3.2.43 \(\int \frac {1}{(-25+4 x^2)^{3/2}} \, dx\) [143]

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

[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 = {197} \begin {gather*} -\frac {x}{25 \sqrt {4 x^2-25}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-25 + 4*x^2)^(-3/2),x]

[Out]

-1/25*x/Sqrt[-25 + 4*x^2]

Rule 197

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 = 0.02, size = 16, normalized size = 1.00 \begin {gather*} -\frac {x}{25 \sqrt {-25+4 x^2}} \end {gather*}

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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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 2.09, size = 33, normalized size = 2.06 \begin {gather*} \text {Piecewise}\left [\left \{\left \{-\frac {x}{25 \sqrt {-25+4 x^2}},\text {Abs}\left [x^2\right ]>\frac {25}{4}\right \}\right \},\frac {I x}{25 \sqrt {25-4 x^2}}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[1/(4*x^2 - 25)^(3/2),x]')

[Out]

Piecewise[{{-x / (25 Sqrt[-25 + 4 x ^ 2]), Abs[x ^ 2] > 25 / 4}}, I x / (25 Sqrt[25 - 4 x ^ 2])]

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Maple [A]
time = 0.06, size = 13, normalized size = 0.81

method	result	size
default	\(-\frac {x}{25 \sqrt {4 x^{2}-25}}\)	\(13\)
trager	\(-\frac {x}{25 \sqrt {4 x^{2}-25}}\)	\(13\)
risch	\(-\frac {x}{25 \sqrt {4 x^{2}-25}}\)	\(13\)
gosper	\(-\frac {\left (2 x -5\right ) \left (5+2 x \right ) x}{25 \left (4 x^{2}-25\right )^{\frac {3}{2}}}\)	\(23\)
meijerg	\(\frac {\left (-\mathrm {signum}\left (-1+\frac {4 x^{2}}{25}\right )\right )^{\frac {3}{2}} x}{125 \mathrm {signum}\left (-1+\frac {4 x^{2}}{25}\right )^{\frac {3}{2}} \sqrt {1-\frac {4 x^{2}}{25}}}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 12, normalized size = 0.75 \begin {gather*} -\frac {x}{25 \, \sqrt {4 \, x^{2} - 25}} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 30 vs. \(2 (12) = 24\).
time = 0.35, size = 30, normalized size = 1.88 \begin {gather*} -\frac {4 \, x^{2} + 2 \, \sqrt {4 \, x^{2} - 25} x - 25}{50 \, {\left (4 \, x^{2} - 25\right )}} \end {gather*}

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 = 0.38, size = 34, normalized size = 2.12 \begin {gather*} \begin {cases} - \frac {x}{25 \sqrt {4 x^{2} - 25}} & \text {for}\: \left |{x^{2}}\right | > \frac {25}{4} \\\frac {i x}{25 \sqrt {25 - 4 x^{2}}} & \text {otherwise} \end {cases} \end {gather*}

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

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Giac [A]
time = 0.00, size = 25, normalized size = 1.56 \begin {gather*} -\frac {2 x \sqrt {4 x^{2}-25}}{50 \left (4 x^{2}-25\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/25*x/sqrt(4*x^2 - 25)

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Mupad [B]
time = 0.25, size = 12, normalized size = 0.75 \begin {gather*} -\frac {x}{25\,\sqrt {4\,x^2-25}} \end {gather*}

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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