3.1658 \(\int \left (a+\frac{b}{x}\right )^3 x^{3/2} \, dx\)

Optimal. Leaf size=45 \[ \frac{2}{5} a^3 x^{5/2}+2 a^2 b x^{3/2}+6 a b^2 \sqrt{x}-\frac{2 b^3}{\sqrt{x}} \]

[Out]

(-2*b^3)/Sqrt[x] + 6*a*b^2*Sqrt[x] + 2*a^2*b*x^(3/2) + (2*a^3*x^(5/2))/5

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Rubi [A]  time = 0.0444258, antiderivative size = 45, 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 \[ \frac{2}{5} a^3 x^{5/2}+2 a^2 b x^{3/2}+6 a b^2 \sqrt{x}-\frac{2 b^3}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^3*x^(3/2),x]

[Out]

(-2*b^3)/Sqrt[x] + 6*a*b^2*Sqrt[x] + 2*a^2*b*x^(3/2) + (2*a^3*x^(5/2))/5

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Rubi in Sympy [A]  time = 6.92123, size = 44, normalized size = 0.98 \[ \frac{2 a^{3} x^{\frac{5}{2}}}{5} + 2 a^{2} b x^{\frac{3}{2}} + 6 a b^{2} \sqrt{x} - \frac{2 b^{3}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**3*x**(5/2)/5 + 2*a**2*b*x**(3/2) + 6*a*b**2*sqrt(x) - 2*b**3/sqrt(x)

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Mathematica [A]  time = 0.0139925, size = 38, normalized size = 0.84 \[ \frac{2 \left (a^3 x^3+5 a^2 b x^2+15 a b^2 x-5 b^3\right )}{5 \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 35, normalized size = 0.8 \[{\frac{2\,{a}^{3}{x}^{3}+10\,{a}^{2}b{x}^{2}+30\,a{b}^{2}x-10\,{b}^{3}}{5}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^3*x^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.4392, size = 49, normalized size = 1.09 \[ \frac{2}{5} \,{\left (a^{3} + \frac{5 \, a^{2} b}{x} + \frac{15 \, a b^{2}}{x^{2}}\right )} x^{\frac{5}{2}} - \frac{2 \, b^{3}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.228005, size = 46, normalized size = 1.02 \[ \frac{2 \,{\left (a^{3} x^{3} + 5 \, a^{2} b x^{2} + 15 \, a b^{2} x - 5 \, b^{3}\right )}}{5 \, \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 6.69417, size = 44, normalized size = 0.98 \[ \frac{2 a^{3} x^{\frac{5}{2}}}{5} + 2 a^{2} b x^{\frac{3}{2}} + 6 a b^{2} \sqrt{x} - \frac{2 b^{3}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**3*x**(5/2)/5 + 2*a**2*b*x**(3/2) + 6*a*b**2*sqrt(x) - 2*b**3/sqrt(x)

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GIAC/XCAS [A]  time = 0.22244, size = 47, normalized size = 1.04 \[ \frac{2}{5} \, a^{3} x^{\frac{5}{2}} + 2 \, a^{2} b x^{\frac{3}{2}} + 6 \, a b^{2} \sqrt{x} - \frac{2 \, b^{3}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*a^3*x^(5/2) + 2*a^2*b*x^(3/2) + 6*a*b^2*sqrt(x) - 2*b^3/sqrt(x)