3.1647 \(\int \left (a+\frac{b}{x}\right ) \sqrt{x} \, dx\)

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

[Out]

2*b*Sqrt[x] + (2*a*x^(3/2))/3

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Rubi [A]  time = 0.0136521, 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 \[ \frac{2}{3} a x^{3/2}+2 b \sqrt{x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)*Sqrt[x],x]

[Out]

2*b*Sqrt[x] + (2*a*x^(3/2))/3

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Rubi in Sympy [A]  time = 2.85467, size = 17, normalized size = 0.89 \[ \frac{2 a x^{\frac{3}{2}}}{3} + 2 b \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a*x**(3/2)/3 + 2*b*sqrt(x)

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Mathematica [A]  time = 0.00542563, size = 16, normalized size = 0.84 \[ \frac{2}{3} \sqrt{x} (a x+3 b) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)*Sqrt[x],x]

[Out]

(2*Sqrt[x]*(3*b + a*x))/3

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Maple [A]  time = 0.005, size = 13, normalized size = 0.7 \[{\frac{2\,ax+6\,b}{3}\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)*x^(1/2),x)

[Out]

2/3*(a*x+3*b)*x^(1/2)

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Maxima [A]  time = 1.43671, size = 18, normalized size = 0.95 \[ \frac{2}{3} \,{\left (a + \frac{3 \, b}{x}\right )} x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(a + 3*b/x)*x^(3/2)

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Fricas [A]  time = 0.224643, size = 16, normalized size = 0.84 \[ \frac{2}{3} \,{\left (a x + 3 \, b\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(a*x + 3*b)*sqrt(x)

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Sympy [A]  time = 0.545696, size = 17, normalized size = 0.89 \[ \frac{2 a x^{\frac{3}{2}}}{3} + 2 b \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)*x**(1/2),x)

[Out]

2*a*x**(3/2)/3 + 2*b*sqrt(x)

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GIAC/XCAS [A]  time = 0.222567, size = 18, normalized size = 0.95 \[ \frac{2}{3} \, a x^{\frac{3}{2}} + 2 \, b \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*a*x^(3/2) + 2*b*sqrt(x)