3.2403 \(\int \frac{x}{\sqrt{2+5 x-3 x^2}} \, dx\)

Optimal. Leaf size=38 \[ -\frac{1}{3} \sqrt{-3 x^2+5 x+2}-\frac{5 \sin ^{-1}\left (\frac{1}{7} (5-6 x)\right )}{6 \sqrt{3}} \]

[Out]

-Sqrt[2 + 5*x - 3*x^2]/3 - (5*ArcSin[(5 - 6*x)/7])/(6*Sqrt[3])

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Rubi [A]  time = 0.0396446, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ -\frac{1}{3} \sqrt{-3 x^2+5 x+2}-\frac{5 \sin ^{-1}\left (\frac{1}{7} (5-6 x)\right )}{6 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[2 + 5*x - 3*x^2],x]

[Out]

-Sqrt[2 + 5*x - 3*x^2]/3 - (5*ArcSin[(5 - 6*x)/7])/(6*Sqrt[3])

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Rubi in Sympy [A]  time = 4.46666, size = 51, normalized size = 1.34 \[ - \frac{\sqrt{- 3 x^{2} + 5 x + 2}}{3} - \frac{5 \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (- 6 x + 5\right )}{6 \sqrt{- 3 x^{2} + 5 x + 2}} \right )}}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(-3*x**2+5*x+2)**(1/2),x)

[Out]

-sqrt(-3*x**2 + 5*x + 2)/3 - 5*sqrt(3)*atan(sqrt(3)*(-6*x + 5)/(6*sqrt(-3*x**2 +
5*x + 2)))/18

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Mathematica [A]  time = 0.0242755, size = 38, normalized size = 1. \[ -\frac{1}{3} \sqrt{-3 x^2+5 x+2}-\frac{5 \sin ^{-1}\left (\frac{1}{7} (5-6 x)\right )}{6 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x/Sqrt[2 + 5*x - 3*x^2],x]

[Out]

-Sqrt[2 + 5*x - 3*x^2]/3 - (5*ArcSin[(5 - 6*x)/7])/(6*Sqrt[3])

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Maple [A]  time = 0.007, size = 27, normalized size = 0.7 \[{\frac{5\,\sqrt{3}}{18}\arcsin \left ( -{\frac{5}{7}}+{\frac{6\,x}{7}} \right ) }-{\frac{1}{3}\sqrt{-3\,{x}^{2}+5\,x+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(-3*x^2+5*x+2)^(1/2),x)

[Out]

5/18*arcsin(-5/7+6/7*x)*3^(1/2)-1/3*(-3*x^2+5*x+2)^(1/2)

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Maxima [A]  time = 0.75064, size = 35, normalized size = 0.92 \[ -\frac{5}{18} \, \sqrt{3} \arcsin \left (-\frac{6}{7} \, x + \frac{5}{7}\right ) - \frac{1}{3} \, \sqrt{-3 \, x^{2} + 5 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(-3*x^2 + 5*x + 2),x, algorithm="maxima")

[Out]

-5/18*sqrt(3)*arcsin(-6/7*x + 5/7) - 1/3*sqrt(-3*x^2 + 5*x + 2)

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Fricas [A]  time = 0.222549, size = 65, normalized size = 1.71 \[ -\frac{1}{18} \, \sqrt{3}{\left (2 \, \sqrt{3} \sqrt{-3 \, x^{2} + 5 \, x + 2} - 5 \, \arctan \left (\frac{\sqrt{3}{\left (6 \, x - 5\right )}}{6 \, \sqrt{-3 \, x^{2} + 5 \, x + 2}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(-3*x^2 + 5*x + 2),x, algorithm="fricas")

[Out]

-1/18*sqrt(3)*(2*sqrt(3)*sqrt(-3*x^2 + 5*x + 2) - 5*arctan(1/6*sqrt(3)*(6*x - 5)
/sqrt(-3*x^2 + 5*x + 2)))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{- \left (x - 2\right ) \left (3 x + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(-3*x**2+5*x+2)**(1/2),x)

[Out]

Integral(x/sqrt(-(x - 2)*(3*x + 1)), x)

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GIAC/XCAS [A]  time = 0.215217, size = 35, normalized size = 0.92 \[ \frac{5}{18} \, \sqrt{3} \arcsin \left (\frac{6}{7} \, x - \frac{5}{7}\right ) - \frac{1}{3} \, \sqrt{-3 \, x^{2} + 5 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(-3*x^2 + 5*x + 2),x, algorithm="giac")

[Out]

5/18*sqrt(3)*arcsin(6/7*x - 5/7) - 1/3*sqrt(-3*x^2 + 5*x + 2)