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

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

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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 = 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 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 (-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*}

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Mathematica [A]  time = 0.01, size = 26, normalized size = 0.60 \[ \frac {x^3-3 x^2+2}{6 \left (x^2-2 x-1\right )^{3/2}} \]

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))

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IntegrateAlgebraic [A]  time = 0.28, size = 27, normalized size = 0.63 \[ \frac {(x-1) \left (x^2-2 x-2\right )}{6 \left (x^2-2 x-1\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-1 - 2*x + x^2)^(-5/2),x]

[Out]

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

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fricas [A]  time = 1.15, size = 61, normalized size = 1.42 \[ \frac {x^{4} - 4 \, x^{3} + 2 \, x^{2} + {\left (x^{3} - 3 \, x^{2} + 2\right )} \sqrt {x^{2} - 2 \, x - 1} + 4 \, x + 1}{6 \, {\left (x^{4} - 4 \, x^{3} + 2 \, x^{2} + 4 \, x + 1\right )}} \]

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)

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giac [A]  time = 0.66, size = 21, normalized size = 0.49 \[ \frac {{\left (x - 3\right )} x^{2} + 2}{6 \, {\left (x^{2} - 2 \, x - 1\right )}^{\frac {3}{2}}} \]

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)

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maple [A]  time = 0.39, size = 23, normalized size = 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	\(-\frac {2 x -2}{12 \left (x^{2}-2 x -1\right )^{\frac {3}{2}}}+\frac {2 x -2}{12 \sqrt {x^{2}-2 x -1}}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [A]  time = 0.42, size = 51, normalized size = 1.19 \[ \frac {x}{6 \, \sqrt {x^{2} - 2 \, x - 1}} - \frac {1}{6 \, \sqrt {x^{2} - 2 \, x - 1}} - \frac {x}{6 \, {\left (x^{2} - 2 \, x - 1\right )}^{\frac {3}{2}}} + \frac {1}{6 \, {\left (x^{2} - 2 \, x - 1\right )}^{\frac {3}{2}}} \]

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)

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mupad [B]  time = 0.28, size = 22, normalized size = 0.51 \[ \frac {x^3-3\,x^2+2}{6\,{\left (x^2-2\,x-1\right )}^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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

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)

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