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

Optimal. Leaf size=43 \[ \frac{1-x}{6 \left (x^2-2 x-1\right )^{3/2}}-\frac{1-x}{6 \sqrt{x^2-2 x-1}} \]

[Out]

(1 - x)/(6*(-1 - 2*x + x^2)^(3/2)) - (1 - x)/(6*Sqrt[-1 - 2*x + x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.0061111, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {614, 613} \[ \frac{1-x}{6 \left (x^2-2 x-1\right )^{3/2}}-\frac{1-x}{6 \sqrt{x^2-2 x-1}} \]

Antiderivative was successfully verified.

[In]

Int[(-1 - 2*x + x^2)^(-5/2),x]

[Out]

(1 - x)/(6*(-1 - 2*x + x^2)^(3/2)) - (1 - x)/(6*Sqrt[-1 - 2*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 (-1-2 x+x^2\right )^{5/2}} \, dx &=\frac{1-x}{6 \left (-1-2 x+x^2\right )^{3/2}}-\frac{1}{3} \int \frac{1}{\left (-1-2 x+x^2\right )^{3/2}} \, dx\\ &=\frac{1-x}{6 \left (-1-2 x+x^2\right )^{3/2}}-\frac{1-x}{6 \sqrt{-1-2 x+x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0138116, size = 26, normalized size = 0.6 \[ \frac{x^3-3 x^2+2}{6 \left (x^2-2 x-1\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 - 2*x + x^2)^(-5/2),x]

[Out]

(2 - 3*x^2 + x^3)/(6*(-1 - 2*x + x^2)^(3/2))

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 23, normalized size = 0.5 \begin{align*}{\frac{{x}^{3}-3\,{x}^{2}+2}{6} \left ({x}^{2}-2\,x-1 \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(x^3-3*x^2+2)/(x^2-2*x-1)^(3/2)

________________________________________________________________________________________

Maxima [A]  time = 0.934023, size = 69, normalized size = 1.6 \begin{align*} \frac{x}{6 \, \sqrt{x^{2} - 2 \, x - 1}} - \frac{1}{6 \, \sqrt{x^{2} - 2 \, x - 1}} - \frac{x}{6 \,{\left (x^{2} - 2 \, x - 1\right )}^{\frac{3}{2}}} + \frac{1}{6 \,{\left (x^{2} - 2 \, x - 1\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x/sqrt(x^2 - 2*x - 1) - 1/6/sqrt(x^2 - 2*x - 1) - 1/6*x/(x^2 - 2*x - 1)^(3/2) + 1/6/(x^2 - 2*x - 1)^(3/2)

________________________________________________________________________________________

Fricas [A]  time = 2.11103, size = 147, normalized size = 3.42 \begin{align*} \frac{x^{4} - 4 \, x^{3} + 2 \, x^{2} +{\left (x^{3} - 3 \, x^{2} + 2\right )} \sqrt{x^{2} - 2 \, x - 1} + 4 \, x + 1}{6 \,{\left (x^{4} - 4 \, x^{3} + 2 \, x^{2} + 4 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((x**2 - 2*x - 1)**(-5/2), x)

________________________________________________________________________________________

Giac [A]  time = 1.06268, size = 28, normalized size = 0.65 \begin{align*} \frac{{\left (x - 3\right )} x^{2} + 2}{6 \,{\left (x^{2} - 2 \, x - 1\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*((x - 3)*x^2 + 2)/(x^2 - 2*x - 1)^(3/2)