3.1752 \(\int \sqrt{a+\frac{b}{x}} x^{3/2} \, dx\)

Optimal. Leaf size=48 \[ \frac{2 x^{5/2} \left (a+\frac{b}{x}\right )^{3/2}}{5 a}-\frac{4 b x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}}{15 a^2} \]

[Out]

(-4*b*(a + b/x)^(3/2)*x^(3/2))/(15*a^2) + (2*(a + b/x)^(3/2)*x^(5/2))/(5*a)

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Rubi [A]  time = 0.051524, 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 \[ \frac{2 x^{5/2} \left (a+\frac{b}{x}\right )^{3/2}}{5 a}-\frac{4 b x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}}{15 a^2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x]*x^(3/2),x]

[Out]

(-4*b*(a + b/x)^(3/2)*x^(3/2))/(15*a^2) + (2*(a + b/x)^(3/2)*x^(5/2))/(5*a)

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Rubi in Sympy [A]  time = 4.21269, size = 39, normalized size = 0.81 \[ \frac{2 x^{\frac{5}{2}} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{5 a} - \frac{4 b x^{\frac{3}{2}} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{15 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**(5/2)*(a + b/x)**(3/2)/(5*a) - 4*b*x**(3/2)*(a + b/x)**(3/2)/(15*a**2)

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Mathematica [A]  time = 0.0344929, size = 41, normalized size = 0.85 \[ \frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}} \left (3 a^2 x^2+a b x-2 b^2\right )}{15 a^2} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[a + b/x]*Sqrt[x]*(-2*b^2 + a*b*x + 3*a^2*x^2))/(15*a^2)

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Maple [A]  time = 0.004, size = 33, normalized size = 0.7 \[{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 3\,ax-2\,b \right ) }{15\,{a}^{2}}\sqrt{{\frac{ax+b}{x}}}\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(1/2)*x^(3/2),x)

[Out]

2/15*(a*x+b)*(3*a*x-2*b)*x^(1/2)*((a*x+b)/x)^(1/2)/a^2

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Maxima [A]  time = 1.45693, size = 47, normalized size = 0.98 \[ \frac{2 \,{\left (3 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}} x^{\frac{5}{2}} - 5 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} b x^{\frac{3}{2}}\right )}}{15 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(3*(a + b/x)^(5/2)*x^(5/2) - 5*(a + b/x)^(3/2)*b*x^(3/2))/a^2

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Fricas [A]  time = 0.237594, size = 50, normalized size = 1.04 \[ \frac{2 \,{\left (3 \, a^{2} x^{2} + a b x - 2 \, b^{2}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{15 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(3*a^2*x^2 + a*b*x - 2*b^2)*sqrt(x)*sqrt((a*x + b)/x)/a^2

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Sympy [A]  time = 91.0332, size = 65, normalized size = 1.35 \[ \frac{2 \sqrt{b} x^{2} \sqrt{\frac{a x}{b} + 1}}{5} + \frac{2 b^{\frac{3}{2}} x \sqrt{\frac{a x}{b} + 1}}{15 a} - \frac{4 b^{\frac{5}{2}} \sqrt{\frac{a x}{b} + 1}}{15 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(b)*x**2*sqrt(a*x/b + 1)/5 + 2*b**(3/2)*x*sqrt(a*x/b + 1)/(15*a) - 4*b**(5
/2)*sqrt(a*x/b + 1)/(15*a**2)

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GIAC/XCAS [A]  time = 0.234291, size = 50, normalized size = 1.04 \[ \frac{2}{15} \,{\left (\frac{2 \, b^{\frac{5}{2}}}{a^{2}} + \frac{3 \,{\left (a x + b\right )}^{\frac{5}{2}} - 5 \,{\left (a x + b\right )}^{\frac{3}{2}} b}{a^{2}}\right )}{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(2*b^(5/2)/a^2 + (3*(a*x + b)^(5/2) - 5*(a*x + b)^(3/2)*b)/a^2)*sign(x)