3.2038 \(\int \frac{1}{\left (a+\frac{b}{x^3}\right )^{3/2} x^4} \, dx\)

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

[Out]

2/(3*b*Sqrt[a + b/x^3])

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Rubi [A]  time = 0.0302301, 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 \[ \frac{2}{3 b \sqrt{a+\frac{b}{x^3}}} \]

Antiderivative was successfully verified.

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

[Out]

2/(3*b*Sqrt[a + b/x^3])

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Rubi in Sympy [A]  time = 2.12479, size = 14, normalized size = 0.78 \[ \frac{2}{3 b \sqrt{a + \frac{b}{x^{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/(3*b*sqrt(a + b/x**3))

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Mathematica [A]  time = 0.0202149, size = 18, normalized size = 1. \[ \frac{2}{3 b \sqrt{a+\frac{b}{x^3}}} \]

Antiderivative was successfully verified.

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

[Out]

2/(3*b*Sqrt[a + b/x^3])

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Maple [A]  time = 0.01, size = 29, normalized size = 1.6 \[{\frac{2\,a{x}^{3}+2\,b}{3\,b{x}^{3}} \left ({\frac{a{x}^{3}+b}{{x}^{3}}} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^3)^(3/2)/x^4,x)

[Out]

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

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Maxima [A]  time = 1.43742, size = 19, normalized size = 1.06 \[ \frac{2}{3 \, \sqrt{a + \frac{b}{x^{3}}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^3)^(3/2)*x^4),x, algorithm="maxima")

[Out]

2/3/(sqrt(a + b/x^3)*b)

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Fricas [A]  time = 0.252417, size = 41, normalized size = 2.28 \[ \frac{2 \, x^{3} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{3 \,{\left (a b x^{3} + b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^3)^(3/2)*x^4),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 10.9752, size = 27, normalized size = 1.5 \[ \begin{cases} \frac{2}{3 b \sqrt{a + \frac{b}{x^{3}}}} & \text{for}\: b \neq 0 \\- \frac{1}{3 a^{\frac{3}{2}} x^{3}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^3)^(3/2)*x^4),x, algorithm="giac")

[Out]

integrate(1/((a + b/x^3)^(3/2)*x^4), x)