3.925 \(\int \frac {4+x}{(6 x-x^2)^{3/2}} \, dx\)

Optimal. Leaf size=22 \[ -\frac {12-7 x}{9 \sqrt {6 x-x^2}} \]

[Out]

1/9*(-12+7*x)/(-x^2+6*x)^(1/2)

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Rubi [A]  time = 0.00, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {636} \[ -\frac {12-7 x}{9 \sqrt {6 x-x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(4 + x)/(6*x - x^2)^(3/2),x]

[Out]

-(12 - 7*x)/(9*Sqrt[6*x - x^2])

Rule 636

Int[((d_.) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(-2*(b*d - 2*a*e + (2*c*
d - b*e)*x))/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] &&
NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {4+x}{\left (6 x-x^2\right )^{3/2}} \, dx &=-\frac {12-7 x}{9 \sqrt {6 x-x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 19, normalized size = 0.86 \[ \frac {7 x-12}{9 \sqrt {-((x-6) x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(4 + x)/(6*x - x^2)^(3/2),x]

[Out]

(-12 + 7*x)/(9*Sqrt[-((-6 + x)*x)])

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fricas [A]  time = 0.71, size = 27, normalized size = 1.23 \[ -\frac {\sqrt {-x^{2} + 6 \, x} {\left (7 \, x - 12\right )}}{9 \, {\left (x^{2} - 6 \, x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4+x)/(-x^2+6*x)^(3/2),x, algorithm="fricas")

[Out]

-1/9*sqrt(-x^2 + 6*x)*(7*x - 12)/(x^2 - 6*x)

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giac [A]  time = 0.52, size = 27, normalized size = 1.23 \[ -\frac {\sqrt {-x^{2} + 6 \, x} {\left (7 \, x - 12\right )}}{9 \, {\left (x^{2} - 6 \, x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4+x)/(-x^2+6*x)^(3/2),x, algorithm="giac")

[Out]

-1/9*sqrt(-x^2 + 6*x)*(7*x - 12)/(x^2 - 6*x)

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maple [A]  time = 0.00, size = 23, normalized size = 1.05 \[ -\frac {\left (x -6\right ) \left (7 x -12\right ) x}{9 \left (-x^{2}+6 x \right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x+4)/(-x^2+6*x)^(3/2),x)

[Out]

-1/9*x*(x-6)*(-12+7*x)/(-x^2+6*x)^(3/2)

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maxima [A]  time = 0.88, size = 28, normalized size = 1.27 \[ \frac {7 \, x}{9 \, \sqrt {-x^{2} + 6 \, x}} - \frac {4}{3 \, \sqrt {-x^{2} + 6 \, x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4+x)/(-x^2+6*x)^(3/2),x, algorithm="maxima")

[Out]

7/9*x/sqrt(-x^2 + 6*x) - 4/3/sqrt(-x^2 + 6*x)

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mupad [B]  time = 3.50, size = 18, normalized size = 0.82 \[ \frac {7\,x-12}{9\,\sqrt {6\,x-x^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 4)/(6*x - x^2)^(3/2),x)

[Out]

(7*x - 12)/(9*(6*x - x^2)^(1/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x + 4}{\left (- x \left (x - 6\right )\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4+x)/(-x**2+6*x)**(3/2),x)

[Out]

Integral((x + 4)/(-x*(x - 6))**(3/2), x)

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