3.1649 \(\int \frac{a+\frac{b}{x}}{x^{3/2}} \, dx\)

Optimal. Leaf size=19 \[ -\frac{2 a}{\sqrt{x}}-\frac{2 b}{3 x^{3/2}} \]

[Out]

(-2*b)/(3*x^(3/2)) - (2*a)/Sqrt[x]

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Rubi [A]  time = 0.0135852, antiderivative size = 19, 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 \[ -\frac{2 a}{\sqrt{x}}-\frac{2 b}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)/x^(3/2),x]

[Out]

(-2*b)/(3*x^(3/2)) - (2*a)/Sqrt[x]

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Rubi in Sympy [A]  time = 2.82218, size = 19, normalized size = 1. \[ - \frac{2 a}{\sqrt{x}} - \frac{2 b}{3 x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)/x**(3/2),x)

[Out]

-2*a/sqrt(x) - 2*b/(3*x**(3/2))

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Mathematica [A]  time = 0.00700763, size = 15, normalized size = 0.79 \[ -\frac{2 (3 a x+b)}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)/x^(3/2),x]

[Out]

(-2*(b + 3*a*x))/(3*x^(3/2))

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Maple [A]  time = 0.003, size = 12, normalized size = 0.6 \[ -{\frac{6\,ax+2\,b}{3}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)/x^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.43532, size = 18, normalized size = 0.95 \[ -\frac{2 \, a}{\sqrt{x}} - \frac{2 \, b}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)/x^(3/2),x, algorithm="maxima")

[Out]

-2*a/sqrt(x) - 2/3*b/x^(3/2)

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Fricas [A]  time = 0.224405, size = 15, normalized size = 0.79 \[ -\frac{2 \,{\left (3 \, a x + b\right )}}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)/x^(3/2),x, algorithm="fricas")

[Out]

-2/3*(3*a*x + b)/x^(3/2)

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Sympy [A]  time = 1.81571, size = 19, normalized size = 1. \[ - \frac{2 a}{\sqrt{x}} - \frac{2 b}{3 x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)/x**(3/2),x)

[Out]

-2*a/sqrt(x) - 2*b/(3*x**(3/2))

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GIAC/XCAS [A]  time = 0.230573, size = 15, normalized size = 0.79 \[ -\frac{2 \,{\left (3 \, a x + b\right )}}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)/x^(3/2),x, algorithm="giac")

[Out]

-2/3*(3*a*x + b)/x^(3/2)