\(\int \frac {1}{(-7+6 x-x^2)^{5/2}} \, dx\) [478]

Optimal result

Integrand size = 14, antiderivative size = 47 \[ \int \frac {1}{\left (-7+6 x-x^2\right )^{5/2}} \, dx=-\frac {3-x}{6 \left (-7+6 x-x^2\right )^{3/2}}-\frac {3-x}{6 \sqrt {-7+6 x-x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {628, 627} \[ \int \frac {1}{\left (-7+6 x-x^2\right )^{5/2}} \, dx=-\frac {3-x}{6 \sqrt {-x^2+6 x-7}}-\frac {3-x}{6 \left (-x^2+6 x-7\right )^{3/2}} \]

[In]

Int[(-7 + 6*x - x^2)^(-5/2),x]

[Out]

-1/6*(3 - x)/(-7 + 6*x - x^2)^(3/2) - (3 - x)/(6*Sqrt[-7 + 6*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 {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] (verified)

Time = 0.13 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.66 \[ \int \frac {1}{\left (-7+6 x-x^2\right )^{5/2}} \, dx=-\frac {-18+24 x-9 x^2+x^3}{6 \left (-7+6 x-x^2\right )^{3/2}} \]

[In]

Integrate[(-7 + 6*x - x^2)^(-5/2),x]

[Out]

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

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 28, normalized size of antiderivative = 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	\(-\frac {-2 x +6}{12 \left (-x^{2}+6 x -7\right )^{\frac {3}{2}}}-\frac {-2 x +6}{12 \sqrt {-x^{2}+6 x -7}}\)	\(40\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (-7+6 x-x^2\right )^{5/2}} \, dx=-\frac {{\left (x^{3} - 9 \, x^{2} + 24 \, x - 18\right )} \sqrt {-x^{2} + 6 \, x - 7}}{6 \, {\left (x^{4} - 12 \, x^{3} + 50 \, x^{2} - 84 \, x + 49\right )}} \]

[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)

Sympy [F]

\[ \int \frac {1}{\left (-7+6 x-x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (- x^{2} + 6 x - 7\right )^{\frac {5}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.26 \[ \int \frac {1}{\left (-7+6 x-x^2\right )^{5/2}} \, dx=\frac {x}{6 \, \sqrt {-x^{2} + 6 \, x - 7}} - \frac {1}{2 \, \sqrt {-x^{2} + 6 \, x - 7}} + \frac {x}{6 \, {\left (-x^{2} + 6 \, x - 7\right )}^{\frac {3}{2}}} - \frac {1}{2 \, {\left (-x^{2} + 6 \, x - 7\right )}^{\frac {3}{2}}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.74 \[ \int \frac {1}{\left (-7+6 x-x^2\right )^{5/2}} \, dx=-\frac {{\left ({\left ({\left (x - 9\right )} x + 24\right )} x - 18\right )} \sqrt {-x^{2} + 6 \, x - 7}}{6 \, {\left (x^{2} - 6 \, x + 7\right )}^{2}} \]

[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

Mupad [B] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.62 \[ \int \frac {1}{\left (-7+6 x-x^2\right )^{5/2}} \, dx=-\frac {\left (4\,x-12\right )\,\left (8\,x^2-48\,x+48\right )}{192\,{\left (-x^2+6\,x-7\right )}^{3/2}} \]

[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))