3.1744 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^{5/2} x^2} \, dx\)

Optimal. Leaf size=18 \[ \frac{2}{3 b \left (a+\frac{b}{x}\right )^{3/2}} \]

[Out]

2/(3*b*(a + b/x)^(3/2))

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Rubi [A]  time = 0.0271003, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{2}{3 b \left (a+\frac{b}{x}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^(5/2)*x^2),x]

[Out]

2/(3*b*(a + b/x)^(3/2))

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Rubi in Sympy [A]  time = 2.20045, size = 12, normalized size = 0.67 \[ \frac{2}{3 b \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**(5/2)/x**2,x)

[Out]

2/(3*b*(a + b/x)**(3/2))

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Mathematica [A]  time = 0.0250144, size = 18, normalized size = 1. \[ \frac{2}{3 b \left (a+\frac{b}{x}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)^(5/2)*x^2),x]

[Out]

2/(3*b*(a + b/x)^(3/2))

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Maple [A]  time = 0.007, size = 25, normalized size = 1.4 \[{\frac{2\,ax+2\,b}{3\,bx} \left ({\frac{ax+b}{x}} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^(5/2)/x^2,x)

[Out]

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

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Maxima [A]  time = 1.43015, size = 19, normalized size = 1.06 \[ \frac{2}{3 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(5/2)*x^2),x, algorithm="maxima")

[Out]

2/3/((a + b/x)^(3/2)*b)

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Fricas [A]  time = 0.237275, size = 32, normalized size = 1.78 \[ \frac{2 \, x}{3 \,{\left (a b x + b^{2}\right )} \sqrt{\frac{a x + b}{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(5/2)*x^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 8.50829, size = 39, normalized size = 2.17 \[ \begin{cases} \frac{2}{3 a b \sqrt{a + \frac{b}{x}} + \frac{3 b^{2} \sqrt{a + \frac{b}{x}}}{x}} & \text{for}\: b \neq 0 \\- \frac{1}{a^{\frac{5}{2}} x} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)**(5/2)/x**2,x)

[Out]

Piecewise((2/(3*a*b*sqrt(a + b/x) + 3*b**2*sqrt(a + b/x)/x), Ne(b, 0)), (-1/(a**
(5/2)*x), True))

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GIAC/XCAS [A]  time = 0.273901, size = 19, normalized size = 1.06 \[ \frac{2}{3 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(5/2)*x^2),x, algorithm="giac")

[Out]

2/3/((a + b/x)^(3/2)*b)