3.1657 \(\int \left (a+\frac{b}{x}\right )^3 x^{5/2} \, dx\)

Optimal. Leaf size=47 \[ \frac{2}{7} a^3 x^{7/2}+\frac{6}{5} a^2 b x^{5/2}+2 a b^2 x^{3/2}+2 b^3 \sqrt{x} \]

[Out]

2*b^3*Sqrt[x] + 2*a*b^2*x^(3/2) + (6*a^2*b*x^(5/2))/5 + (2*a^3*x^(7/2))/7

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Rubi [A]  time = 0.0454168, antiderivative size = 47, 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 \[ \frac{2}{7} a^3 x^{7/2}+\frac{6}{5} a^2 b x^{5/2}+2 a b^2 x^{3/2}+2 b^3 \sqrt{x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^3*x^(5/2),x]

[Out]

2*b^3*Sqrt[x] + 2*a*b^2*x^(3/2) + (6*a^2*b*x^(5/2))/5 + (2*a^3*x^(7/2))/7

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Rubi in Sympy [A]  time = 6.84821, size = 46, normalized size = 0.98 \[ \frac{2 a^{3} x^{\frac{7}{2}}}{7} + \frac{6 a^{2} b x^{\frac{5}{2}}}{5} + 2 a b^{2} x^{\frac{3}{2}} + 2 b^{3} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**3*x**(5/2),x)

[Out]

2*a**3*x**(7/2)/7 + 6*a**2*b*x**(5/2)/5 + 2*a*b**2*x**(3/2) + 2*b**3*sqrt(x)

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Mathematica [A]  time = 0.013087, size = 39, normalized size = 0.83 \[ \frac{2}{35} \sqrt{x} \left (5 a^3 x^3+21 a^2 b x^2+35 a b^2 x+35 b^3\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^3*x^(5/2),x]

[Out]

(2*Sqrt[x]*(35*b^3 + 35*a*b^2*x + 21*a^2*b*x^2 + 5*a^3*x^3))/35

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Maple [A]  time = 0.007, size = 36, normalized size = 0.8 \[{\frac{10\,{a}^{3}{x}^{3}+42\,{a}^{2}b{x}^{2}+70\,a{b}^{2}x+70\,{b}^{3}}{35}\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^3*x^(5/2),x)

[Out]

2/35*(5*a^3*x^3+21*a^2*b*x^2+35*a*b^2*x+35*b^3)*x^(1/2)

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Maxima [A]  time = 1.44526, size = 50, normalized size = 1.06 \[ \frac{2}{35} \,{\left (5 \, a^{3} + \frac{21 \, a^{2} b}{x} + \frac{35 \, a b^{2}}{x^{2}} + \frac{35 \, b^{3}}{x^{3}}\right )} x^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^3*x^(5/2),x, algorithm="maxima")

[Out]

2/35*(5*a^3 + 21*a^2*b/x + 35*a*b^2/x^2 + 35*b^3/x^3)*x^(7/2)

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Fricas [A]  time = 0.228466, size = 47, normalized size = 1. \[ \frac{2}{35} \,{\left (5 \, a^{3} x^{3} + 21 \, a^{2} b x^{2} + 35 \, a b^{2} x + 35 \, b^{3}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^3*x^(5/2),x, algorithm="fricas")

[Out]

2/35*(5*a^3*x^3 + 21*a^2*b*x^2 + 35*a*b^2*x + 35*b^3)*sqrt(x)

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Sympy [A]  time = 14.8014, size = 46, normalized size = 0.98 \[ \frac{2 a^{3} x^{\frac{7}{2}}}{7} + \frac{6 a^{2} b x^{\frac{5}{2}}}{5} + 2 a b^{2} x^{\frac{3}{2}} + 2 b^{3} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**3*x**(5/2),x)

[Out]

2*a**3*x**(7/2)/7 + 6*a**2*b*x**(5/2)/5 + 2*a*b**2*x**(3/2) + 2*b**3*sqrt(x)

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GIAC/XCAS [A]  time = 0.223068, size = 47, normalized size = 1. \[ \frac{2}{7} \, a^{3} x^{\frac{7}{2}} + \frac{6}{5} \, a^{2} b x^{\frac{5}{2}} + 2 \, a b^{2} x^{\frac{3}{2}} + 2 \, b^{3} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^3*x^(5/2),x, algorithm="giac")

[Out]

2/7*a^3*x^(7/2) + 6/5*a^2*b*x^(5/2) + 2*a*b^2*x^(3/2) + 2*b^3*sqrt(x)