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

Optimal. Leaf size=43 \[ -\frac{1}{12} \sqrt{-3 x^2+5 x+2} (5-6 x)-\frac{49 \sin ^{-1}\left (\frac{1}{7} (5-6 x)\right )}{24 \sqrt{3}} \]

[Out]

-((5 - 6*x)*Sqrt[2 + 5*x - 3*x^2])/12 - (49*ArcSin[(5 - 6*x)/7])/(24*Sqrt[3])

_______________________________________________________________________________________

Rubi [A]  time = 0.0288746, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ -\frac{1}{12} \sqrt{-3 x^2+5 x+2} (5-6 x)-\frac{49 \sin ^{-1}\left (\frac{1}{7} (5-6 x)\right )}{24 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[2 + 5*x - 3*x^2],x]

[Out]

-((5 - 6*x)*Sqrt[2 + 5*x - 3*x^2])/12 - (49*ArcSin[(5 - 6*x)/7])/(24*Sqrt[3])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.07932, size = 56, normalized size = 1.3 \[ - \frac{\left (- 6 x + 5\right ) \sqrt{- 3 x^{2} + 5 x + 2}}{12} - \frac{49 \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (- 6 x + 5\right )}{6 \sqrt{- 3 x^{2} + 5 x + 2}} \right )}}{72} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-6*x + 5)*sqrt(-3*x**2 + 5*x + 2)/12 - 49*sqrt(3)*atan(sqrt(3)*(-6*x + 5)/(6*s
qrt(-3*x**2 + 5*x + 2)))/72

_______________________________________________________________________________________

Mathematica [A]  time = 0.0336152, size = 44, normalized size = 1.02 \[ \left (\frac{x}{2}-\frac{5}{12}\right ) \sqrt{-3 x^2+5 x+2}-\frac{49 \sin ^{-1}\left (\frac{1}{7} (5-6 x)\right )}{24 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(-5/12 + x/2)*Sqrt[2 + 5*x - 3*x^2] - (49*ArcSin[(5 - 6*x)/7])/(24*Sqrt[3])

_______________________________________________________________________________________

Maple [A]  time = 0.004, size = 32, normalized size = 0.7 \[{\frac{49\,\sqrt{3}}{72}\arcsin \left ( -{\frac{5}{7}}+{\frac{6\,x}{7}} \right ) }-{\frac{5-6\,x}{12}\sqrt{-3\,{x}^{2}+5\,x+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-3*x^2+5*x+2)^(1/2),x)

[Out]

49/72*arcsin(-5/7+6/7*x)*3^(1/2)-1/12*(5-6*x)*(-3*x^2+5*x+2)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.820811, size = 55, normalized size = 1.28 \[ \frac{1}{2} \, \sqrt{-3 \, x^{2} + 5 \, x + 2} x - \frac{49}{72} \, \sqrt{3} \arcsin \left (-\frac{6}{7} \, x + \frac{5}{7}\right ) - \frac{5}{12} \, \sqrt{-3 \, x^{2} + 5 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(-3*x^2 + 5*x + 2)*x - 49/72*sqrt(3)*arcsin(-6/7*x + 5/7) - 5/12*sqrt(-3
*x^2 + 5*x + 2)

_______________________________________________________________________________________

Fricas [A]  time = 0.224465, size = 72, normalized size = 1.67 \[ \frac{1}{72} \, \sqrt{3}{\left (2 \, \sqrt{3} \sqrt{-3 \, x^{2} + 5 \, x + 2}{\left (6 \, x - 5\right )} + 49 \, \arctan \left (\frac{\sqrt{3}{\left (6 \, x - 5\right )}}{6 \, \sqrt{-3 \, x^{2} + 5 \, x + 2}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/72*sqrt(3)*(2*sqrt(3)*sqrt(-3*x^2 + 5*x + 2)*(6*x - 5) + 49*arctan(1/6*sqrt(3)
*(6*x - 5)/sqrt(-3*x^2 + 5*x + 2)))

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{- 3 x^{2} + 5 x + 2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(-3*x**2 + 5*x + 2), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.21008, size = 42, normalized size = 0.98 \[ \frac{1}{12} \, \sqrt{-3 \, x^{2} + 5 \, x + 2}{\left (6 \, x - 5\right )} + \frac{49}{72} \, \sqrt{3} \arcsin \left (\frac{6}{7} \, x - \frac{5}{7}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*sqrt(-3*x^2 + 5*x + 2)*(6*x - 5) + 49/72*sqrt(3)*arcsin(6/7*x - 5/7)