3.1566 \(\int \frac{\left (a+\frac{b}{x}\right )^2}{x^4} \, dx\)

Optimal. Leaf size=30 \[ -\frac{a^2}{3 x^3}-\frac{a b}{2 x^4}-\frac{b^2}{5 x^5} \]

[Out]

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

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Rubi [A]  time = 0.0350996, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{a^2}{3 x^3}-\frac{a b}{2 x^4}-\frac{b^2}{5 x^5} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^2/x^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 6.15237, size = 26, normalized size = 0.87 \[ - \frac{a^{2}}{3 x^{3}} - \frac{a b}{2 x^{4}} - \frac{b^{2}}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**2/x**4,x)

[Out]

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

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Mathematica [A]  time = 0.0109626, size = 30, normalized size = 1. \[ -\frac{a^2}{3 x^3}-\frac{a b}{2 x^4}-\frac{b^2}{5 x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^2/x^4,x]

[Out]

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

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Maple [A]  time = 0.007, size = 25, normalized size = 0.8 \[ -{\frac{{b}^{2}}{5\,{x}^{5}}}-{\frac{ab}{2\,{x}^{4}}}-{\frac{{a}^{2}}{3\,{x}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^2/x^4,x)

[Out]

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

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Maxima [A]  time = 1.44078, size = 32, normalized size = 1.07 \[ -\frac{10 \, a^{2} x^{2} + 15 \, a b x + 6 \, b^{2}}{30 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^2/x^4,x, algorithm="maxima")

[Out]

-1/30*(10*a^2*x^2 + 15*a*b*x + 6*b^2)/x^5

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Fricas [A]  time = 0.21789, size = 32, normalized size = 1.07 \[ -\frac{10 \, a^{2} x^{2} + 15 \, a b x + 6 \, b^{2}}{30 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^2/x^4,x, algorithm="fricas")

[Out]

-1/30*(10*a^2*x^2 + 15*a*b*x + 6*b^2)/x^5

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Sympy [A]  time = 1.29355, size = 26, normalized size = 0.87 \[ - \frac{10 a^{2} x^{2} + 15 a b x + 6 b^{2}}{30 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**2/x**4,x)

[Out]

-(10*a**2*x**2 + 15*a*b*x + 6*b**2)/(30*x**5)

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GIAC/XCAS [A]  time = 0.226533, size = 32, normalized size = 1.07 \[ -\frac{10 \, a^{2} x^{2} + 15 \, a b x + 6 \, b^{2}}{30 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^2/x^4,x, algorithm="giac")

[Out]

-1/30*(10*a^2*x^2 + 15*a*b*x + 6*b^2)/x^5