3.2.31 \(\int \frac {1}{(5-4 x-x^2)^{5/2}} \, dx\) [131]

Optimal. Leaf size=43 \[ \frac {2+x}{27 \left (5-4 x-x^2\right )^{3/2}}+\frac {2 (2+x)}{243 \sqrt {5-4 x-x^2}} \]

[Out]

1/27*(2+x)/(-x^2-4*x+5)^(3/2)+2/243*(2+x)/(-x^2-4*x+5)^(1/2)

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Rubi [A]
time = 0.00, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {628, 627} \begin {gather*} \frac {2 (x+2)}{243 \sqrt {-x^2-4 x+5}}+\frac {x+2}{27 \left (-x^2-4 x+5\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(5 - 4*x - x^2)^(-5/2),x]

[Out]

(2 + x)/(27*(5 - 4*x - x^2)^(3/2)) + (2*(2 + x))/(243*Sqrt[5 - 4*x - x^2])

Rule 627

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

Rule 628

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1
)*(b^2 - 4*a*c))), x] - Dist[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

Rubi steps

\begin {align*} \int \frac {1}{\left (5-4 x-x^2\right )^{5/2}} \, dx &=\frac {2+x}{27 \left (5-4 x-x^2\right )^{3/2}}+\frac {2}{27} \int \frac {1}{\left (5-4 x-x^2\right )^{3/2}} \, dx\\ &=\frac {2+x}{27 \left (5-4 x-x^2\right )^{3/2}}+\frac {2 (2+x)}{243 \sqrt {5-4 x-x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.14, size = 43, normalized size = 1.00 \begin {gather*} \frac {\sqrt {5-4 x-x^2} \left (38+3 x-12 x^2-2 x^3\right )}{243 (-1+x)^2 (5+x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5 - 4*x - x^2)^(-5/2),x]

[Out]

(Sqrt[5 - 4*x - x^2]*(38 + 3*x - 12*x^2 - 2*x^3))/(243*(-1 + x)^2*(5 + x)^2)

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Maple [A]
time = 0.49, size = 40, normalized size = 0.93

method	result	size
gosper	\(\frac {\left (x +5\right ) \left (x -1\right ) \left (2 x^{3}+12 x^{2}-3 x -38\right )}{243 \left (-x^{2}-4 x +5\right )^{\frac {5}{2}}}\)	\(36\)
default	\(-\frac {-2 x -4}{54 \left (-x^{2}-4 x +5\right )^{\frac {3}{2}}}-\frac {-2 x -4}{243 \sqrt {-x^{2}-4 x +5}}\)	\(40\)
trager	\(-\frac {\left (2 x^{3}+12 x^{2}-3 x -38\right ) \sqrt {-x^{2}-4 x +5}}{243 \left (x^{2}+4 x -5\right )^{2}}\)	\(40\)
risch	\(\frac {2 x^{3}+12 x^{2}-3 x -38}{243 \left (x^{2}+4 x -5\right ) \sqrt {-x^{2}-4 x +5}}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-x^2-4*x+5)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/54*(-2*x-4)/(-x^2-4*x+5)^(3/2)-1/243*(-2*x-4)/(-x^2-4*x+5)^(1/2)

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Maxima [A]
time = 0.28, size = 59, normalized size = 1.37 \begin {gather*} \frac {2 \, x}{243 \, \sqrt {-x^{2} - 4 \, x + 5}} + \frac {4}{243 \, \sqrt {-x^{2} - 4 \, x + 5}} + \frac {x}{27 \, {\left (-x^{2} - 4 \, x + 5\right )}^{\frac {3}{2}}} + \frac {2}{27 \, {\left (-x^{2} - 4 \, x + 5\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^2-4*x+5)^(5/2),x, algorithm="maxima")

[Out]

2/243*x/sqrt(-x^2 - 4*x + 5) + 4/243/sqrt(-x^2 - 4*x + 5) + 1/27*x/(-x^2 - 4*x + 5)^(3/2) + 2/27/(-x^2 - 4*x +
5)^(3/2)

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Fricas [A]
time = 2.10, size = 49, normalized size = 1.14 \begin {gather*} -\frac {{\left (2 \, x^{3} + 12 \, x^{2} - 3 \, x - 38\right )} \sqrt {-x^{2} - 4 \, x + 5}}{243 \, {\left (x^{4} + 8 \, x^{3} + 6 \, x^{2} - 40 \, x + 25\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^2-4*x+5)^(5/2),x, algorithm="fricas")

[Out]

-1/243*(2*x^3 + 12*x^2 - 3*x - 38)*sqrt(-x^2 - 4*x + 5)/(x^4 + 8*x^3 + 6*x^2 - 40*x + 25)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (- x^{2} - 4 x + 5\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x**2-4*x+5)**(5/2),x)

[Out]

Integral((-x**2 - 4*x + 5)**(-5/2), x)

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Giac [A]
time = 1.63, size = 36, normalized size = 0.84 \begin {gather*} -\frac {{\left ({\left (2 \, {\left (x + 6\right )} x - 3\right )} x - 38\right )} \sqrt {-x^{2} - 4 \, x + 5}}{243 \, {\left (x^{2} + 4 \, x - 5\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^2-4*x+5)^(5/2),x, algorithm="giac")

[Out]

-1/243*((2*(x + 6)*x - 3)*x - 38)*sqrt(-x^2 - 4*x + 5)/(x^2 + 4*x - 5)^2

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Mupad [B]
time = 0.03, size = 29, normalized size = 0.67 \begin {gather*} -\frac {\left (4\,x+8\right )\,\left (8\,x^2+32\,x-76\right )}{3888\,{\left (-x^2-4\,x+5\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(5 - x^2 - 4*x)^(5/2),x)

[Out]

-((4*x + 8)*(32*x + 8*x^2 - 76))/(3888*(5 - x^2 - 4*x)^(3/2))

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