3.7.61 \(\int (1+x) \sqrt {1-x^2} \, dx\)

Optimal. Leaf size=38 \[ -\frac {1}{3} \left (1-x^2\right )^{3/2}+\frac {1}{2} x \sqrt {1-x^2}+\frac {1}{2} \sin ^{-1}(x) \]

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Rubi [A]  time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {641, 195, 216} \begin {gather*} -\frac {1}{3} \left (1-x^2\right )^{3/2}+\frac {1}{2} x \sqrt {1-x^2}+\frac {1}{2} \sin ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x)*Sqrt[1 - x^2],x]

[Out]

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

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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]

Rule 641

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

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 31, normalized size = 0.82 \begin {gather*} \frac {1}{6} \left (\sqrt {1-x^2} \left (2 x^2+3 x-2\right )+3 \sin ^{-1}(x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x)*Sqrt[1 - x^2],x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.14, size = 46, normalized size = 1.21 \begin {gather*} \frac {1}{6} \sqrt {1-x^2} \left (2 x^2+3 x-2\right )-\tan ^{-1}\left (\frac {\sqrt {1-x^2}}{x+1}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 + x)*Sqrt[1 - x^2],x]

[Out]

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

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fricas [A]  time = 0.39, size = 40, normalized size = 1.05 \begin {gather*} \frac {1}{6} \, {\left (2 \, x^{2} + 3 \, x - 2\right )} \sqrt {-x^{2} + 1} - \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.16, size = 25, normalized size = 0.66 \begin {gather*} \frac {1}{6} \, {\left ({\left (2 \, x + 3\right )} x - 2\right )} \sqrt {-x^{2} + 1} + \frac {1}{2} \, \arcsin \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 29, normalized size = 0.76 \begin {gather*} \frac {\sqrt {-x^{2}+1}\, x}{2}+\frac {\arcsin \relax (x )}{2}-\frac {\left (-x^{2}+1\right )^{\frac {3}{2}}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 3.01, size = 28, normalized size = 0.74 \begin {gather*} -\frac {1}{3} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} + \frac {1}{2} \, \sqrt {-x^{2} + 1} x + \frac {1}{2} \, \arcsin \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.03, size = 25, normalized size = 0.66 \begin {gather*} \frac {\mathrm {asin}\relax (x)}{2}+\sqrt {1-x^2}\,\left (\frac {x^2}{3}+\frac {x}{2}-\frac {1}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x^2)^(1/2)*(x + 1),x)

[Out]

asin(x)/2 + (1 - x^2)^(1/2)*(x/2 + x^2/3 - 1/3)

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sympy [A]  time = 0.21, size = 39, normalized size = 1.03 \begin {gather*} \frac {x^{2} \sqrt {1 - x^{2}}}{3} + \frac {x \sqrt {1 - x^{2}}}{2} - \frac {\sqrt {1 - x^{2}}}{3} + \frac {\operatorname {asin}{\relax (x )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2*sqrt(1 - x**2)/3 + x*sqrt(1 - x**2)/2 - sqrt(1 - x**2)/3 + asin(x)/2

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