\(\int \frac {1}{(-3-2 x+x^2)^{5/2}} \, dx\) [229]

Optimal result

Integrand size = 12, antiderivative size = 43 \[ \int \frac {1}{\left (-3-2 x+x^2\right )^{5/2}} \, dx=\frac {1-x}{12 \left (-3-2 x+x^2\right )^{3/2}}-\frac {1-x}{24 \sqrt {-3-2 x+x^2}} \]

[Out]

1/12*(1-x)/(x^2-2*x-3)^(3/2)+1/24*(-1+x)/(x^2-2*x-3)^(1/2)

Rubi [A] (verified)

Time = 0.01 (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} \[ \int \frac {1}{\left (-3-2 x+x^2\right )^{5/2}} \, dx=\frac {1-x}{12 \left (x^2-2 x-3\right )^{3/2}}-\frac {1-x}{24 \sqrt {x^2-2 x-3}} \]

[In]

Int[(-3 - 2*x + x^2)^(-5/2),x]

[Out]

(1 - x)/(12*(-3 - 2*x + x^2)^(3/2)) - (1 - x)/(24*Sqrt[-3 - 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}{12 \left (-3-2 x+x^2\right )^{3/2}}-\frac {1}{6} \int \frac {1}{\left (-3-2 x+x^2\right )^{3/2}} \, dx \\ & = \frac {1-x}{12 \left (-3-2 x+x^2\right )^{3/2}}-\frac {1-x}{24 \sqrt {-3-2 x+x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (-3-2 x+x^2\right )^{5/2}} \, dx=\frac {\sqrt {-3-2 x+x^2} \left (5-3 x-3 x^2+x^3\right )}{24 (-3+x)^2 (1+x)^2} \]

[In]

Integrate[(-3 - 2*x + x^2)^(-5/2),x]

[Out]

(Sqrt[-3 - 2*x + x^2]*(5 - 3*x - 3*x^2 + x^3))/(24*(-3 + x)^2*(1 + x)^2)

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.60

method	result	size
trager	\(\frac {x^{3}-3 x^{2}-3 x +5}{24 \left (x^{2}-2 x -3\right )^{\frac {3}{2}}}\)	\(26\)
risch	\(\frac {x^{3}-3 x^{2}-3 x +5}{24 \left (x^{2}-2 x -3\right )^{\frac {3}{2}}}\)	\(26\)
gosper	\(\frac {\left (1+x \right ) \left (-3+x \right ) \left (x^{3}-3 x^{2}-3 x +5\right )}{24 \left (x^{2}-2 x -3\right )^{\frac {5}{2}}}\)	\(32\)
default	\(-\frac {-2+2 x}{24 \left (x^{2}-2 x -3\right )^{\frac {3}{2}}}+\frac {-2+2 x}{48 \sqrt {x^{2}-2 x -3}}\)	\(36\)

[In]

int(1/(x^2-2*x-3)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/24*(x^3-3*x^2-3*x+5)/(x^2-2*x-3)^(3/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (31) = 62\).

Time = 0.24 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.49 \[ \int \frac {1}{\left (-3-2 x+x^2\right )^{5/2}} \, dx=\frac {x^{4} - 4 \, x^{3} - 2 \, x^{2} + {\left (x^{3} - 3 \, x^{2} - 3 \, x + 5\right )} \sqrt {x^{2} - 2 \, x - 3} + 12 \, x + 9}{24 \, {\left (x^{4} - 4 \, x^{3} - 2 \, x^{2} + 12 \, x + 9\right )}} \]

[In]

integrate(1/(x^2-2*x-3)^(5/2),x, algorithm="fricas")

[Out]

1/24*(x^4 - 4*x^3 - 2*x^2 + (x^3 - 3*x^2 - 3*x + 5)*sqrt(x^2 - 2*x - 3) + 12*x + 9)/(x^4 - 4*x^3 - 2*x^2 + 12*
x + 9)

Sympy [F]

\[ \int \frac {1}{\left (-3-2 x+x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (x^{2} - 2 x - 3\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(1/(x**2-2*x-3)**(5/2),x)

[Out]

Integral((x**2 - 2*x - 3)**(-5/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.19 \[ \int \frac {1}{\left (-3-2 x+x^2\right )^{5/2}} \, dx=\frac {x}{24 \, \sqrt {x^{2} - 2 \, x - 3}} - \frac {1}{24 \, \sqrt {x^{2} - 2 \, x - 3}} - \frac {x}{12 \, {\left (x^{2} - 2 \, x - 3\right )}^{\frac {3}{2}}} + \frac {1}{12 \, {\left (x^{2} - 2 \, x - 3\right )}^{\frac {3}{2}}} \]

[In]

integrate(1/(x^2-2*x-3)^(5/2),x, algorithm="maxima")

[Out]

1/24*x/sqrt(x^2 - 2*x - 3) - 1/24/sqrt(x^2 - 2*x - 3) - 1/12*x/(x^2 - 2*x - 3)^(3/2) + 1/12/(x^2 - 2*x - 3)^(3
/2)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.53 \[ \int \frac {1}{\left (-3-2 x+x^2\right )^{5/2}} \, dx=\frac {{\left ({\left (x - 3\right )} x - 3\right )} x + 5}{24 \, {\left (x^{2} - 2 \, x - 3\right )}^{\frac {3}{2}}} \]

[In]

integrate(1/(x^2-2*x-3)^(5/2),x, algorithm="giac")

[Out]

1/24*(((x - 3)*x - 3)*x + 5)/(x^2 - 2*x - 3)^(3/2)

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.63 \[ \int \frac {1}{\left (-3-2 x+x^2\right )^{5/2}} \, dx=-\frac {\left (4\,x-4\right )\,\left (-8\,x^2+16\,x+40\right )}{768\,{\left (x^2-2\,x-3\right )}^{3/2}} \]

[In]

int(1/(x^2 - 2*x - 3)^(5/2),x)

[Out]

-((4*x - 4)*(16*x - 8*x^2 + 40))/(768*(x^2 - 2*x - 3)^(3/2))