### 3.2414 $$\int \frac{x}{\sqrt{2+5 x-3 x^2}} \, dx$$

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 = 0.0127859, 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, Rules used = {640, 619, 216} $-\frac{1}{3} \sqrt{-3 x^2+5 x+2}-\frac{5 \sin ^{-1}\left (\frac{1}{7} (5-6 x)\right )}{6 \sqrt{3}}$

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])

Rule 640

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

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{x}{\sqrt{2+5 x-3 x^2}} \, dx &=-\frac{1}{3} \sqrt{2+5 x-3 x^2}+\frac{5}{6} \int \frac{1}{\sqrt{2+5 x-3 x^2}} \, dx\\ &=-\frac{1}{3} \sqrt{2+5 x-3 x^2}-\frac{5 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{49}}} \, dx,x,5-6 x\right )}{42 \sqrt{3}}\\ &=-\frac{1}{3} \sqrt{2+5 x-3 x^2}-\frac{5 \sin ^{-1}\left (\frac{1}{7} (5-6 x)\right )}{6 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0089675, 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 = 0.04, size = 27, normalized size = 0.7 \begin{align*}{\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}} \end{align*}

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 = 1.44108, size = 35, normalized size = 0.92 \begin{align*} -\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} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-3*x^2+5*x+2)^(1/2),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 [B]  time = 2.49295, size = 155, normalized size = 4.08 \begin{align*} -\frac{5}{18} \, \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 ) - \frac{1}{3} \, \sqrt{-3 \, x^{2} + 5 \, x + 2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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 [A]  time = 1.1524, size = 35, normalized size = 0.92 \begin{align*} \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} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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