3.2115 \(\int \frac{a+b \sqrt{x}}{x^2} \, dx\)

Optimal. Leaf size=15 \[ -\frac{a}{x}-\frac{2 b}{\sqrt{x}} \]

[Out]

-(a/x) - (2*b)/Sqrt[x]

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Rubi [A]  time = 0.0164164, antiderivative size = 15, 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 \[ -\frac{a}{x}-\frac{2 b}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])/x^2,x]

[Out]

-(a/x) - (2*b)/Sqrt[x]

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Rubi in Sympy [A]  time = 2.81284, size = 12, normalized size = 0.8 \[ - \frac{a}{x} - \frac{2 b}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a/x - 2*b/sqrt(x)

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Mathematica [A]  time = 0.00598624, size = 15, normalized size = 1. \[ -\frac{a}{x}-\frac{2 b}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])/x^2,x]

[Out]

-(a/x) - (2*b)/Sqrt[x]

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Maple [A]  time = 0.002, size = 14, normalized size = 0.9 \[ -{\frac{a}{x}}-2\,{\frac{b}{\sqrt{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/2))/x^2,x)

[Out]

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

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Maxima [A]  time = 1.43654, size = 18, normalized size = 1.2 \[ -\frac{2 \, b \sqrt{x} + a}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*b*sqrt(x) + a)/x

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Fricas [A]  time = 0.232864, size = 18, normalized size = 1.2 \[ -\frac{2 \, b \sqrt{x} + a}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*b*sqrt(x) + a)/x

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Sympy [A]  time = 1.55826, size = 12, normalized size = 0.8 \[ - \frac{a}{x} - \frac{2 b}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a/x - 2*b/sqrt(x)

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GIAC/XCAS [A]  time = 0.216703, size = 18, normalized size = 1.2 \[ -\frac{2 \, b \sqrt{x} + a}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*b*sqrt(x) + a)/x