - (a^2*(b + a*x)^9)/(49
5*b^3*x^9)

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Rubi in Sympy [A]  time = 7.6067, size = 48, normalized size = 0.86 \[ - \frac{a^{2} \left (a x + b\right )^{9}}{495 b^{3} x^{9}} + \frac{a \left (a x + b\right )^{9}}{55 b^{2} x^{10}} - \frac{\left (a x + b\right )^{9}}{11 b x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**8/x**4,x)

[Out]

-a**2*(a*x + b)**9/(495*b**3*x**9) + a*(a*x + b)**9/(55*b**2*x**10) - (a*x + b)*
*9/(11*b*x**11)

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Mathematica [A]  time = 0.0137641, size = 102, normalized size = 1.82 \[ -\frac{a^8}{3 x^3}-\frac{2 a^7 b}{x^4}-\frac{28 a^6 b^2}{5 x^5}-\frac{28 a^5 b^3}{3 x^6}-\frac{10 a^4 b^4}{x^7}-\frac{7 a^3 b^5}{x^8}-\frac{28 a^2 b^6}{9 x^9}-\frac{4 a b^7}{5 x^{10}}-\frac{b^8}{11 x^{11}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^8/x^4,x]

[Out]

-b^8/(11*x^11) - (4*a*b^7)/(5*x^10) - (28*a^2*b^6)/(9*x^9) - (7*a^3*b^5)/x^8 - (
10*a^4*b^4)/x^7 - (28*a^5*b^3)/(3*x^6) - (28*a^6*b^2)/(5*x^5) - (2*a^7*b)/x^4 -
a^8/(3*x^3)

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Maple [A]  time = 0.009, size = 91, normalized size = 1.6 \[ -{\frac{28\,{a}^{5}{b}^{3}}{3\,{x}^{6}}}-2\,{\frac{{a}^{7}b}{{x}^{4}}}-{\frac{4\,a{b}^{7}}{5\,{x}^{10}}}-7\,{\frac{{a}^{3}{b}^{5}}{{x}^{8}}}-{\frac{{b}^{8}}{11\,{x}^{11}}}-{\frac{28\,{a}^{2}{b}^{6}}{9\,{x}^{9}}}-{\frac{{a}^{8}}{3\,{x}^{3}}}-{\frac{28\,{a}^{6}{b}^{2}}{5\,{x}^{5}}}-10\,{\frac{{a}^{4}{b}^{4}}{{x}^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^8/x^4,x)

[Out]

-28/3*a^5*b^3/x^6-2*a^7*b/x^4-4/5*a*b^7/x^10-7*a^3*b^5/x^8-1/11*b^8/x^11-28/9*a^
2*b^6/x^9-1/3*a^8/x^3-28/5*a^6*b^2/x^5-10*a^4*b^4/x^7

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Maxima [A]  time = 1.44453, size = 122, normalized size = 2.18 \[ -\frac{165 \, a^{8} x^{8} + 990 \, a^{7} b x^{7} + 2772 \, a^{6} b^{2} x^{6} + 4620 \, a^{5} b^{3} x^{5} + 4950 \, a^{4} b^{4} x^{4} + 3465 \, a^{3} b^{5} x^{3} + 1540 \, a^{2} b^{6} x^{2} + 396 \, a b^{7} x + 45 \, b^{8}}{495 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^8/x^4,x, algorithm="maxima")

[Out]

-1/495*(165*a^8*x^8 + 990*a^7*b*x^7 + 2772*a^6*b^2*x^6 + 4620*a^5*b^3*x^5 + 4950
*a^4*b^4*x^4 + 3465*a^3*b^5*x^3 + 1540*a^2*b^6*x^2 + 396*a*b^7*x + 45*b^8)/x^11

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Fricas [A]  time = 0.213238, size = 122, normalized size = 2.18 \[ -\frac{165 \, a^{8} x^{8} + 990 \, a^{7} b x^{7} + 2772 \, a^{6} b^{2} x^{6} + 4620 \, a^{5} b^{3} x^{5} + 4950 \, a^{4} b^{4} x^{4} + 3465 \, a^{3} b^{5} x^{3} + 1540 \, a^{2} b^{6} x^{2} + 396 \, a b^{7} x + 45 \, b^{8}}{495 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^8/x^4,x, algorithm="fricas")

[Out]

-1/495*(165*a^8*x^8 + 990*a^7*b*x^7 + 2772*a^6*b^2*x^6 + 4620*a^5*b^3*x^5 + 4950
*a^4*b^4*x^4 + 3465*a^3*b^5*x^3 + 1540*a^2*b^6*x^2 + 396*a*b^7*x + 45*b^8)/x^11

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Sympy [A]  time = 3.34295, size = 97, normalized size = 1.73 \[ - \frac{165 a^{8} x^{8} + 990 a^{7} b x^{7} + 2772 a^{6} b^{2} x^{6} + 4620 a^{5} b^{3} x^{5} + 4950 a^{4} b^{4} x^{4} + 3465 a^{3} b^{5} x^{3} + 1540 a^{2} b^{6} x^{2} + 396 a b^{7} x + 45 b^{8}}{495 x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**8/x**4,x)

[Out]

-(165*a**8*x**8 + 990*a**7*b*x**7 + 2772*a**6*b**2*x**6 + 4620*a**5*b**3*x**5 +
4950*a**4*b**4*x**4 + 3465*a**3*b**5*x**3 + 1540*a**2*b**6*x**2 + 396*a*b**7*x +
45*b**8)/(495*x**11)

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GIAC/XCAS [A]  time = 0.225963, size = 122, normalized size = 2.18 \[ -\frac{165 \, a^{8} x^{8} + 990 \, a^{7} b x^{7} + 2772 \, a^{6} b^{2} x^{6} + 4620 \, a^{5} b^{3} x^{5} + 4950 \, a^{4} b^{4} x^{4} + 3465 \, a^{3} b^{5} x^{3} + 1540 \, a^{2} b^{6} x^{2} + 396 \, a b^{7} x + 45 \, b^{8}}{495 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^8/x^4,x, algorithm="giac")

[Out]

-1/495*(165*a^8*x^8 + 990*a^7*b*x^7 + 2772*a^6*b^2*x^6 + 4620*a^5*b^3*x^5 + 4950
*a^4*b^4*x^4 + 3465*a^3*b^5*x^3 + 1540*a^2*b^6*x^2 + 396*a*b^7*x + 45*b^8)/x^11