3.419 \(\int x^3 (\sqrt{1-x}+\sqrt{1+x})^2 \, dx\)

Optimal. Leaf size=38 \[ \frac{x^4}{2}+\frac{2}{5} \left (1-x^2\right )^{5/2}-\frac{2}{3} \left (1-x^2\right )^{3/2} \]

[Out]

x^4/2 - (2*(1 - x^2)^(3/2))/3 + (2*(1 - x^2)^(5/2))/5

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Rubi [A]  time = 0.113624, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {6742, 266, 43} \[ \frac{x^4}{2}+\frac{2}{5} \left (1-x^2\right )^{5/2}-\frac{2}{3} \left (1-x^2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(Sqrt[1 - x] + Sqrt[1 + x])^2,x]

[Out]

x^4/2 - (2*(1 - x^2)^(3/2))/3 + (2*(1 - x^2)^(5/2))/5

Rule 6742

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

Rule 266

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x^3 \left (\sqrt{1-x}+\sqrt{1+x}\right )^2 \, dx &=\int \left (2 x^3+2 x^3 \sqrt{1-x^2}\right ) \, dx\\ &=\frac{x^4}{2}+2 \int x^3 \sqrt{1-x^2} \, dx\\ &=\frac{x^4}{2}+\operatorname{Subst}\left (\int \sqrt{1-x} x \, dx,x,x^2\right )\\ &=\frac{x^4}{2}+\operatorname{Subst}\left (\int \left (\sqrt{1-x}-(1-x)^{3/2}\right ) \, dx,x,x^2\right )\\ &=\frac{x^4}{2}-\frac{2}{3} \left (1-x^2\right )^{3/2}+\frac{2}{5} \left (1-x^2\right )^{5/2}\\ \end{align*}

Mathematica [A]  time = 0.0536561, size = 38, normalized size = 1. \[ \frac{x^4}{2}+\frac{2}{5} \left (1-x^2\right )^{5/2}-\frac{2}{3} \left (1-x^2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(Sqrt[1 - x] + Sqrt[1 + x])^2,x]

[Out]

x^4/2 - (2*(1 - x^2)^(3/2))/3 + (2*(1 - x^2)^(5/2))/5

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Maple [A]  time = 0.004, size = 33, normalized size = 0.9 \begin{align*}{\frac{{x}^{4}}{2}}+{\frac{ \left ( 2\,{x}^{2}-2 \right ) \left ( 3\,{x}^{2}+2 \right ) }{15}\sqrt{1-x}\sqrt{1+x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^4+2/15*(1+x)^(1/2)*(1-x)^(1/2)*(x^2-1)*(3*x^2+2)

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Maxima [A]  time = 1.69087, size = 42, normalized size = 1.11 \begin{align*} \frac{1}{2} \, x^{4} - \frac{2}{5} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} x^{2} - \frac{4}{15} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*((1-x)^(1/2)+(1+x)^(1/2))^2,x, algorithm="maxima")

[Out]

1/2*x^4 - 2/5*(-x^2 + 1)^(3/2)*x^2 - 4/15*(-x^2 + 1)^(3/2)

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Fricas [A]  time = 1.24945, size = 80, normalized size = 2.11 \begin{align*} \frac{1}{2} \, x^{4} + \frac{2}{15} \,{\left (3 \, x^{4} - x^{2} - 2\right )} \sqrt{x + 1} \sqrt{-x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*((1-x)^(1/2)+(1+x)^(1/2))^2,x, algorithm="fricas")

[Out]

1/2*x^4 + 2/15*(3*x^4 - x^2 - 2)*sqrt(x + 1)*sqrt(-x + 1)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*((1-x)**(1/2)+(1+x)**(1/2))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.11482, size = 76, normalized size = 2. \begin{align*} \frac{1}{2} \,{\left (x + 1\right )}^{4} - 2 \,{\left (x + 1\right )}^{3} + \frac{2}{15} \,{\left ({\left (3 \,{\left (x + 1\right )}{\left (x - 3\right )} + 17\right )}{\left (x + 1\right )} - 10\right )}{\left (x + 1\right )}^{\frac{3}{2}} \sqrt{-x + 1} + 3 \,{\left (x + 1\right )}^{2} - 2 \, x - 2 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*((1-x)^(1/2)+(1+x)^(1/2))^2,x, algorithm="giac")

[Out]

1/2*(x + 1)^4 - 2*(x + 1)^3 + 2/15*((3*(x + 1)*(x - 3) + 17)*(x + 1) - 10)*(x + 1)^(3/2)*sqrt(-x + 1) + 3*(x +
1)^2 - 2*x - 2