3.1646 \(\int \left (a+\frac{b}{x}\right ) x^{3/2} \, dx\)

Optimal. Leaf size=21 \[ \frac{2}{5} a x^{5/2}+\frac{2}{3} b x^{3/2} \]

[Out]

(2*b*x^(3/2))/3 + (2*a*x^(5/2))/5

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Rubi [A]  time = 0.0138204, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{2}{5} a x^{5/2}+\frac{2}{3} b x^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)*x^(3/2),x]

[Out]

(2*b*x^(3/2))/3 + (2*a*x^(5/2))/5

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Rubi in Sympy [A]  time = 2.78427, size = 19, normalized size = 0.9 \[ \frac{2 a x^{\frac{5}{2}}}{5} + \frac{2 b x^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a*x**(5/2)/5 + 2*b*x**(3/2)/3

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Mathematica [A]  time = 0.00625823, size = 17, normalized size = 0.81 \[ \frac{2}{15} x^{3/2} (3 a x+5 b) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)*x^(3/2),x]

[Out]

(2*x^(3/2)*(5*b + 3*a*x))/15

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Maple [A]  time = 0.003, size = 14, normalized size = 0.7 \[{\frac{6\,ax+10\,b}{15}{x}^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)*x^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.43811, size = 20, normalized size = 0.95 \[ \frac{2}{15} \,{\left (3 \, a + \frac{5 \, b}{x}\right )} x^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(3*a + 5*b/x)*x^(5/2)

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Fricas [A]  time = 0.227104, size = 22, normalized size = 1.05 \[ \frac{2}{15} \,{\left (3 \, a x^{2} + 5 \, b x\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(3*a*x^2 + 5*b*x)*sqrt(x)

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Sympy [A]  time = 1.83358, size = 19, normalized size = 0.9 \[ \frac{2 a x^{\frac{5}{2}}}{5} + \frac{2 b x^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)*x**(3/2),x)

[Out]

2*a*x**(5/2)/5 + 2*b*x**(3/2)/3

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GIAC/XCAS [A]  time = 0.229559, size = 18, normalized size = 0.86 \[ \frac{2}{5} \, a x^{\frac{5}{2}} + \frac{2}{3} \, b x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*a*x^(5/2) + 2/3*b*x^(3/2)