3.1648 \(\int \frac{a+\frac{b}{x}}{\sqrt{x}} \, dx\)

Optimal. Leaf size=17 \[ 2 a \sqrt{x}-\frac{2 b}{\sqrt{x}} \]

[Out]

(-2*b)/Sqrt[x] + 2*a*Sqrt[x]

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Rubi [A]  time = 0.0140354, antiderivative size = 17, 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 \[ 2 a \sqrt{x}-\frac{2 b}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)/Sqrt[x],x]

[Out]

(-2*b)/Sqrt[x] + 2*a*Sqrt[x]

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Rubi in Sympy [A]  time = 2.83582, size = 15, normalized size = 0.88 \[ 2 a \sqrt{x} - \frac{2 b}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)/x**(1/2),x)

[Out]

2*a*sqrt(x) - 2*b/sqrt(x)

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Mathematica [A]  time = 0.00744568, size = 14, normalized size = 0.82 \[ \frac{2 (a x-b)}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)/Sqrt[x],x]

[Out]

(2*(-b + a*x))/Sqrt[x]

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Maple [A]  time = 0.004, size = 13, normalized size = 0.8 \[ 2\,{\frac{ax-b}{\sqrt{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)/x^(1/2),x)

[Out]

2*(a*x-b)/x^(1/2)

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Maxima [A]  time = 1.43797, size = 18, normalized size = 1.06 \[ 2 \, a \sqrt{x} - \frac{2 \, b}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a*sqrt(x) - 2*b/sqrt(x)

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Fricas [A]  time = 0.22516, size = 16, normalized size = 0.94 \[ \frac{2 \,{\left (a x - b\right )}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(a*x - b)/sqrt(x)

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Sympy [A]  time = 1.3605, size = 15, normalized size = 0.88 \[ 2 a \sqrt{x} - \frac{2 b}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)/x**(1/2),x)

[Out]

2*a*sqrt(x) - 2*b/sqrt(x)

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GIAC/XCAS [A]  time = 0.21857, size = 18, normalized size = 1.06 \[ 2 \, a \sqrt{x} - \frac{2 \, b}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a*sqrt(x) - 2*b/sqrt(x)