3.135 \(\int \frac {1}{x^2 \sqrt {-9+16 x^2}} \, dx\)

Optimal. Leaf size=18 \[ \frac {\sqrt {16 x^2-9}}{9 x} \]

[Out]

1/9*(16*x^2-9)^(1/2)/x

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Rubi [A]  time = 0.00, 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 = {264} \[ \frac {\sqrt {16 x^2-9}}{9 x} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*Sqrt[-9 + 16*x^2]),x]

[Out]

Sqrt[-9 + 16*x^2]/(9*x)

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 {1}{x^2 \sqrt {-9+16 x^2}} \, dx &=\frac {\sqrt {-9+16 x^2}}{9 x}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 18, normalized size = 1.00 \[ \frac {\sqrt {16 x^2-9}}{9 x} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*Sqrt[-9 + 16*x^2]),x]

[Out]

Sqrt[-9 + 16*x^2]/(9*x)

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fricas [A]  time = 0.40, size = 18, normalized size = 1.00 \[ \frac {4 \, x + \sqrt {16 \, x^{2} - 9}}{9 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(16*x^2-9)^(1/2),x, algorithm="fricas")

[Out]

1/9*(4*x + sqrt(16*x^2 - 9))/x

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giac [A]  time = 0.99, size = 23, normalized size = 1.28 \[ \frac {8}{{\left (4 \, x - \sqrt {16 \, x^{2} - 9}\right )}^{2} + 9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(16*x^2-9)^(1/2),x, algorithm="giac")

[Out]

8/((4*x - sqrt(16*x^2 - 9))^2 + 9)

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maple [A]  time = 0.00, size = 25, normalized size = 1.39 \[ \frac {\left (4 x -3\right ) \left (4 x +3\right )}{9 \sqrt {16 x^{2}-9}\, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(16*x^2-9)^(1/2),x)

[Out]

1/9/x*(4*x-3)*(4*x+3)/(16*x^2-9)^(1/2)

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maxima [A]  time = 0.98, size = 14, normalized size = 0.78 \[ \frac {\sqrt {16 \, x^{2} - 9}}{9 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(16*x^2-9)^(1/2),x, algorithm="maxima")

[Out]

1/9*sqrt(16*x^2 - 9)/x

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mupad [B]  time = 0.25, size = 14, normalized size = 0.78 \[ \frac {\sqrt {16\,x^2-9}}{9\,x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(16*x^2 - 9)^(1/2)),x)

[Out]

(16*x^2 - 9)^(1/2)/(9*x)

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sympy [A]  time = 0.82, size = 37, normalized size = 2.06 \[ \begin {cases} \frac {4 i \sqrt {-1 + \frac {9}{16 x^{2}}}}{9} & \text {for}\: \frac {9}{16 \left |{x^{2}}\right |} > 1 \\\frac {4 \sqrt {1 - \frac {9}{16 x^{2}}}}{9} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(16*x**2-9)**(1/2),x)

[Out]

Piecewise((4*I*sqrt(-1 + 9/(16*x**2))/9, 9/(16*Abs(x**2)) > 1), (4*sqrt(1 - 9/(16*x**2))/9, True))

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