3.1992 \(\int \frac{\sqrt{a+\frac{b}{x^3}}}{x^4} \, dx\)

Optimal. Leaf size=18 \[ -\frac{2 \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b} \]

[Out]

(-2*(a + b/x^3)^(3/2))/(9*b)

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Rubi [A]  time = 0.029736, 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 \[ -\frac{2 \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x^3]/x^4,x]

[Out]

(-2*(a + b/x^3)^(3/2))/(9*b)

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Rubi in Sympy [A]  time = 2.1002, size = 15, normalized size = 0.83 \[ - \frac{2 \left (a + \frac{b}{x^{3}}\right )^{\frac{3}{2}}}{9 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**3)**(1/2)/x**4,x)

[Out]

-2*(a + b/x**3)**(3/2)/(9*b)

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Mathematica [A]  time = 0.033716, size = 18, normalized size = 1. \[ -\frac{2 \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b/x^3]/x^4,x]

[Out]

(-2*(a + b/x^3)^(3/2))/(9*b)

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Maple [A]  time = 0.008, size = 29, normalized size = 1.6 \[ -{\frac{2\,a{x}^{3}+2\,b}{9\,b{x}^{3}}\sqrt{{\frac{a{x}^{3}+b}{{x}^{3}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^3)^(1/2)/x^4,x)

[Out]

-2/9/x^3*(a*x^3+b)/b*((a*x^3+b)/x^3)^(1/2)

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Maxima [A]  time = 1.44299, size = 19, normalized size = 1.06 \[ -\frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}}}{9 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x^3)/x^4,x, algorithm="maxima")

[Out]

-2/9*(a + b/x^3)^(3/2)/b

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Fricas [A]  time = 0.243108, size = 38, normalized size = 2.11 \[ -\frac{2 \,{\left (a x^{3} + b\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{9 \, b x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x^3)/x^4,x, algorithm="fricas")

[Out]

-2/9*(a*x^3 + b)*sqrt((a*x^3 + b)/x^3)/(b*x^3)

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Sympy [A]  time = 4.29441, size = 46, normalized size = 2.56 \[ - \frac{2 a^{\frac{3}{2}} \sqrt{1 + \frac{b}{a x^{3}}}}{9 b} - \frac{2 \sqrt{a} \sqrt{1 + \frac{b}{a x^{3}}}}{9 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**3)**(1/2)/x**4,x)

[Out]

-2*a**(3/2)*sqrt(1 + b/(a*x**3))/(9*b) - 2*sqrt(a)*sqrt(1 + b/(a*x**3))/(9*x**3)

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GIAC/XCAS [A]  time = 0.237493, size = 19, normalized size = 1.06 \[ -\frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}}}{9 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a + b/x^3)/x^4,x, algorithm="giac")

[Out]

-2/9*(a + b/x^3)^(3/2)/b