3.11 \(\int \sqrt{5 x-9 x^2} \, dx\)

Optimal. Leaf size=35 \[ -\frac{1}{36} \sqrt{5 x-9 x^2} (5-18 x)-\frac{25}{216} \sin ^{-1}\left (1-\frac{18 x}{5}\right ) \]

[Out]

-((5 - 18*x)*Sqrt[5*x - 9*x^2])/36 - (25*ArcSin[1 - (18*x)/5])/216

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Rubi [A]  time = 0.0097332, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {612, 619, 216} \[ -\frac{1}{36} \sqrt{5 x-9 x^2} (5-18 x)-\frac{25}{216} \sin ^{-1}\left (1-\frac{18 x}{5}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[5*x - 9*x^2],x]

[Out]

-((5 - 18*x)*Sqrt[5*x - 9*x^2])/36 - (25*ArcSin[1 - (18*x)/5])/216

Rule 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \sqrt{5 x-9 x^2} \, dx &=-\frac{1}{36} (5-18 x) \sqrt{5 x-9 x^2}+\frac{25}{72} \int \frac{1}{\sqrt{5 x-9 x^2}} \, dx\\ &=-\frac{1}{36} (5-18 x) \sqrt{5 x-9 x^2}-\frac{5}{216} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{25}}} \, dx,x,5-18 x\right )\\ &=-\frac{1}{36} (5-18 x) \sqrt{5 x-9 x^2}-\frac{25}{216} \sin ^{-1}\left (1-\frac{18 x}{5}\right )\\ \end{align*}

Mathematica [A]  time = 0.0360542, size = 58, normalized size = 1.66 \[ \frac{-3 x \left (162 x^2-135 x+25\right )-25 \sqrt{5-9 x} \sqrt{x} \sin ^{-1}\left (\sqrt{1-\frac{9 x}{5}}\right )}{108 \sqrt{-x (9 x-5)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[5*x - 9*x^2],x]

[Out]

(-3*x*(25 - 135*x + 162*x^2) - 25*Sqrt[5 - 9*x]*Sqrt[x]*ArcSin[Sqrt[1 - (9*x)/5]])/(108*Sqrt[-(x*(-5 + 9*x))])

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Maple [A]  time = 0.044, size = 28, normalized size = 0.8 \begin{align*}{\frac{25}{216}\arcsin \left ( -1+{\frac{18\,x}{5}} \right ) }-{\frac{5-18\,x}{36}\sqrt{-9\,{x}^{2}+5\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-9*x^2+5*x)^(1/2),x)

[Out]

25/216*arcsin(-1+18/5*x)-1/36*(5-18*x)*(-9*x^2+5*x)^(1/2)

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Maxima [A]  time = 1.77616, size = 49, normalized size = 1.4 \begin{align*} \frac{1}{2} \, \sqrt{-9 \, x^{2} + 5 \, x} x - \frac{5}{36} \, \sqrt{-9 \, x^{2} + 5 \, x} - \frac{25}{216} \, \arcsin \left (-\frac{18}{5} \, x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-9*x^2+5*x)^(1/2),x, algorithm="maxima")

[Out]

1/2*sqrt(-9*x^2 + 5*x)*x - 5/36*sqrt(-9*x^2 + 5*x) - 25/216*arcsin(-18/5*x + 1)

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Fricas [A]  time = 2.07538, size = 105, normalized size = 3. \begin{align*} \frac{1}{36} \, \sqrt{-9 \, x^{2} + 5 \, x}{\left (18 \, x - 5\right )} - \frac{25}{108} \, \arctan \left (\frac{\sqrt{-9 \, x^{2} + 5 \, x}}{3 \, x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-9*x^2+5*x)^(1/2),x, algorithm="fricas")

[Out]

1/36*sqrt(-9*x^2 + 5*x)*(18*x - 5) - 25/108*arctan(1/3*sqrt(-9*x^2 + 5*x)/x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{- 9 x^{2} + 5 x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-9*x**2+5*x)**(1/2),x)

[Out]

Integral(sqrt(-9*x**2 + 5*x), x)

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Giac [A]  time = 1.22833, size = 36, normalized size = 1.03 \begin{align*} \frac{1}{36} \, \sqrt{-9 \, x^{2} + 5 \, x}{\left (18 \, x - 5\right )} + \frac{25}{216} \, \arcsin \left (\frac{18}{5} \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-9*x^2+5*x)^(1/2),x, algorithm="giac")

[Out]

1/36*sqrt(-9*x^2 + 5*x)*(18*x - 5) + 25/216*arcsin(18/5*x - 1)