### 3.143 $$\int \frac{1}{(-25+4 x^2)^{3/2}} \, dx$$

Optimal. Leaf size=16 $-\frac{x}{25 \sqrt{4 x^2-25}}$

[Out]

-x/(25*Sqrt[-25 + 4*x^2])

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Rubi [A]  time = 0.0014153, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.091, Rules used = {191} $-\frac{x}{25 \sqrt{4 x^2-25}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(-25 + 4*x^2)^(-3/2),x]

[Out]

-x/(25*Sqrt[-25 + 4*x^2])

Rule 191

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

Rubi steps

\begin{align*} \int \frac{1}{\left (-25+4 x^2\right )^{3/2}} \, dx &=-\frac{x}{25 \sqrt{-25+4 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0038095, size = 16, normalized size = 1. $-\frac{x}{25 \sqrt{4 x^2-25}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(-25 + 4*x^2)^(-3/2),x]

[Out]

-x/(25*Sqrt[-25 + 4*x^2])

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Maple [A]  time = 0.002, size = 23, normalized size = 1.4 \begin{align*} -{\frac{ \left ( 2\,x-5 \right ) \left ( 5+2\,x \right ) x}{25} \left ( 4\,{x}^{2}-25 \right ) ^{-{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(4*x^2-25)^(3/2),x)

[Out]

-1/25*(2*x-5)*(5+2*x)*x/(4*x^2-25)^(3/2)

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Maxima [A]  time = 0.920857, size = 16, normalized size = 1. \begin{align*} -\frac{x}{25 \, \sqrt{4 \, x^{2} - 25}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4*x^2-25)^(3/2),x, algorithm="maxima")

[Out]

-1/25*x/sqrt(4*x^2 - 25)

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Fricas [B]  time = 2.06223, size = 76, normalized size = 4.75 \begin{align*} -\frac{4 \, x^{2} + 2 \, \sqrt{4 \, x^{2} - 25} x - 25}{50 \,{\left (4 \, x^{2} - 25\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4*x^2-25)^(3/2),x, algorithm="fricas")

[Out]

-1/50*(4*x^2 + 2*sqrt(4*x^2 - 25)*x - 25)/(4*x^2 - 25)

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Sympy [A]  time = 0.721938, size = 36, normalized size = 2.25 \begin{align*} \begin{cases} - \frac{x}{25 \sqrt{4 x^{2} - 25}} & \text{for}\: \frac{4 \left |{x^{2}}\right |}{25} > 1 \\\frac{i x}{25 \sqrt{25 - 4 x^{2}}} & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4*x**2-25)**(3/2),x)

[Out]

Piecewise((-x/(25*sqrt(4*x**2 - 25)), 4*Abs(x**2)/25 > 1), (I*x/(25*sqrt(25 - 4*x**2)), True))

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Giac [A]  time = 1.06364, size = 16, normalized size = 1. \begin{align*} -\frac{x}{25 \, \sqrt{4 \, x^{2} - 25}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4*x^2-25)^(3/2),x, algorithm="giac")

[Out]

-1/25*x/sqrt(4*x^2 - 25)