3.131 \(\int \frac {1}{(5-4 x-x^2)^{5/2}} \, dx\)

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

[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 = 0.01, 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 = {614, 613} \[ \frac {2 (x+2)}{243 \sqrt {-x^2-4 x+5}}+\frac {x+2}{27 \left (-x^2-4 x+5\right )^{3/2}} \]

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 613

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 614

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 = 0.01, size = 31, normalized size = 0.72 \[ -\frac {(x+2) \left (2 x^2+8 x-19\right )}{243 \left (-x^2-4 x+5\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(5 - 4*x - x^2)^(-5/2),x]

[Out]

-1/243*((2 + x)*(-19 + 8*x + 2*x^2))/(5 - 4*x - x^2)^(3/2)

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fricas [A]  time = 0.89, size = 49, normalized size = 1.14 \[ -\frac {{\left (2 \, x^{3} + 12 \, x^{2} - 3 \, x - 38\right )} \sqrt {-x^{2} - 4 \, x + 5}}{243 \, {\left (x^{4} + 8 \, x^{3} + 6 \, x^{2} - 40 \, x + 25\right )}} \]

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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giac [A]  time = 0.78, size = 36, normalized size = 0.84 \[ -\frac {{\left ({\left (2 \, {\left (x + 6\right )} x - 3\right )} x - 38\right )} \sqrt {-x^{2} - 4 \, x + 5}}{243 \, {\left (x^{2} + 4 \, x - 5\right )}^{2}} \]

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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maple [A]  time = 0.04, size = 36, normalized size = 0.84 \[ \frac {\left (x +5\right ) \left (x -1\right ) \left (2 x^{3}+12 x^{2}-3 x -38\right )}{243 \left (-x^{2}-4 x +5\right )^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.41, size = 59, normalized size = 1.37 \[ \frac {2 \, x}{243 \, \sqrt {-x^{2} - 4 \, x + 5}} + \frac {4}{243 \, \sqrt {-x^{2} - 4 \, x + 5}} + \frac {x}{27 \, {\left (-x^{2} - 4 \, x + 5\right )}^{\frac {3}{2}}} + \frac {2}{27 \, {\left (-x^{2} - 4 \, x + 5\right )}^{\frac {3}{2}}} \]

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

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

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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