3.1660 \(\int \frac{\left (a+\frac{b}{x}\right )^3}{\sqrt{x}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0437228, 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 \[ 2 a^3 \sqrt{x}-\frac{6 a^2 b}{\sqrt{x}}-\frac{2 a b^2}{x^{3/2}}-\frac{2 b^3}{5 x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^3/Sqrt[x],x]

[Out]

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

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Rubi in Sympy [A]  time = 6.88763, size = 44, normalized size = 0.98 \[ 2 a^{3} \sqrt{x} - \frac{6 a^{2} b}{\sqrt{x}} - \frac{2 a b^{2}}{x^{\frac{3}{2}}} - \frac{2 b^{3}}{5 x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0148408, size = 39, normalized size = 0.87 \[ \frac{2 \left (5 a^3 x^3-15 a^2 b x^2-5 a b^2 x-b^3\right )}{5 x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 36, normalized size = 0.8 \[{\frac{10\,{a}^{3}{x}^{3}-30\,{a}^{2}b{x}^{2}-10\,a{b}^{2}x-2\,{b}^{3}}{5}{x}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^3/x^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.43719, size = 47, normalized size = 1.04 \[ 2 \, a^{3} \sqrt{x} - \frac{6 \, a^{2} b}{\sqrt{x}} - \frac{2 \, a b^{2}}{x^{\frac{3}{2}}} - \frac{2 \, b^{3}}{5 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.225973, size = 47, normalized size = 1.04 \[ \frac{2 \,{\left (5 \, a^{3} x^{3} - 15 \, a^{2} b x^{2} - 5 \, a b^{2} x - b^{3}\right )}}{5 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.6717, size = 44, normalized size = 0.98 \[ 2 a^{3} \sqrt{x} - \frac{6 a^{2} b}{\sqrt{x}} - \frac{2 a b^{2}}{x^{\frac{3}{2}}} - \frac{2 b^{3}}{5 x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.222939, size = 46, normalized size = 1.02 \[ 2 \, a^{3} \sqrt{x} - \frac{2 \,{\left (15 \, a^{2} b x^{2} + 5 \, a b^{2} x + b^{3}\right )}}{5 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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