3.1994 \(\int \frac{\sqrt{a+\frac{b}{x^3}}}{x^4} \, dx\)

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 = 0.0059425, 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} \[ -\frac{2 \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b/x^3]/x^4,x]

[Out]

(-2*(a + b/x^3)^(3/2))/(9*b)

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{\sqrt{a+\frac{b}{x^3}}}{x^4} \, dx &=-\frac{2 \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b}\\ \end{align*}

Mathematica [A]  time = 0.0111442, 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 = 0.004, size = 29, normalized size = 1.6 \begin{align*} -{\frac{2\,a{x}^{3}+2\,b}{9\,b{x}^{3}}\sqrt{{\frac{a{x}^{3}+b}{{x}^{3}}}}} \end{align*}

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 = 0.957996, size = 19, normalized size = 1.06 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}}}{9 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/9*(a + b/x^3)^(3/2)/b

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Fricas [A]  time = 1.50823, size = 65, normalized size = 3.61 \begin{align*} -\frac{2 \,{\left (a x^{3} + b\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{9 \, b x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^3)^(1/2)/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 [B]  time = 1.13011, size = 46, normalized size = 2.56 \begin{align*} - \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}} \end{align*}

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 [A]  time = 1.59336, size = 19, normalized size = 1.06 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}}}{9 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/9*(a + b/x^3)^(3/2)/b