### 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]

Sqrt[-9 + 16*x^2]/(9*x)

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Rubi [A]  time = 0.0030559, 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 = {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*}

Mathematica [A]  time = 0.0032676, size = 18, normalized size = 1. $\frac{\sqrt{16 x^2-9}}{9 x}$

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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Maple [A]  time = 0.003, size = 25, normalized size = 1.4 \begin{align*}{\frac{ \left ( 4\,x-3 \right ) \left ( 3+4\,x \right ) }{9\,x}{\frac{1}{\sqrt{16\,{x}^{2}-9}}}} \end{align*}

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

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Maxima [A]  time = 1.4093, size = 19, normalized size = 1.06 \begin{align*} \frac{\sqrt{16 \, x^{2} - 9}}{9 \, x} \end{align*}

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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Fricas [A]  time = 1.98513, size = 43, normalized size = 2.39 \begin{align*} \frac{4 \, x + \sqrt{16 \, x^{2} - 9}}{9 \, x} \end{align*}

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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Sympy [A]  time = 0.743997, size = 37, normalized size = 2.06 \begin{align*} \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} \end{align*}

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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Giac [A]  time = 1.05403, size = 31, normalized size = 1.72 \begin{align*} \frac{8}{{\left (4 \, x - \sqrt{16 \, x^{2} - 9}\right )}^{2} + 9} \end{align*}

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)