3.235 \(\int \frac{1}{\sqrt{4+3 x-2 x^2}} \, dx\)

Optimal. Leaf size=19 \[ -\frac{\sin ^{-1}\left (\frac{3-4 x}{\sqrt{41}}\right )}{\sqrt{2}} \]

[Out]

-(ArcSin[(3 - 4*x)/Sqrt[41]]/Sqrt[2])

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Rubi [A]  time = 0.0137482, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {619, 216} \[ -\frac{\sin ^{-1}\left (\frac{3-4 x}{\sqrt{41}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[4 + 3*x - 2*x^2],x]

[Out]

-(ArcSin[(3 - 4*x)/Sqrt[41]]/Sqrt[2])

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 \frac{1}{\sqrt{4+3 x-2 x^2}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{41}}} \, dx,x,3-4 x\right )}{\sqrt{82}}\\ &=-\frac{\sin ^{-1}\left (\frac{3-4 x}{\sqrt{41}}\right )}{\sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.00997, size = 19, normalized size = 1. \[ -\frac{\sin ^{-1}\left (\frac{3-4 x}{\sqrt{41}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[4 + 3*x - 2*x^2],x]

[Out]

-(ArcSin[(3 - 4*x)/Sqrt[41]]/Sqrt[2])

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Maple [A]  time = 0.002, size = 15, normalized size = 0.8 \begin{align*}{\frac{\sqrt{2}}{2}\arcsin \left ({\frac{4\,\sqrt{41}}{41} \left ( x-{\frac{3}{4}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-2*x^2+3*x+4)^(1/2),x)

[Out]

1/2*2^(1/2)*arcsin(4/41*41^(1/2)*(x-3/4))

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Maxima [A]  time = 1.44799, size = 22, normalized size = 1.16 \begin{align*} -\frac{1}{2} \, \sqrt{2} \arcsin \left (-\frac{1}{41} \, \sqrt{41}{\left (4 \, x - 3\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-2*x^2+3*x+4)^(1/2),x, algorithm="maxima")

[Out]

-1/2*sqrt(2)*arcsin(-1/41*sqrt(41)*(4*x - 3))

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Fricas [B]  time = 1.75203, size = 93, normalized size = 4.89 \begin{align*} -\sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-2 \, x^{2} + 3 \, x + 4} - 2 \, \sqrt{2}}{2 \, x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-2*x^2+3*x+4)^(1/2),x, algorithm="fricas")

[Out]

-sqrt(2)*arctan(1/2*(sqrt(2)*sqrt(-2*x^2 + 3*x + 4) - 2*sqrt(2))/x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- 2 x^{2} + 3 x + 4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-2*x**2+3*x+4)**(1/2),x)

[Out]

Integral(1/sqrt(-2*x**2 + 3*x + 4), x)

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Giac [A]  time = 1.08688, size = 22, normalized size = 1.16 \begin{align*} \frac{1}{2} \, \sqrt{2} \arcsin \left (\frac{1}{41} \, \sqrt{41}{\left (4 \, x - 3\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-2*x^2+3*x+4)^(1/2),x, algorithm="giac")

[Out]

1/2*sqrt(2)*arcsin(1/41*sqrt(41)*(4*x - 3))