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])

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Rubi [A]  time = 0.0122895, 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, Rules used = {612, 619, 216} \[ -\frac{1}{12} \sqrt{-3 x^2+5 x+2} (5-6 x)-\frac{49 \sin ^{-1}\left (\frac{1}{7} (5-6 x)\right )}{24 \sqrt{3}} \]

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])

Rule 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

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 \sqrt{2+5 x-3 x^2} \, dx &=-\frac{1}{12} (5-6 x) \sqrt{2+5 x-3 x^2}+\frac{49}{24} \int \frac{1}{\sqrt{2+5 x-3 x^2}} \, dx\\ &=-\frac{1}{12} (5-6 x) \sqrt{2+5 x-3 x^2}-\frac{7 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{49}}} \, dx,x,5-6 x\right )}{24 \sqrt{3}}\\ &=-\frac{1}{12} (5-6 x) \sqrt{2+5 x-3 x^2}-\frac{49 \sin ^{-1}\left (\frac{1}{7} (5-6 x)\right )}{24 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0193787, 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])

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Maple [A]  time = 0.046, size = 32, normalized size = 0.7 \begin{align*}{\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}} \end{align*}

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)

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Maxima [A]  time = 1.7193, size = 55, normalized size = 1.28 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [A]  time = 1.90065, size = 170, normalized size = 3.95 \begin{align*} \frac{1}{12} \, \sqrt{-3 \, x^{2} + 5 \, x + 2}{\left (6 \, x - 5\right )} - \frac{49}{72} \, \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((-3*x^2+5*x+2)^(1/2),x, algorithm="fricas")

[Out]

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

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

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)

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Giac [A]  time = 1.29829, size = 42, normalized size = 0.98 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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