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]

-1/25*x/(4*x^2-25)^(1/2)

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Rubi [A]  time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 1.00 \[ -\frac {x}{25 \sqrt {4 x^2-25}} \]

Antiderivative was successfully verified.

[In]

Integrate[(-25 + 4*x^2)^(-3/2),x]

[Out]

-1/25*x/Sqrt[-25 + 4*x^2]

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fricas [B]  time = 0.40, size = 30, normalized size = 1.88 \[ -\frac {4 \, x^{2} + 2 \, \sqrt {4 \, x^{2} - 25} x - 25}{50 \, {\left (4 \, x^{2} - 25\right )}} \]

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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giac [A]  time = 1.04, size = 12, normalized size = 0.75 \[ -\frac {x}{25 \, \sqrt {4 \, x^{2} - 25}} \]

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)

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maple [A]  time = 0.00, size = 23, normalized size = 1.44 \[ -\frac {\left (2 x -5\right ) \left (2 x +5\right ) x}{25 \left (4 x^{2}-25\right )^{\frac {3}{2}}} \]

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

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maxima [A]  time = 0.43, size = 12, normalized size = 0.75 \[ -\frac {x}{25 \, \sqrt {4 \, x^{2} - 25}} \]

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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mupad [B]  time = 0.25, size = 12, normalized size = 0.75 \[ -\frac {x}{25\,\sqrt {4\,x^2-25}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(4*x^2 - 25)^(3/2),x)

[Out]

-x/(25*(4*x^2 - 25)^(1/2))

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sympy [A]  time = 0.78, size = 36, normalized size = 2.25 \[ \begin {cases} - \frac {x}{25 \sqrt {4 x^{2} - 25}} & \text {for}\: \frac {4 \left |{x^{2}}\right |}{25} > 1 \\\frac {i x}{25 \sqrt {25 - 4 x^{2}}} & \text {otherwise} \end {cases} \]

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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