3.1797 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^{5/2} x^{3/2}} \, dx\)

Optimal. Leaf size=46 \[ -\frac{4 b}{3 a^2 x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}}-\frac{2}{a \sqrt{x} \left (a+\frac{b}{x}\right )^{3/2}} \]

[Out]

(-4*b)/(3*a^2*(a + b/x)^(3/2)*x^(3/2)) - 2/(a*(a + b/x)^(3/2)*Sqrt[x])

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Rubi [A]  time = 0.0532577, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{4 b}{3 a^2 x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}}-\frac{2}{a \sqrt{x} \left (a+\frac{b}{x}\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-4*b)/(3*a^2*(a + b/x)^(3/2)*x^(3/2)) - 2/(a*(a + b/x)^(3/2)*Sqrt[x])

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Rubi in Sympy [A]  time = 4.40001, size = 39, normalized size = 0.85 \[ - \frac{2}{a \sqrt{x} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} - \frac{4 b}{3 a^{2} x^{\frac{3}{2}} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/(a*sqrt(x)*(a + b/x)**(3/2)) - 4*b/(3*a**2*x**(3/2)*(a + b/x)**(3/2))

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Mathematica [A]  time = 0.0467815, size = 38, normalized size = 0.83 \[ -\frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}} (3 a x+2 b)}{3 a^2 (a x+b)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 33, normalized size = 0.7 \[ -{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 3\,ax+2\,b \right ) }{3\,{a}^{2}}{x}^{-{\frac{5}{2}}} \left ({\frac{ax+b}{x}} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^(5/2)/x^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.44978, size = 42, normalized size = 0.91 \[ -\frac{2 \,{\left (3 \,{\left (a + \frac{b}{x}\right )} x - b\right )}}{3 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} a^{2} x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.23224, size = 50, normalized size = 1.09 \[ -\frac{2 \,{\left (3 \, a x + 2 \, b\right )}}{3 \,{\left (a^{3} x + a^{2} b\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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(1/(a+b/x)**(5/2)/x**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.242552, size = 39, normalized size = 0.85 \[ -\frac{2 \,{\left (3 \, a x + 2 \, b\right )}}{3 \,{\left (a x + b\right )}^{\frac{3}{2}} a^{2}} + \frac{4}{3 \, a^{2} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(3*a*x + 2*b)/((a*x + b)^(3/2)*a^2) + 4/3/(a^2*sqrt(b))