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]

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

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Rubi [A]  time = 0.00, antiderivative size = 23, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 23, normalized size = 1.00 \[ \frac {\left (x^2-a^2\right )^{3/2}}{3 a^2 x^3} \]

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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fricas [A]  time = 0.40, size = 23, normalized size = 1.00 \[ \frac {x^{3} + {\left (-a^{2} + x^{2}\right )}^{\frac {3}{2}}}{3 \, a^{2} x^{3}} \]

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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giac [B]  time = 0.98, size = 48, normalized size = 2.09 \[ \frac {2 \, {\left (a^{4} + 3 \, {\left (x - \sqrt {-a^{2} + x^{2}}\right )}^{4}\right )}}{3 \, {\left (a^{2} + {\left (x - \sqrt {-a^{2} + x^{2}}\right )}^{2}\right )}^{3}} \]

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

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maple [A]  time = 0.01, size = 28, normalized size = 1.22 \[ -\frac {\left (a +x \right ) \left (a -x \right ) \sqrt {-a^{2}+x^{2}}}{3 a^{2} x^{3}} \]

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 [A]  time = 0.97, size = 19, normalized size = 0.83 \[ \frac {{\left (-a^{2} + x^{2}\right )}^{\frac {3}{2}}}{3 \, a^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(-a^2 + x^2)^(3/2)/(a^2*x^3)

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mupad [B]  time = 0.34, size = 19, normalized size = 0.83 \[ \frac {{\left (x^2-a^2\right )}^{3/2}}{3\,a^2\,x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 - a^2)^(1/2)/x^4,x)

[Out]

(x^2 - a^2)^(3/2)/(3*a^2*x^3)

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sympy [A]  time = 0.76, size = 76, normalized size = 3.30 \[ \begin {cases} - \frac {i \sqrt {\frac {a^{2}}{x^{2}} - 1}}{3 x^{2}} + \frac {i \sqrt {\frac {a^{2}}{x^{2}} - 1}}{3 a^{2}} & \text {for}\: \left |{\frac {a^{2}}{x^{2}}}\right | > 1 \\- \frac {\sqrt {- \frac {a^{2}}{x^{2}} + 1}}{3 x^{2}} + \frac {\sqrt {- \frac {a^{2}}{x^{2}} + 1}}{3 a^{2}} & \text {otherwise} \end {cases} \]

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/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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