3.1652 \(\int \left (a+\frac{b}{x}\right )^2 x^{3/2} \, dx\)

Optimal. Leaf size=34 \[ \frac{2}{5} a^2 x^{5/2}+\frac{4}{3} a b x^{3/2}+2 b^2 \sqrt{x} \]

[Out]

2*b^2*Sqrt[x] + (4*a*b*x^(3/2))/3 + (2*a^2*x^(5/2))/5

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Rubi [A]  time = 0.0335745, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{2}{5} a^2 x^{5/2}+\frac{4}{3} a b x^{3/2}+2 b^2 \sqrt{x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^2*x^(3/2),x]

[Out]

2*b^2*Sqrt[x] + (4*a*b*x^(3/2))/3 + (2*a^2*x^(5/2))/5

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Rubi in Sympy [A]  time = 5.45849, size = 32, normalized size = 0.94 \[ \frac{2 a^{2} x^{\frac{5}{2}}}{5} + \frac{4 a b x^{\frac{3}{2}}}{3} + 2 b^{2} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.010315, size = 28, normalized size = 0.82 \[ \frac{2}{15} \sqrt{x} \left (3 a^2 x^2+10 a b x+15 b^2\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[x]*(15*b^2 + 10*a*b*x + 3*a^2*x^2))/15

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Maple [A]  time = 0.006, size = 25, normalized size = 0.7 \[{\frac{6\,{a}^{2}{x}^{2}+20\,abx+30\,{b}^{2}}{15}\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^2*x^(3/2),x)

[Out]

2/15*(3*a^2*x^2+10*a*b*x+15*b^2)*x^(1/2)

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Maxima [A]  time = 1.4503, size = 35, normalized size = 1.03 \[ \frac{2}{15} \,{\left (3 \, a^{2} + \frac{10 \, a b}{x} + \frac{15 \, b^{2}}{x^{2}}\right )} x^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(3*a^2 + 10*a*b/x + 15*b^2/x^2)*x^(5/2)

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Fricas [A]  time = 0.227482, size = 32, normalized size = 0.94 \[ \frac{2}{15} \,{\left (3 \, a^{2} x^{2} + 10 \, a b x + 15 \, b^{2}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(3*a^2*x^2 + 10*a*b*x + 15*b^2)*sqrt(x)

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Sympy [A]  time = 3.29041, size = 32, normalized size = 0.94 \[ \frac{2 a^{2} x^{\frac{5}{2}}}{5} + \frac{4 a b x^{\frac{3}{2}}}{3} + 2 b^{2} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.228209, size = 32, normalized size = 0.94 \[ \frac{2}{5} \, a^{2} x^{\frac{5}{2}} + \frac{4}{3} \, a b x^{\frac{3}{2}} + 2 \, b^{2} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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