3.2.30 \(\int \frac {x}{(5-4 x-x^2)^{3/2}} \, dx\) [130]

Optimal. Leaf size=23 \[ \frac {5-2 x}{9 \sqrt {5-4 x-x^2}} \]

[Out]

1/9*(5-2*x)/(-x^2-4*x+5)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {650} \begin {gather*} \frac {5-2 x}{9 \sqrt {-x^2-4 x+5}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(5 - 4*x - x^2)^(3/2),x]

[Out]

(5 - 2*x)/(9*Sqrt[5 - 4*x - x^2])

Rule 650

Int[((d_.) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[-2*((b*d - 2*a*e + (2*c*
d - b*e)*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] &&
NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {x}{\left (5-4 x-x^2\right )^{3/2}} \, dx &=\frac {5-2 x}{9 \sqrt {5-4 x-x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.08, size = 33, normalized size = 1.43 \begin {gather*} \frac {(-5+2 x) \sqrt {5-4 x-x^2}}{9 (-1+x) (5+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(5 - 4*x - x^2)^(3/2),x]

[Out]

((-5 + 2*x)*Sqrt[5 - 4*x - x^2])/(9*(-1 + x)*(5 + x))

________________________________________________________________________________________

Maple [A]
time = 0.50, size = 33, normalized size = 1.43

method	result	size
risch	\(-\frac {2 x -5}{9 \sqrt {-x^{2}-4 x +5}}\)	\(20\)
gosper	\(\frac {\left (x +5\right ) \left (x -1\right ) \left (2 x -5\right )}{9 \left (-x^{2}-4 x +5\right )^{\frac {3}{2}}}\)	\(26\)
trager	\(\frac {\left (2 x -5\right ) \sqrt {-x^{2}-4 x +5}}{9 x^{2}+36 x -45}\)	\(30\)
default	\(\frac {1}{\sqrt {-x^{2}-4 x +5}}+\frac {-2 x -4}{9 \sqrt {-x^{2}-4 x +5}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(-x^2-4*x+5)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/(-x^2-4*x+5)^(1/2)+1/9*(-2*x-4)/(-x^2-4*x+5)^(1/2)

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 30, normalized size = 1.30 \begin {gather*} -\frac {2 \, x}{9 \, \sqrt {-x^{2} - 4 \, x + 5}} + \frac {5}{9 \, \sqrt {-x^{2} - 4 \, x + 5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x^2-4*x+5)^(3/2),x, algorithm="maxima")

[Out]

-2/9*x/sqrt(-x^2 - 4*x + 5) + 5/9/sqrt(-x^2 - 4*x + 5)

________________________________________________________________________________________

Fricas [A]
time = 1.29, size = 29, normalized size = 1.26 \begin {gather*} \frac {\sqrt {-x^{2} - 4 \, x + 5} {\left (2 \, x - 5\right )}}{9 \, {\left (x^{2} + 4 \, x - 5\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x^2-4*x+5)^(3/2),x, algorithm="fricas")

[Out]

1/9*sqrt(-x^2 - 4*x + 5)*(2*x - 5)/(x^2 + 4*x - 5)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\left (- \left (x - 1\right ) \left (x + 5\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x**2-4*x+5)**(3/2),x)

[Out]

Integral(x/(-(x - 1)*(x + 5))**(3/2), x)

________________________________________________________________________________________

Giac [A]
time = 1.63, size = 29, normalized size = 1.26 \begin {gather*} \frac {\sqrt {-x^{2} - 4 \, x + 5} {\left (2 \, x - 5\right )}}{9 \, {\left (x^{2} + 4 \, x - 5\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x^2-4*x+5)^(3/2),x, algorithm="giac")

[Out]

1/9*sqrt(-x^2 - 4*x + 5)*(2*x - 5)/(x^2 + 4*x - 5)

________________________________________________________________________________________

Mupad [B]
time = 0.05, size = 19, normalized size = 0.83 \begin {gather*} -\frac {2\,x-5}{9\,\sqrt {-x^2-4\,x+5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(5 - x^2 - 4*x)^(3/2),x)

[Out]

-(2*x - 5)/(9*(5 - x^2 - 4*x)^(1/2))

________________________________________________________________________________________