3.133 \(\int \frac{\sqrt{-a^2+x^2}}{x^4} \, dx\)

Optimal. Leaf size=23 \[ \frac{\left (x^2-a^2\right )^{3/2}}{3 a^2 x^3} \]

[Out]

(-a^2 + x^2)^(3/2)/(3*a^2*x^3)

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Rubi [A]  time = 0.0040181, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {264} \[ \frac{\left (x^2-a^2\right )^{3/2}}{3 a^2 x^3} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-a^2 + x^2]/x^4,x]

[Out]

(-a^2 + x^2)^(3/2)/(3*a^2*x^3)

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0044987, size = 23, normalized size = 1. \[ \frac{\left (x^2-a^2\right )^{3/2}}{3 a^2 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-a^2 + x^2]/x^4,x]

[Out]

(-a^2 + x^2)^(3/2)/(3*a^2*x^3)

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Maple [A]  time = 0.004, size = 28, normalized size = 1.2 \begin{align*} -{\frac{ \left ( a+x \right ) \left ( a-x \right ) }{3\,{a}^{2}{x}^{3}}\sqrt{-{a}^{2}+{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a^2+x^2)^(1/2)/x^4,x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.0589, size = 57, normalized size = 2.48 \begin{align*} \frac{x^{3} +{\left (-a^{2} + x^{2}\right )}^{\frac{3}{2}}}{3 \, a^{2} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.628714, size = 78, normalized size = 3.39 \begin{align*} \begin{cases} - \frac{i \sqrt{\frac{a^{2}}{x^{2}} - 1}}{3 x^{2}} + \frac{i \sqrt{\frac{a^{2}}{x^{2}} - 1}}{3 a^{2}} & \text{for}\: \frac{\left |{a^{2}}\right |}{\left |{x^{2}}\right |} > 1 \\- \frac{\sqrt{- \frac{a^{2}}{x^{2}} + 1}}{3 x^{2}} + \frac{\sqrt{- \frac{a^{2}}{x^{2}} + 1}}{3 a^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a**2+x**2)**(1/2)/x**4,x)

[Out]

Piecewise((-I*sqrt(a**2/x**2 - 1)/(3*x**2) + I*sqrt(a**2/x**2 - 1)/(3*a**2), Abs(a**2)/Abs(x**2) > 1), (-sqrt(
-a**2/x**2 + 1)/(3*x**2) + sqrt(-a**2/x**2 + 1)/(3*a**2), True))

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Giac [B]  time = 1.06475, size = 65, normalized size = 2.83 \begin{align*} \frac{2 \,{\left (a^{4} + 3 \,{\left (x - \sqrt{-a^{2} + x^{2}}\right )}^{4}\right )}}{3 \,{\left (a^{2} +{\left (x - \sqrt{-a^{2} + x^{2}}\right )}^{2}\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(a^4 + 3*(x - sqrt(-a^2 + x^2))^4)/(a^2 + (x - sqrt(-a^2 + x^2))^2)^3