\(\int \frac {1}{(-1-2 x+x^2)^{5/2}} \, dx\) [469]

Optimal result

Integrand size = 12, antiderivative size = 43 \[ \int \frac {1}{\left (-1-2 x+x^2\right )^{5/2}} \, dx=\frac {1-x}{6 \left (-1-2 x+x^2\right )^{3/2}}-\frac {1-x}{6 \sqrt {-1-2 x+x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {628, 627} \[ \int \frac {1}{\left (-1-2 x+x^2\right )^{5/2}} \, dx=\frac {1-x}{6 \left (x^2-2 x-1\right )^{3/2}}-\frac {1-x}{6 \sqrt {x^2-2 x-1}} \]

[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 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*} \text {integral}& = \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] (verified)

Time = 0.15 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.60 \[ \int \frac {1}{\left (-1-2 x+x^2\right )^{5/2}} \, dx=\frac {2-3 x^2+x^3}{6 \left (-1-2 x+x^2\right )^{3/2}} \]

[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] (verified)

Time = 0.35 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.53

method	result	size
gosper	\(\frac {x^{3}-3 x^{2}+2}{6 \left (x^{2}-2 x -1\right )^{\frac {3}{2}}}\)	\(23\)
trager	\(\frac {x^{3}-3 x^{2}+2}{6 \left (x^{2}-2 x -1\right )^{\frac {3}{2}}}\)	\(23\)
risch	\(\frac {x^{3}-3 x^{2}+2}{6 \left (x^{2}-2 x -1\right )^{\frac {3}{2}}}\)	\(23\)
default	\(-\frac {-2+2 x}{12 \left (x^{2}-2 x -1\right )^{\frac {3}{2}}}+\frac {-2+2 x}{12 \sqrt {x^{2}-2 x -1}}\)	\(36\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.42 \[ \int \frac {1}{\left (-1-2 x+x^2\right )^{5/2}} \, dx=\frac {x^{4} - 4 \, x^{3} + 2 \, x^{2} + {\left (x^{3} - 3 \, x^{2} + 2\right )} \sqrt {x^{2} - 2 \, x - 1} + 4 \, x + 1}{6 \, {\left (x^{4} - 4 \, x^{3} + 2 \, x^{2} + 4 \, x + 1\right )}} \]

[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]

\[ \int \frac {1}{\left (-1-2 x+x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (x^{2} - 2 x - 1\right )^{\frac {5}{2}}}\, dx \]

[In]

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

[Out]

Integral((x**2 - 2*x - 1)**(-5/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.19 \[ \int \frac {1}{\left (-1-2 x+x^2\right )^{5/2}} \, dx=\frac {x}{6 \, \sqrt {x^{2} - 2 \, x - 1}} - \frac {1}{6 \, \sqrt {x^{2} - 2 \, x - 1}} - \frac {x}{6 \, {\left (x^{2} - 2 \, x - 1\right )}^{\frac {3}{2}}} + \frac {1}{6 \, {\left (x^{2} - 2 \, x - 1\right )}^{\frac {3}{2}}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.49 \[ \int \frac {1}{\left (-1-2 x+x^2\right )^{5/2}} \, dx=\frac {{\left (x - 3\right )} x^{2} + 2}{6 \, {\left (x^{2} - 2 \, x - 1\right )}^{\frac {3}{2}}} \]

[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)

Mupad [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.51 \[ \int \frac {1}{\left (-1-2 x+x^2\right )^{5/2}} \, dx=\frac {x^3-3\,x^2+2}{6\,{\left (x^2-2\,x-1\right )}^{3/2}} \]

[In]

int(1/(x^2 - 2*x - 1)^(5/2),x)

[Out]

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