3.1650 \(\int \frac{a+\frac{b}{x}}{x^{5/2}} \, dx\)

Optimal. Leaf size=21 \[ -\frac{2 a}{3 x^{3/2}}-\frac{2 b}{5 x^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.014278, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{2 a}{3 x^{3/2}}-\frac{2 b}{5 x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)/x^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 2.83557, size = 20, normalized size = 0.95 \[ - \frac{2 a}{3 x^{\frac{3}{2}}} - \frac{2 b}{5 x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)/x**(5/2),x)

[Out]

-2*a/(3*x**(3/2)) - 2*b/(5*x**(5/2))

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Mathematica [A]  time = 0.00752632, size = 17, normalized size = 0.81 \[ -\frac{2 (5 a x+3 b)}{15 x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)/x^(5/2),x]

[Out]

(-2*(3*b + 5*a*x))/(15*x^(5/2))

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Maple [A]  time = 0.004, size = 14, normalized size = 0.7 \[ -{\frac{10\,ax+6\,b}{15}{x}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)/x^(5/2),x)

[Out]

-2/15*(5*a*x+3*b)/x^(5/2)

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Maxima [A]  time = 1.44323, size = 18, normalized size = 0.86 \[ -\frac{2 \, a}{3 \, x^{\frac{3}{2}}} - \frac{2 \, b}{5 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)/x^(5/2),x, algorithm="maxima")

[Out]

-2/3*a/x^(3/2) - 2/5*b/x^(5/2)

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Fricas [A]  time = 0.230219, size = 18, normalized size = 0.86 \[ -\frac{2 \,{\left (5 \, a x + 3 \, b\right )}}{15 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)/x^(5/2),x, algorithm="fricas")

[Out]

-2/15*(5*a*x + 3*b)/x^(5/2)

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Sympy [A]  time = 2.99475, size = 20, normalized size = 0.95 \[ - \frac{2 a}{3 x^{\frac{3}{2}}} - \frac{2 b}{5 x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)/x**(5/2),x)

[Out]

-2*a/(3*x**(3/2)) - 2*b/(5*x**(5/2))

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GIAC/XCAS [A]  time = 0.221512, size = 18, normalized size = 0.86 \[ -\frac{2 \,{\left (5 \, a x + 3 \, b\right )}}{15 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)/x^(5/2),x, algorithm="giac")

[Out]

-2/15*(5*a*x + 3*b)/x^(5/2)