3.2020 \(\int \frac{1}{\sqrt{a+\frac{b}{x^3}} x^4} \, dx\)

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 = 0.0058873, 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, Rules used = {261} \[ -\frac{2 \sqrt{a+\frac{b}{x^3}}}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b/x^3]*x^4),x]

[Out]

(-2*Sqrt[a + b/x^3])/(3*b)

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{a+\frac{b}{x^3}} x^4} \, dx &=-\frac{2 \sqrt{a+\frac{b}{x^3}}}{3 b}\\ \end{align*}

Mathematica [A]  time = 0.0079881, 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 = 0.004, size = 29, normalized size = 1.6 \begin{align*} -{\frac{2\,a{x}^{3}+2\,b}{3\,b{x}^{3}}{\frac{1}{\sqrt{{\frac{a{x}^{3}+b}{{x}^{3}}}}}}} \end{align*}

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 = 0.980899, size = 19, normalized size = 1.06 \begin{align*} -\frac{2 \, \sqrt{a + \frac{b}{x^{3}}}}{3 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*sqrt(a + b/x^3)/b

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Fricas [A]  time = 1.52495, size = 41, normalized size = 2.28 \begin{align*} -\frac{2 \, \sqrt{\frac{a x^{3} + b}{x^{3}}}}{3 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*sqrt((a*x^3 + b)/x^3)/b

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Sympy [A]  time = 1.74299, size = 29, normalized size = 1.61 \begin{align*} \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} \end{align*}

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 [A]  time = 1.11277, size = 19, normalized size = 1.06 \begin{align*} -\frac{2 \, \sqrt{a + \frac{b}{x^{3}}}}{3 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*sqrt(a + b/x^3)/b