3.2018 \(\int \frac{1}{\sqrt{a+\frac{b}{x^3}} x^4} \, dx\)

Optimal. Leaf size=18 \[ -\frac{2 \sqrt{a+\frac{b}{x^3}}}{3 b} \]

[Out]

(-2*Sqrt[a + b/x^3])/(3*b)

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Rubi [A]  time = 0.0295821, 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 \[ -\frac{2 \sqrt{a+\frac{b}{x^3}}}{3 b} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[a + b/x^3]*x^4),x]

[Out]

(-2*Sqrt[a + b/x^3])/(3*b)

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Rubi in Sympy [A]  time = 2.11858, size = 15, normalized size = 0.83 \[ - \frac{2 \sqrt{a + \frac{b}{x^{3}}}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(a + b/x**3)/(3*b)

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Mathematica [A]  time = 0.0210587, size = 18, normalized size = 1. \[ -\frac{2 \sqrt{a+\frac{b}{x^3}}}{3 b} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[a + b/x^3]*x^4),x]

[Out]

(-2*Sqrt[a + b/x^3])/(3*b)

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Maple [A]  time = 0.007, size = 29, normalized size = 1.6 \[ -{\frac{2\,a{x}^{3}+2\,b}{3\,b{x}^{3}}{\frac{1}{\sqrt{{\frac{a{x}^{3}+b}{{x}^{3}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(a+b/x^3)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.42252, size = 19, normalized size = 1.06 \[ -\frac{2 \, \sqrt{a + \frac{b}{x^{3}}}}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x^3)*x^4),x, algorithm="maxima")

[Out]

-2/3*sqrt(a + b/x^3)/b

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Fricas [A]  time = 0.241184, size = 24, normalized size = 1.33 \[ -\frac{2 \, \sqrt{\frac{a x^{3} + b}{x^{3}}}}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x^3)*x^4),x, algorithm="fricas")

[Out]

-2/3*sqrt((a*x^3 + b)/x^3)/b

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Sympy [A]  time = 6.28361, size = 29, normalized size = 1.61 \[ \begin{cases} - \frac{2 \sqrt{a + \frac{b}{x^{3}}}}{3 b} & \text{for}\: b \neq 0 \\- \frac{1}{3 \sqrt{a} x^{3}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.224881, size = 19, normalized size = 1.06 \[ -\frac{2 \, \sqrt{a + \frac{b}{x^{3}}}}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x^3)*x^4),x, algorithm="giac")

[Out]

-2/3*sqrt(a + b/x^3)/b