3.1786 \(\int \frac{\sqrt{x}}{\left (a+\frac{b}{x}\right )^{3/2}} \, dx\)

Optimal. Leaf size=74 \[ -\frac{16 b^2}{3 a^3 \sqrt{x} \sqrt{a+\frac{b}{x}}}-\frac{8 b \sqrt{x}}{3 a^2 \sqrt{a+\frac{b}{x}}}+\frac{2 x^{3/2}}{3 a \sqrt{a+\frac{b}{x}}} \]

[Out]

(-16*b^2)/(3*a^3*Sqrt[a + b/x]*Sqrt[x]) - (8*b*Sqrt[x])/(3*a^2*Sqrt[a + b/x]) +
(2*x^(3/2))/(3*a*Sqrt[a + b/x])

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Rubi [A]  time = 0.0818078, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{16 b^2}{3 a^3 \sqrt{x} \sqrt{a+\frac{b}{x}}}-\frac{8 b \sqrt{x}}{3 a^2 \sqrt{a+\frac{b}{x}}}+\frac{2 x^{3/2}}{3 a \sqrt{a+\frac{b}{x}}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]/(a + b/x)^(3/2),x]

[Out]

(-16*b^2)/(3*a^3*Sqrt[a + b/x]*Sqrt[x]) - (8*b*Sqrt[x])/(3*a^2*Sqrt[a + b/x]) +
(2*x^(3/2))/(3*a*Sqrt[a + b/x])

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Rubi in Sympy [A]  time = 6.73129, size = 63, normalized size = 0.85 \[ \frac{2 x^{\frac{3}{2}}}{3 a \sqrt{a + \frac{b}{x}}} - \frac{8 b \sqrt{x}}{3 a^{2} \sqrt{a + \frac{b}{x}}} - \frac{16 b^{2}}{3 a^{3} \sqrt{x} \sqrt{a + \frac{b}{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(1/2)/(a+b/x)**(3/2),x)

[Out]

2*x**(3/2)/(3*a*sqrt(a + b/x)) - 8*b*sqrt(x)/(3*a**2*sqrt(a + b/x)) - 16*b**2/(3
*a**3*sqrt(x)*sqrt(a + b/x))

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Mathematica [A]  time = 0.0510546, size = 48, normalized size = 0.65 \[ \frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}} \left (a^2 x^2-4 a b x-8 b^2\right )}{3 a^3 (a x+b)} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x]/(a + b/x)^(3/2),x]

[Out]

(2*Sqrt[a + b/x]*Sqrt[x]*(-8*b^2 - 4*a*b*x + a^2*x^2))/(3*a^3*(b + a*x))

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Maple [A]  time = 0.007, size = 43, normalized size = 0.6 \[{\frac{ \left ( 2\,ax+2\,b \right ) \left ({a}^{2}{x}^{2}-4\,abx-8\,{b}^{2} \right ) }{3\,{a}^{3}}{x}^{-{\frac{3}{2}}} \left ({\frac{ax+b}{x}} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)/(a+b/x)^(3/2),x)

[Out]

2/3*(a*x+b)*(a^2*x^2-4*a*b*x-8*b^2)/a^3/x^(3/2)/((a*x+b)/x)^(3/2)

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Maxima [A]  time = 1.44394, size = 74, normalized size = 1. \[ \frac{2 \,{\left ({\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} x^{\frac{3}{2}} - 6 \, \sqrt{a + \frac{b}{x}} b \sqrt{x}\right )}}{3 \, a^{3}} - \frac{2 \, b^{2}}{\sqrt{a + \frac{b}{x}} a^{3} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*((a + b/x)^(3/2)*x^(3/2) - 6*sqrt(a + b/x)*b*sqrt(x))/a^3 - 2*b^2/(sqrt(a +
b/x)*a^3*sqrt(x))

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Fricas [A]  time = 0.227952, size = 50, normalized size = 0.68 \[ \frac{2 \,{\left (a^{2} x^{2} - 4 \, a b x - 8 \, b^{2}\right )}}{3 \, a^{3} \sqrt{x} \sqrt{\frac{a x + b}{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(a^2*x^2 - 4*a*b*x - 8*b^2)/(a^3*sqrt(x)*sqrt((a*x + b)/x))

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Sympy [A]  time = 16.802, size = 206, normalized size = 2.78 \[ \frac{2 a^{3} b^{\frac{9}{2}} x^{3} \sqrt{\frac{a x}{b} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x + 3 a^{3} b^{6}} - \frac{6 a^{2} b^{\frac{11}{2}} x^{2} \sqrt{\frac{a x}{b} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x + 3 a^{3} b^{6}} - \frac{24 a b^{\frac{13}{2}} x \sqrt{\frac{a x}{b} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x + 3 a^{3} b^{6}} - \frac{16 b^{\frac{15}{2}} \sqrt{\frac{a x}{b} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x + 3 a^{3} b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(1/2)/(a+b/x)**(3/2),x)

[Out]

2*a**3*b**(9/2)*x**3*sqrt(a*x/b + 1)/(3*a**5*b**4*x**2 + 6*a**4*b**5*x + 3*a**3*
b**6) - 6*a**2*b**(11/2)*x**2*sqrt(a*x/b + 1)/(3*a**5*b**4*x**2 + 6*a**4*b**5*x
+ 3*a**3*b**6) - 24*a*b**(13/2)*x*sqrt(a*x/b + 1)/(3*a**5*b**4*x**2 + 6*a**4*b**
5*x + 3*a**3*b**6) - 16*b**(15/2)*sqrt(a*x/b + 1)/(3*a**5*b**4*x**2 + 6*a**4*b**
5*x + 3*a**3*b**6)

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GIAC/XCAS [A]  time = 0.234486, size = 59, normalized size = 0.8 \[ \frac{16 \, b^{\frac{3}{2}}}{3 \, a^{3}} + \frac{2 \,{\left ({\left (a x + b\right )}^{\frac{3}{2}} - 6 \, \sqrt{a x + b} b - \frac{3 \, b^{2}}{\sqrt{a x + b}}\right )}}{3 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

16/3*b^(3/2)/a^3 + 2/3*((a*x + b)^(3/2) - 6*sqrt(a*x + b)*b - 3*b^2/sqrt(a*x + b
))/a^3