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

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

[Out]

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

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Rubi [A]  time = 0.0062876, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {641, 195, 216} \[ -\frac{1}{3} \left (1-x^2\right )^{3/2}+\frac{1}{2} x \sqrt{1-x^2}+\frac{1}{2} \sin ^{-1}(x) \]

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 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]

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]

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*}

Mathematica [A]  time = 0.0094504, size = 31, normalized size = 0.82 \[ \frac{1}{6} \left (\sqrt{1-x^2} \left (2 x^2+3 x-2\right )+3 \sin ^{-1}(x)\right ) \]

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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Maple [A]  time = 0.04, size = 29, normalized size = 0.8 \begin{align*} -{\frac{1}{3} \left ( -{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{\frac{\arcsin \left ( x \right ) }{2}}+{\frac{x}{2}\sqrt{-{x}^{2}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.51298, size = 38, normalized size = 1. \begin{align*} -\frac{1}{3} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} + \frac{1}{2} \, \sqrt{-x^{2} + 1} x + \frac{1}{2} \, \arcsin \left (x\right ) \end{align*}

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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Fricas [A]  time = 2.3934, size = 96, normalized size = 2.53 \begin{align*} \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{align*}

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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Sympy [A]  time = 0.194033, size = 39, normalized size = 1.03 \begin{align*} \frac{x^{2} \sqrt{1 - x^{2}}}{3} + \frac{x \sqrt{1 - x^{2}}}{2} - \frac{\sqrt{1 - x^{2}}}{3} + \frac{\operatorname{asin}{\left (x \right )}}{2} \end{align*}

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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Giac [A]  time = 1.19394, size = 34, normalized size = 0.89 \begin{align*} \frac{1}{6} \,{\left ({\left (2 \, x + 3\right )} x - 2\right )} \sqrt{-x^{2} + 1} + \frac{1}{2} \, \arcsin \left (x\right ) \end{align*}

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)