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

Optimal. Leaf size=19 \[ x^2-\frac {2}{3} \left (1-x^2\right )^{3/2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {6742, 261} \[ x^2-\frac {2}{3} \left (1-x^2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^2 - (2*(1 - x^2)^(3/2))/3

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 19, normalized size = 1.00 \[ x^2-\frac {2}{3} \left (1-x^2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^2 - (2*(1 - x^2)^(3/2))/3

________________________________________________________________________________________

fricas [A]  time = 0.49, size = 23, normalized size = 1.21 \[ x^{2} + \frac {2}{3} \, {\left (x^{2} - 1\right )} \sqrt {x + 1} \sqrt {-x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [B]  time = 0.25, size = 51, normalized size = 2.68 \[ {\left (x + 1\right )}^{2} + \frac {1}{3} \, {\left ({\left (2 \, x - 5\right )} {\left (x + 1\right )} + 9\right )} \sqrt {x + 1} \sqrt {-x + 1} + \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} - 2 \, x - 2 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x + 1)^2 + 1/3*((2*x - 5)*(x + 1) + 9)*sqrt(x + 1)*sqrt(-x + 1) + sqrt(x + 1)*(x - 2)*sqrt(-x + 1) - 2*x - 2

________________________________________________________________________________________

maple [A]  time = 0.00, size = 24, normalized size = 1.26 \[ x^{2}+\frac {2 \sqrt {x +1}\, \sqrt {-x +1}\, \left (x^{2}-1\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.34, size = 15, normalized size = 0.79 \[ x^{2} - \frac {2}{3} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2 - 2/3*(-x^2 + 1)^(3/2)

________________________________________________________________________________________

mupad [B]  time = 2.98, size = 33, normalized size = 1.74 \[ x^2-\frac {\sqrt {1-x}\,\left (-\frac {2\,x^3}{3}-\frac {2\,x^2}{3}+\frac {2\,x}{3}+\frac {2}{3}\right )}{\sqrt {x+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 106.39, size = 110, normalized size = 5.79 \[ - \frac {x^{3}}{3} - x + \frac {\left (x + 1\right )^{3}}{3} - 4 \left (\begin {cases} \frac {x \sqrt {1 - x} \sqrt {x + 1}}{4} + \frac {\operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{2} & \text {for}\: x \geq -1 \wedge x < 1 \end {cases}\right ) + 4 \left (\begin {cases} \frac {x \sqrt {1 - x} \sqrt {x + 1}}{4} - \frac {\left (1 - x\right )^{\frac {3}{2}} \left (x + 1\right )^{\frac {3}{2}}}{6} + \frac {\operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{2} & \text {for}\: x \geq -1 \wedge x < 1 \end {cases}\right ) - 1 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x**3/3 - x + (x + 1)**3/3 - 4*Piecewise((x*sqrt(1 - x)*sqrt(x + 1)/4 + asin(sqrt(2)*sqrt(x + 1)/2)/2, (x >= -
1) & (x < 1))) + 4*Piecewise((x*sqrt(1 - x)*sqrt(x + 1)/4 - (1 - x)**(3/2)*(x + 1)**(3/2)/6 + asin(sqrt(2)*sqr
t(x + 1)/2)/2, (x >= -1) & (x < 1))) - 1

________________________________________________________________________________________