### 3.149 $$\int \frac{1}{(5-4 x-x^2)^{5/2}} \, dx$$

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]

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

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Rubi [A]  time = 0.0075282, antiderivative size = 43, normalized size of antiderivative = 1., 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 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]

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]

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*}

Mathematica [A]  time = 0.0093429, 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]

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

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Maple [A]  time = 0.001, size = 36, normalized size = 0.8 \begin{align*}{\frac{ \left ( 5+x \right ) \left ( -1+x \right ) \left ( 2\,{x}^{3}+12\,{x}^{2}-3\,x-38 \right ) }{243} \left ( -{x}^{2}-4\,x+5 \right ) ^{-{\frac{5}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.921316, size = 80, normalized size = 1.86 \begin{align*} \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{align*}

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 = 2.30562, size = 123, normalized size = 2.86 \begin{align*} -\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{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (- x^{2} - 4 x + 5\right )^{\frac{5}{2}}}\, dx \end{align*}

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 = 1.07263, size = 49, normalized size = 1.14 \begin{align*} -\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{align*}

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