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

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

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Rubi [A]  time = 0.01, antiderivative size = 47, normalized size of antiderivative = 1.00, 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 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 (-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*}

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Mathematica [A]  time = 0.01, 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]

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

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IntegrateAlgebraic [A]  time = 0.32, size = 55, normalized size = 1.17 \[ \frac {\sqrt {-x^2+6 x-7} \left (-x^3+9 x^2-24 x+18\right )}{6 \left (-x+\sqrt {2}+3\right )^2 \left (x+\sqrt {2}-3\right )^2} \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-7 + 6*x - x^2)^(-5/2),x]

[Out]

(Sqrt[-7 + 6*x - x^2]*(18 - 24*x + 9*x^2 - x^3))/(6*(3 + Sqrt[2] - x)^2*(-3 + Sqrt[2] + x)^2)

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fricas [A]  time = 1.32, size = 47, normalized size = 1.00 \[ -\frac {{\left (x^{3} - 9 \, x^{2} + 24 \, x - 18\right )} \sqrt {-x^{2} + 6 \, x - 7}}{6 \, {\left (x^{4} - 12 \, x^{3} + 50 \, x^{2} - 84 \, x + 49\right )}} \]

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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giac [A]  time = 0.65, size = 35, normalized size = 0.74 \[ -\frac {{\left ({\left ({\left (x - 9\right )} x + 24\right )} x - 18\right )} \sqrt {-x^{2} + 6 \, x - 7}}{6 \, {\left (x^{2} - 6 \, x + 7\right )}^{2}} \]

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

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maple [A]  time = 0.44, size = 28, normalized size = 0.60

method	result	size
gosper	\(-\frac {x^{3}-9 x^{2}+24 x -18}{6 \left (-x^{2}+6 x -7\right )^{\frac {3}{2}}}\)	\(28\)
trager	\(-\frac {\left (x^{3}-9 x^{2}+24 x -18\right ) \sqrt {-x^{2}+6 x -7}}{6 \left (x^{2}-6 x +7\right )^{2}}\)	\(38\)
risch	\(\frac {x^{3}-9 x^{2}+24 x -18}{6 \left (x^{2}-6 x +7\right ) \sqrt {-x^{2}+6 x -7}}\)	\(38\)
default	\(-\frac {-2 x +6}{12 \left (-x^{2}+6 x -7\right )^{\frac {3}{2}}}-\frac {-2 x +6}{12 \sqrt {-x^{2}+6 x -7}}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.46, size = 59, normalized size = 1.26 \[ \frac {x}{6 \, \sqrt {-x^{2} + 6 \, x - 7}} - \frac {1}{2 \, \sqrt {-x^{2} + 6 \, x - 7}} + \frac {x}{6 \, {\left (-x^{2} + 6 \, x - 7\right )}^{\frac {3}{2}}} - \frac {1}{2 \, {\left (-x^{2} + 6 \, x - 7\right )}^{\frac {3}{2}}} \]

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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mupad [B]  time = 0.29, size = 29, normalized size = 0.62 \[ -\frac {\left (4\,x-12\right )\,\left (8\,x^2-48\,x+48\right )}{192\,{\left (-x^2+6\,x-7\right )}^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(6*x - x^2 - 7)^(5/2),x)

[Out]

-((4*x - 12)*(8*x^2 - 48*x + 48))/(192*(6*x - x^2 - 7)^(3/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (- x^{2} + 6 x - 7\right )^{\frac {5}{2}}}\, dx \]

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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