3.1778 \(\int \frac{\sqrt{x}}{\sqrt{a+\frac{b}{x}}} \, dx\)

Optimal. Leaf size=48 \[ \frac{2 x^{3/2} \sqrt{a+\frac{b}{x}}}{3 a}-\frac{4 b \sqrt{x} \sqrt{a+\frac{b}{x}}}{3 a^2} \]

[Out]

(-4*b*Sqrt[a + b/x]*Sqrt[x])/(3*a^2) + (2*Sqrt[a + b/x]*x^(3/2))/(3*a)

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Rubi [A]  time = 0.0518068, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{2 x^{3/2} \sqrt{a+\frac{b}{x}}}{3 a}-\frac{4 b \sqrt{x} \sqrt{a+\frac{b}{x}}}{3 a^2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]/Sqrt[a + b/x],x]

[Out]

(-4*b*Sqrt[a + b/x]*Sqrt[x])/(3*a^2) + (2*Sqrt[a + b/x]*x^(3/2))/(3*a)

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Rubi in Sympy [A]  time = 4.36381, size = 39, normalized size = 0.81 \[ \frac{2 x^{\frac{3}{2}} \sqrt{a + \frac{b}{x}}}{3 a} - \frac{4 b \sqrt{x} \sqrt{a + \frac{b}{x}}}{3 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**(3/2)*sqrt(a + b/x)/(3*a) - 4*b*sqrt(x)*sqrt(a + b/x)/(3*a**2)

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Mathematica [A]  time = 0.0356013, size = 30, normalized size = 0.62 \[ \frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}} (a x-2 b)}{3 a^2} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x]/Sqrt[a + b/x],x]

[Out]

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

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Maple [A]  time = 0.003, size = 32, normalized size = 0.7 \[{\frac{ \left ( 2\,ax+2\,b \right ) \left ( ax-2\,b \right ) }{3\,{a}^{2}}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{{\frac{ax+b}{x}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)/(a+b/x)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.44007, size = 46, normalized size = 0.96 \[ \frac{2 \,{\left ({\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} x^{\frac{3}{2}} - 3 \, \sqrt{a + \frac{b}{x}} b \sqrt{x}\right )}}{3 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.231161, size = 35, normalized size = 0.73 \[ \frac{2 \,{\left (a x - 2 \, b\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{3 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 5.74571, size = 42, normalized size = 0.88 \[ \frac{2 \sqrt{b} x \sqrt{\frac{a x}{b} + 1}}{3 a} - \frac{4 b^{\frac{3}{2}} \sqrt{\frac{a x}{b} + 1}}{3 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(b)*x*sqrt(a*x/b + 1)/(3*a) - 4*b**(3/2)*sqrt(a*x/b + 1)/(3*a**2)

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GIAC/XCAS [A]  time = 0.227837, size = 43, normalized size = 0.9 \[ \frac{4 \, b^{\frac{3}{2}}}{3 \, a^{2}} + \frac{2 \,{\left ({\left (a x + b\right )}^{\frac{3}{2}} - 3 \, \sqrt{a x + b} b\right )}}{3 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/3*b^(3/2)/a^2 + 2/3*((a*x + b)^(3/2) - 3*sqrt(a*x + b)*b)/a^2