3.25.18 \(\int \frac {x}{\sqrt {-2+5 x-3 x^2}} \, dx\) [2418]

Optimal. Leaf size=34 \[ -\frac {1}{3} \sqrt {-2+5 x-3 x^2}-\frac {5 \sin ^{-1}(5-6 x)}{6 \sqrt {3}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, 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 = {654, 633, 222} \begin {gather*} -\frac {5 \text {ArcSin}(5-6 x)}{6 \sqrt {3}}-\frac {1}{3} \sqrt {-3 x^2+5 x-2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[-2 + 5*x - 3*x^2],x]

[Out]

-1/3*Sqrt[-2 + 5*x - 3*x^2] - (5*ArcSin[5 - 6*x])/(6*Sqrt[3])

Rule 222

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]

Rule 633

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 654

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]

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

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Mathematica [A]
time = 0.07, size = 51, normalized size = 1.50 \begin {gather*} \frac {1}{9} \left (-3 \sqrt {-2+5 x-3 x^2}-5 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {-6+15 x-9 x^2}}{-2+3 x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[-2 + 5*x - 3*x^2],x]

[Out]

(-3*Sqrt[-2 + 5*x - 3*x^2] - 5*Sqrt[3]*ArcTan[Sqrt[-6 + 15*x - 9*x^2]/(-2 + 3*x)])/9

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Maple [A]
time = 0.83, size = 27, normalized size = 0.79

method	result	size
default	\(\frac {5 \arcsin \left (-5+6 x \right ) \sqrt {3}}{18}-\frac {\sqrt {-3 x^{2}+5 x -2}}{3}\)	\(27\)
risch	\(\frac {3 x^{2}-5 x +2}{3 \sqrt {-3 x^{2}+5 x -2}}+\frac {5 \arcsin \left (-5+6 x \right ) \sqrt {3}}{18}\)	\(37\)
trager	\(-\frac {\sqrt {-3 x^{2}+5 x -2}}{3}+\frac {5 \RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (-6 x \RootOf \left (\textit {\_Z}^{2}+3\right )+6 \sqrt {-3 x^{2}+5 x -2}+5 \RootOf \left (\textit {\_Z}^{2}+3\right )\right )}{18}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.50, size = 26, normalized size = 0.76 \begin {gather*} \frac {5}{18} \, \sqrt {3} \arcsin \left (6 \, x - 5\right ) - \frac {1}{3} \, \sqrt {-3 \, x^{2} + 5 \, x - 2} \end {gather*}

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*x - 5) - 1/3*sqrt(-3*x^2 + 5*x - 2)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (26) = 52\).
time = 2.29, size = 55, normalized size = 1.62 \begin {gather*} -\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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {- \left (x - 1\right ) \left (3 x - 2\right )}}\, dx \end {gather*}

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 - 1)*(3*x - 2)), x)

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Giac [A]
time = 1.19, size = 26, normalized size = 0.76 \begin {gather*} \frac {5}{18} \, \sqrt {3} \arcsin \left (6 \, x - 5\right ) - \frac {1}{3} \, \sqrt {-3 \, x^{2} + 5 \, x - 2} \end {gather*}

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*x - 5) - 1/3*sqrt(-3*x^2 + 5*x - 2)

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Mupad [B]
time = 1.14, size = 46, normalized size = 1.35 \begin {gather*} -\frac {\sqrt {-3\,x^2+5\,x-2}}{3}-\frac {\sqrt {3}\,\ln \left (\sqrt {-3\,x^2+5\,x-2}+\frac {\sqrt {3}\,\left (3\,x-\frac {5}{2}\right )\,1{}\mathrm {i}}{3}\right )\,5{}\mathrm {i}}{18} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(5*x - 3*x^2 - 2)^(1/2),x)

[Out]

- (3^(1/2)*log((5*x - 3*x^2 - 2)^(1/2) + (3^(1/2)*(3*x - 5/2)*1i)/3)*5i)/18 - (5*x - 3*x^2 - 2)^(1/2)/3

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