3.478 \(\int \frac{1}{(-7+6 x-x^2)^{5/2}} \, dx\)

Optimal. Leaf size=47 \[ -\frac{3-x}{6 \sqrt{-x^2+6 x-7}}-\frac{3-x}{6 \left (-x^2+6 x-7\right )^{3/2}} \]

[Out]

-(3 - x)/(6*(-7 + 6*x - x^2)^(3/2)) - (3 - x)/(6*Sqrt[-7 + 6*x - x^2])

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Rubi [A]  time = 0.0074342, antiderivative size = 47, 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} \[ -\frac{3-x}{6 \sqrt{-x^2+6 x-7}}-\frac{3-x}{6 \left (-x^2+6 x-7\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(-7 + 6*x - x^2)^(-5/2),x]

[Out]

-(3 - x)/(6*(-7 + 6*x - x^2)^(3/2)) - (3 - x)/(6*Sqrt[-7 + 6*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 (-7+6 x-x^2\right )^{5/2}} \, dx &=-\frac{3-x}{6 \left (-7+6 x-x^2\right )^{3/2}}+\frac{1}{3} \int \frac{1}{\left (-7+6 x-x^2\right )^{3/2}} \, dx\\ &=-\frac{3-x}{6 \left (-7+6 x-x^2\right )^{3/2}}-\frac{3-x}{6 \sqrt{-7+6 x-x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0110864, size = 29, normalized size = 0.62 \[ -\frac{(x-3) \left (x^2-6 x+6\right )}{6 \left (-x^2+6 x-7\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(-7 + 6*x - x^2)^(-5/2),x]

[Out]

-((-3 + x)*(6 - 6*x + x^2))/(6*(-7 + 6*x - x^2)^(3/2))

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Maple [A]  time = 0.003, size = 28, normalized size = 0.6 \begin{align*} -{\frac{{x}^{3}-9\,{x}^{2}+24\,x-18}{6} \left ( -{x}^{2}+6\,x-7 \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-x^2+6*x-7)^(5/2),x)

[Out]

-1/6*(x^3-9*x^2+24*x-18)/(-x^2+6*x-7)^(3/2)

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Maxima [A]  time = 0.932898, size = 80, normalized size = 1.7 \begin{align*} \frac{x}{6 \, \sqrt{-x^{2} + 6 \, x - 7}} - \frac{1}{2 \, \sqrt{-x^{2} + 6 \, x - 7}} + \frac{x}{6 \,{\left (-x^{2} + 6 \, x - 7\right )}^{\frac{3}{2}}} - \frac{1}{2 \,{\left (-x^{2} + 6 \, x - 7\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^2+6*x-7)^(5/2),x, algorithm="maxima")

[Out]

1/6*x/sqrt(-x^2 + 6*x - 7) - 1/2/sqrt(-x^2 + 6*x - 7) + 1/6*x/(-x^2 + 6*x - 7)^(3/2) - 1/2/(-x^2 + 6*x - 7)^(3
/2)

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Fricas [A]  time = 2.11454, size = 120, normalized size = 2.55 \begin{align*} -\frac{{\left (x^{3} - 9 \, x^{2} + 24 \, x - 18\right )} \sqrt{-x^{2} + 6 \, x - 7}}{6 \,{\left (x^{4} - 12 \, x^{3} + 50 \, x^{2} - 84 \, x + 49\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^2+6*x-7)^(5/2),x, algorithm="fricas")

[Out]

-1/6*(x^3 - 9*x^2 + 24*x - 18)*sqrt(-x^2 + 6*x - 7)/(x^4 - 12*x^3 + 50*x^2 - 84*x + 49)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (- x^{2} + 6 x - 7\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x**2+6*x-7)**(5/2),x)

[Out]

Integral((-x**2 + 6*x - 7)**(-5/2), x)

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Giac [A]  time = 1.08344, size = 47, normalized size = 1. \begin{align*} -\frac{{\left ({\left ({\left (x - 9\right )} x + 24\right )} x - 18\right )} \sqrt{-x^{2} + 6 \, x - 7}}{6 \,{\left (x^{2} - 6 \, x + 7\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^2+6*x-7)^(5/2),x, algorithm="giac")

[Out]

-1/6*(((x - 9)*x + 24)*x - 18)*sqrt(-x^2 + 6*x - 7)/(x^2 - 6*x + 7)^2