3.1745 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^{5/2} x^3} \, dx\)

Optimal. Leaf size=36 \[ \frac{2}{b^2 \sqrt{a+\frac{b}{x}}}-\frac{2 a}{3 b^2 \left (a+\frac{b}{x}\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0575557, antiderivative size = 36, 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 \[ \frac{2}{b^2 \sqrt{a+\frac{b}{x}}}-\frac{2 a}{3 b^2 \left (a+\frac{b}{x}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^(5/2)*x^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 6.80363, size = 29, normalized size = 0.81 \[ - \frac{2 a}{3 b^{2} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} + \frac{2}{b^{2} \sqrt{a + \frac{b}{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**(5/2)/x**3,x)

[Out]

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

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Mathematica [A]  time = 0.0378262, size = 34, normalized size = 0.94 \[ \frac{2 x \sqrt{a+\frac{b}{x}} (2 a x+3 b)}{3 b^2 (a x+b)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)^(5/2)*x^3),x]

[Out]

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

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Maple [A]  time = 0.007, size = 33, normalized size = 0.9 \[{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 2\,ax+3\,b \right ) }{3\,{b}^{2}{x}^{2}} \left ({\frac{ax+b}{x}} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^(5/2)/x^3,x)

[Out]

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

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Maxima [A]  time = 1.43444, size = 41, normalized size = 1.14 \[ \frac{2}{\sqrt{a + \frac{b}{x}} b^{2}} - \frac{2 \, a}{3 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(5/2)*x^3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.233373, size = 45, normalized size = 1.25 \[ \frac{2 \,{\left (2 \, a x + 3 \, b\right )}}{3 \,{\left (a b^{2} x + b^{3}\right )} \sqrt{\frac{a x + b}{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(5/2)*x^3),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 11.2442, size = 82, normalized size = 2.28 \[ \begin{cases} \frac{4 a x}{3 a b^{2} x \sqrt{a + \frac{b}{x}} + 3 b^{3} \sqrt{a + \frac{b}{x}}} + \frac{6 b}{3 a b^{2} x \sqrt{a + \frac{b}{x}} + 3 b^{3} \sqrt{a + \frac{b}{x}}} & \text{for}\: b \neq 0 \\- \frac{1}{2 a^{\frac{5}{2}} x^{2}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)**(5/2)/x**3,x)

[Out]

Piecewise((4*a*x/(3*a*b**2*x*sqrt(a + b/x) + 3*b**3*sqrt(a + b/x)) + 6*b/(3*a*b*
*2*x*sqrt(a + b/x) + 3*b**3*sqrt(a + b/x)), Ne(b, 0)), (-1/(2*a**(5/2)*x**2), Tr
ue))

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GIAC/XCAS [A]  time = 0.269592, size = 49, normalized size = 1.36 \[ -\frac{2 \,{\left (a - \frac{3 \,{\left (a x + b\right )}}{x}\right )} x}{3 \,{\left (a x + b\right )} b^{2} \sqrt{\frac{a x + b}{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(5/2)*x^3),x, algorithm="giac")

[Out]

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