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]

-(12 - 7*x)/(9*Sqrt[6*x - x^2])

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Rubi [A]  time = 0.0047028, antiderivative size = 22, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0100448, 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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Maple [A]  time = 0.004, size = 23, normalized size = 1.1 \begin{align*} -{\frac{x \left ( -6+x \right ) \left ( -12+7\,x \right ) }{9} \left ( -{x}^{2}+6\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.03403, size = 38, normalized size = 1.73 \begin{align*} \frac{7 \, x}{9 \, \sqrt{-x^{2} + 6 \, x}} - \frac{4}{3 \, \sqrt{-x^{2} + 6 \, x}} \end{align*}

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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Fricas [A]  time = 1.4443, size = 62, normalized size = 2.82 \begin{align*} -\frac{\sqrt{-x^{2} + 6 \, x}{\left (7 \, x - 12\right )}}{9 \,{\left (x^{2} - 6 \, x\right )}} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x + 4}{\left (- x \left (x - 6\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

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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Giac [A]  time = 1.12045, size = 36, normalized size = 1.64 \begin{align*} -\frac{\sqrt{-x^{2} + 6 \, x}{\left (7 \, x - 12\right )}}{9 \,{\left (x^{2} - 6 \, x\right )}} \end{align*}

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)