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 = 0.0239242, 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 \[ -\frac{1}{36} \sqrt{5 x-9 x^2} (5-18 x)-\frac{25}{216} \sin ^{-1}\left (1-\frac{18 x}{5}\right ) \]

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

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Rubi in Sympy [A]  time = 1.72949, size = 29, normalized size = 0.83 \[ - \frac{\left (- 18 x + 5\right ) \sqrt{- 9 x^{2} + 5 x}}{36} + \frac{25 \operatorname{asin}{\left (\frac{18 x}{5} - 1 \right )}}{216} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-9*x**2+5*x)**(1/2),x)

[Out]

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

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Mathematica [B]  time = 0.0387896, size = 72, normalized size = 2.06 \[ \frac{\sqrt{-x (9 x-5)} \left (3 \sqrt{x} \sqrt{9 x-5} (18 x-5)-25 \log \left (3 \sqrt{x}+\sqrt{9 x-5}\right )\right )}{108 \sqrt{x} \sqrt{9 x-5}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[5*x - 9*x^2],x]

[Out]

(Sqrt[-(x*(-5 + 9*x))]*(3*Sqrt[x]*Sqrt[-5 + 9*x]*(-5 + 18*x) - 25*Log[3*Sqrt[x]
+ Sqrt[-5 + 9*x]]))/(108*Sqrt[x]*Sqrt[-5 + 9*x])

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Maple [A]  time = 0.005, size = 28, normalized size = 0.8 \[{\frac{25}{216}\arcsin \left ( -1+{\frac{18\,x}{5}} \right ) }-{\frac{5-18\,x}{36}\sqrt{-9\,{x}^{2}+5\,x}} \]

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 = 0.851527, size = 49, normalized size = 1.4 \[ \frac{1}{2} \, \sqrt{-9 \, x^{2} + 5 \, x} x - \frac{5}{36} \, \sqrt{-9 \, x^{2} + 5 \, x} - \frac{25}{216} \, \arcsin \left (-\frac{18}{5} \, x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-9*x^2 + 5*x),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 = 0.211145, size = 51, normalized size = 1.46 \[ \frac{1}{36} \, \sqrt{-9 \, x^{2} + 5 \, x}{\left (18 \, x - 5\right )} - \frac{25}{108} \, \arctan \left (\frac{\sqrt{-9 \, x^{2} + 5 \, x}}{3 \, x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-9*x^2 + 5*x),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. \[ \int \sqrt{- 9 x^{2} + 5 x}\, dx \]

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/XCAS [A]  time = 0.21081, size = 36, normalized size = 1.03 \[ \frac{1}{36} \, \sqrt{-9 \, x^{2} + 5 \, x}{\left (18 \, x - 5\right )} + \frac{25}{216} \, \arcsin \left (\frac{18}{5} \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-9*x^2 + 5*x),x, algorithm="giac")

[Out]

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