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

Optimal result

Integrand size = 11, antiderivative size = 16 \[ \int \frac {1}{\left (-25+4 x^2\right )^{3/2}} \, dx=-\frac {x}{25 \sqrt {-25+4 x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {197} \[ \int \frac {1}{\left (-25+4 x^2\right )^{3/2}} \, dx=-\frac {x}{25 \sqrt {4 x^2-25}} \]

[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*} \text {integral}& = -\frac {x}{25 \sqrt {-25+4 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (-25+4 x^2\right )^{3/2}} \, dx=-\frac {x}{25 \sqrt {-25+4 x^2}} \]

[In]

Integrate[(-25 + 4*x^2)^(-3/2),x]

[Out]

-1/25*x/Sqrt[-25 + 4*x^2]

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 13, normalized size of antiderivative = 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	\(-\frac {x}{25 \sqrt {4 x^{2}-25}}\)	\(13\)
pseudoelliptic	\(-\frac {x}{25 \sqrt {4 x^{2}-25}}\)	\(13\)
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	\(\frac {{\left (-\operatorname {signum}\left (-1+\frac {4 x^{2}}{25}\right )\right )}^{\frac {3}{2}} x}{125 \operatorname {signum}\left (-1+\frac {4 x^{2}}{25}\right )^{\frac {3}{2}} \sqrt {1-\frac {4 x^{2}}{25}}}\)	\(35\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 30 vs. \(2 (12) = 24\).

Time = 0.23 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.88 \[ \int \frac {1}{\left (-25+4 x^2\right )^{3/2}} \, dx=-\frac {4 \, x^{2} + 2 \, \sqrt {4 \, x^{2} - 25} x - 25}{50 \, {\left (4 \, x^{2} - 25\right )}} \]

[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)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.46 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.12 \[ \int \frac {1}{\left (-25+4 x^2\right )^{3/2}} \, dx=\begin {cases} - \frac {x}{25 \sqrt {4 x^{2} - 25}} & \text {for}\: \left |{x^{2}}\right | > \frac {25}{4} \\\frac {i x}{25 \sqrt {25 - 4 x^{2}}} & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.44 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\left (-25+4 x^2\right )^{3/2}} \, dx=-\frac {x}{25 \, \sqrt {4 \, x^{2} - 25}} \]

[In]

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

[Out]

-1/25*x/sqrt(4*x^2 - 25)

Giac [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\left (-25+4 x^2\right )^{3/2}} \, dx=-\frac {x}{25 \, \sqrt {4 \, x^{2} - 25}} \]

[In]

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

[Out]

-1/25*x/sqrt(4*x^2 - 25)

Mupad [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\left (-25+4 x^2\right )^{3/2}} \, dx=-\frac {x}{25\,\sqrt {4\,x^2-25}} \]

[In]

int(1/(4*x^2 - 25)^(3/2),x)

[Out]

-x/(25*(4*x^2 - 25)^(1/2))