3.120 \(\int \frac{1}{\sqrt{2+5 x-3 x^2}} \, dx\)

Optimal. Leaf size=17 \[ -\frac{\sin ^{-1}\left (\frac{1}{7} (5-6 x)\right )}{\sqrt{3}} \]

[Out]

-(ArcSin[(5 - 6*x)/7]/Sqrt[3])

________________________________________________________________________________________

Rubi [A]  time = 0.006666, antiderivative size = 17, 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} \[ -\frac{\sin ^{-1}\left (\frac{1}{7} (5-6 x)\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[2 + 5*x - 3*x^2],x]

[Out]

-(ArcSin[(5 - 6*x)/7]/Sqrt[3])

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{2+5 x-3 x^2}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{49}}} \, dx,x,5-6 x\right )}{7 \sqrt{3}}\\ &=-\frac{\sin ^{-1}\left (\frac{1}{7} (5-6 x)\right )}{\sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0061719, size = 17, normalized size = 1. \[ -\frac{\sin ^{-1}\left (\frac{1}{7} (5-6 x)\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[2 + 5*x - 3*x^2],x]

[Out]

-(ArcSin[(5 - 6*x)/7]/Sqrt[3])

________________________________________________________________________________________

Maple [A]  time = 0.049, size = 12, normalized size = 0.7 \begin{align*}{\frac{\sqrt{3}}{3}\arcsin \left ( -{\frac{5}{7}}+{\frac{6\,x}{7}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-3*x^2+5*x+2)^(1/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.70508, size = 15, normalized size = 0.88 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arcsin \left (-\frac{6}{7} \, x + \frac{5}{7}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x^2+5*x+2)^(1/2),x, algorithm="maxima")

[Out]

-1/3*sqrt(3)*arcsin(-6/7*x + 5/7)

________________________________________________________________________________________

Fricas [B]  time = 1.9463, size = 115, normalized size = 6.76 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{\sqrt{3} \sqrt{-3 \, x^{2} + 5 \, x + 2}{\left (6 \, x - 5\right )}}{6 \,{\left (3 \, x^{2} - 5 \, x - 2\right )}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x^2+5*x+2)^(1/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- 3 x^{2} + 5 x + 2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(-3*x**2 + 5*x + 2), x)

________________________________________________________________________________________

Giac [A]  time = 1.3186, size = 15, normalized size = 0.88 \begin{align*} \frac{1}{3} \, \sqrt{3} \arcsin \left (\frac{6}{7} \, x - \frac{5}{7}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x^2+5*x+2)^(1/2),x, algorithm="giac")

[Out]

1/3*sqrt(3)*arcsin(6/7*x - 5/7)