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]

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

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Rubi [A]  time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 1.00 \[ -\frac {2 \left (a+\frac {b}{x^3}\right )^{3/2}}{9 b} \]

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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fricas [A]  time = 0.87, size = 28, normalized size = 1.56 \[ -\frac {2 \, {\left (a x^{3} + b\right )} \sqrt {\frac {a x^{3} + b}{x^{3}}}}{9 \, b x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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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giac [A]  time = 0.18, size = 14, normalized size = 0.78 \[ -\frac {2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}}}{9 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/9*(a + b/x^3)^(3/2)/b

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maple [A]  time = 0.00, size = 29, normalized size = 1.61 \[ -\frac {2 \left (a \,x^{3}+b \right ) \sqrt {\frac {a \,x^{3}+b}{x^{3}}}}{9 b \,x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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 = 0.79, size = 14, normalized size = 0.78 \[ -\frac {2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}}}{9 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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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mupad [B]  time = 1.43, size = 24, normalized size = 1.33 \[ -\frac {2\,\sqrt {a+\frac {b}{x^3}}\,\left (a\,x^3+b\right )}{9\,b\,x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x^3)^(1/2)/x^4,x)

[Out]

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

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sympy [B]  time = 1.23, 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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