3.25.28 \(\int x \sqrt {8+2 x-x^2} \, dx\) [2428]

Optimal. Leaf size=56 \[ -\frac {1}{2} (1-x) \sqrt {8+2 x-x^2}-\frac {1}{3} \left (8+2 x-x^2\right )^{3/2}-\frac {9}{2} \sin ^{-1}\left (\frac {1-x}{3}\right ) \]

[Out]

-1/3*(-x^2+2*x+8)^(3/2)+9/2*arcsin(-1/3+1/3*x)-1/2*(1-x)*(-x^2+2*x+8)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {654, 626, 633, 222} \begin {gather*} -\frac {9}{2} \text {ArcSin}\left (\frac {1-x}{3}\right )-\frac {1}{3} \left (-x^2+2 x+8\right )^{3/2}-\frac {1}{2} (1-x) \sqrt {-x^2+2 x+8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[8 + 2*x - x^2],x]

[Out]

-1/2*((1 - x)*Sqrt[8 + 2*x - x^2]) - (8 + 2*x - x^2)^(3/2)/3 - (9*ArcSin[(1 - x)/3])/2

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 626

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 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 x \sqrt {8+2 x-x^2} \, dx &=-\frac {1}{3} \left (8+2 x-x^2\right )^{3/2}+\int \sqrt {8+2 x-x^2} \, dx\\ &=-\frac {1}{2} (1-x) \sqrt {8+2 x-x^2}-\frac {1}{3} \left (8+2 x-x^2\right )^{3/2}+\frac {9}{2} \int \frac {1}{\sqrt {8+2 x-x^2}} \, dx\\ &=-\frac {1}{2} (1-x) \sqrt {8+2 x-x^2}-\frac {1}{3} \left (8+2 x-x^2\right )^{3/2}-\frac {3}{4} \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{36}}} \, dx,x,2-2 x\right )\\ &=-\frac {1}{2} (1-x) \sqrt {8+2 x-x^2}-\frac {1}{3} \left (8+2 x-x^2\right )^{3/2}-\frac {9}{2} \sin ^{-1}\left (\frac {1-x}{3}\right )\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 52, normalized size = 0.93 \begin {gather*} \frac {1}{6} \sqrt {8+2 x-x^2} \left (-19-x+2 x^2\right )-9 \tan ^{-1}\left (\frac {\sqrt {8+2 x-x^2}}{2+x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*Sqrt[8 + 2*x - x^2],x]

[Out]

(Sqrt[8 + 2*x - x^2]*(-19 - x + 2*x^2))/6 - 9*ArcTan[Sqrt[8 + 2*x - x^2]/(2 + x)]

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Maple [A]
time = 0.70, size = 43, normalized size = 0.77

method	result	size
risch	\(-\frac {\left (2 x^{2}-x -19\right ) \left (x^{2}-2 x -8\right )}{6 \sqrt {-x^{2}+2 x +8}}+\frac {9 \arcsin \left (-\frac {1}{3}+\frac {x}{3}\right )}{2}\)	\(42\)
default	\(-\frac {\left (-x^{2}+2 x +8\right )^{\frac {3}{2}}}{3}-\frac {\left (2-2 x \right ) \sqrt {-x^{2}+2 x +8}}{4}+\frac {9 \arcsin \left (-\frac {1}{3}+\frac {x}{3}\right )}{2}\)	\(43\)
trager	\(\left (\frac {1}{3} x^{2}-\frac {1}{6} x -\frac {19}{6}\right ) \sqrt {-x^{2}+2 x +8}+\frac {9 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-x \RootOf \left (\textit {\_Z}^{2}+1\right )+\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {-x^{2}+2 x +8}\right )}{2}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(-x^2+2*x+8)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/3*(-x^2+2*x+8)^(3/2)-1/4*(2-2*x)*(-x^2+2*x+8)^(1/2)+9/2*arcsin(-1/3+1/3*x)

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Maxima [A]
time = 0.50, size = 52, normalized size = 0.93 \begin {gather*} -\frac {1}{3} \, {\left (-x^{2} + 2 \, x + 8\right )}^{\frac {3}{2}} + \frac {1}{2} \, \sqrt {-x^{2} + 2 \, x + 8} x - \frac {1}{2} \, \sqrt {-x^{2} + 2 \, x + 8} - \frac {9}{2} \, \arcsin \left (-\frac {1}{3} \, x + \frac {1}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-x^2+2*x+8)^(1/2),x, algorithm="maxima")

[Out]

-1/3*(-x^2 + 2*x + 8)^(3/2) + 1/2*sqrt(-x^2 + 2*x + 8)*x - 1/2*sqrt(-x^2 + 2*x + 8) - 9/2*arcsin(-1/3*x + 1/3)

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Fricas [A]
time = 2.92, size = 54, normalized size = 0.96 \begin {gather*} \frac {1}{6} \, {\left (2 \, x^{2} - x - 19\right )} \sqrt {-x^{2} + 2 \, x + 8} - \frac {9}{2} \, \arctan \left (\frac {\sqrt {-x^{2} + 2 \, x + 8} {\left (x - 1\right )}}{x^{2} - 2 \, x - 8}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-x^2+2*x+8)^(1/2),x, algorithm="fricas")

[Out]

1/6*(2*x^2 - x - 19)*sqrt(-x^2 + 2*x + 8) - 9/2*arctan(sqrt(-x^2 + 2*x + 8)*(x - 1)/(x^2 - 2*x - 8))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \sqrt {- \left (x - 4\right ) \left (x + 2\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-x**2+2*x+8)**(1/2),x)

[Out]

Integral(x*sqrt(-(x - 4)*(x + 2)), x)

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Giac [A]
time = 1.45, size = 32, normalized size = 0.57 \begin {gather*} \frac {1}{6} \, {\left ({\left (2 \, x - 1\right )} x - 19\right )} \sqrt {-x^{2} + 2 \, x + 8} + \frac {9}{2} \, \arcsin \left (\frac {1}{3} \, x - \frac {1}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-x^2+2*x+8)^(1/2),x, algorithm="giac")

[Out]

1/6*((2*x - 1)*x - 19)*sqrt(-x^2 + 2*x + 8) + 9/2*arcsin(1/3*x - 1/3)

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Mupad [B]
time = 0.08, size = 47, normalized size = 0.84 \begin {gather*} -\frac {\sqrt {-x^2+2\,x+8}\,\left (-8\,x^2+4\,x+76\right )}{24}-\frac {\ln \left (x-1-\sqrt {-x^2+2\,x+8}\,1{}\mathrm {i}\right )\,9{}\mathrm {i}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(2*x - x^2 + 8)^(1/2),x)

[Out]

- (log(x - (2*x - x^2 + 8)^(1/2)*1i - 1)*9i)/2 - ((2*x - x^2 + 8)^(1/2)*(4*x - 8*x^2 + 76))/24

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