3.1761 \(\int \left (a+\frac{b}{x}\right )^{3/2} x^{3/2} \, dx\)

Optimal. Leaf size=23 \[ \frac{2 x^{5/2} \left (a+\frac{b}{x}\right )^{5/2}}{5 a} \]

[Out]

(2*(a + b/x)^(5/2)*x^(5/2))/(5*a)

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Rubi [A]  time = 0.0262079, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{2 x^{5/2} \left (a+\frac{b}{x}\right )^{5/2}}{5 a} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^(3/2)*x^(3/2),x]

[Out]

(2*(a + b/x)^(5/2)*x^(5/2))/(5*a)

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Rubi in Sympy [A]  time = 2.67117, size = 17, normalized size = 0.74 \[ \frac{2 x^{\frac{5}{2}} \left (a + \frac{b}{x}\right )^{\frac{5}{2}}}{5 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**(5/2)*(a + b/x)**(5/2)/(5*a)

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Mathematica [A]  time = 0.0373606, size = 30, normalized size = 1.3 \[ \frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}} (a x+b)^2}{5 a} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 25, normalized size = 1.1 \[{\frac{2\,ax+2\,b}{5\,a} \left ({\frac{ax+b}{x}} \right ) ^{{\frac{3}{2}}}{x}^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(3/2)*x^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.44039, size = 23, normalized size = 1. \[ \frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}} x^{\frac{5}{2}}}{5 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*(a + b/x)^(5/2)*x^(5/2)/a

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Fricas [A]  time = 0.241455, size = 47, normalized size = 2.04 \[ \frac{2 \,{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{5 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.226788, size = 16, normalized size = 0.7 \[ \frac{2 \,{\left (a x + b\right )}^{\frac{5}{2}}}{5 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*(a*x + b)^(5/2)/a