3.1798 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^{5/2} x^{5/2}} \, dx\)

Optimal. Leaf size=23 \[ -\frac{2}{3 a x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}} \]

[Out]

-2/(3*a*(a + b/x)^(3/2)*x^(3/2))

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Rubi [A]  time = 0.0261001, 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 \[ -\frac{2}{3 a x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-2/(3*a*(a + b/x)^(3/2)*x^(3/2))

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Rubi in Sympy [A]  time = 2.66811, size = 19, normalized size = 0.83 \[ - \frac{2}{3 a x^{\frac{3}{2}} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/(3*a*x**(3/2)*(a + b/x)**(3/2))

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Mathematica [A]  time = 0.0361117, size = 32, normalized size = 1.39 \[ -\frac{2 \sqrt{x} \sqrt{\frac{a x+b}{x}}}{3 a (a x+b)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 25, normalized size = 1.1 \[ -{\frac{2\,ax+2\,b}{3\,a} \left ({\frac{ax+b}{x}} \right ) ^{-{\frac{5}{2}}}{x}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^(5/2)/x^(5/2),x)

[Out]

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

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Maxima [A]  time = 1.45287, size = 23, normalized size = 1. \[ -\frac{2}{3 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} a x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3/((a + b/x)^(3/2)*a*x^(3/2))

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Fricas [A]  time = 0.233572, size = 36, normalized size = 1.57 \[ -\frac{2}{3 \,{\left (a^{2} x + a b\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3/((a^2*x + a*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**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.228608, size = 28, normalized size = 1.22 \[ -\frac{2}{3 \,{\left (a x + b\right )}^{\frac{3}{2}} a} + \frac{2}{3 \, a b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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