3.125 \(\int x^3 \sqrt {4-x^2} \, dx\)

Optimal. Leaf size=31 \[ \frac {1}{5} \left (4-x^2\right )^{5/2}-\frac {4}{3} \left (4-x^2\right )^{3/2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac {1}{5} \left (4-x^2\right )^{5/2}-\frac {4}{3} \left (4-x^2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[x^3*Sqrt[4 - x^2],x]

[Out]

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

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])

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]]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 22, normalized size = 0.71 \[ -\frac {1}{15} \left (4-x^2\right )^{3/2} \left (3 x^2+8\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*Sqrt[4 - x^2],x]

[Out]

-1/15*((4 - x^2)^(3/2)*(8 + 3*x^2))

________________________________________________________________________________________

fricas [A]  time = 0.41, size = 23, normalized size = 0.74 \[ \frac {1}{15} \, {\left (3 \, x^{4} - 4 \, x^{2} - 32\right )} \sqrt {-x^{2} + 4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(-x^2+4)^(1/2),x, algorithm="fricas")

[Out]

1/15*(3*x^4 - 4*x^2 - 32)*sqrt(-x^2 + 4)

________________________________________________________________________________________

giac [A]  time = 0.88, size = 30, normalized size = 0.97 \[ \frac {1}{5} \, {\left (x^{2} - 4\right )}^{2} \sqrt {-x^{2} + 4} - \frac {4}{3} \, {\left (-x^{2} + 4\right )}^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(-x^2+4)^(1/2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.00, size = 25, normalized size = 0.81 \[ \frac {\left (x -2\right ) \left (x +2\right ) \left (3 x^{2}+8\right ) \sqrt {-x^{2}+4}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(x-2)*(x+2)*(3*x^2+8)*(-x^2+4)^(1/2)

________________________________________________________________________________________

maxima [A]  time = 0.97, size = 26, normalized size = 0.84 \[ -\frac {1}{5} \, {\left (-x^{2} + 4\right )}^{\frac {3}{2}} x^{2} - \frac {8}{15} \, {\left (-x^{2} + 4\right )}^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(-x^2+4)^(1/2),x, algorithm="maxima")

[Out]

-1/5*(-x^2 + 4)^(3/2)*x^2 - 8/15*(-x^2 + 4)^(3/2)

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 23, normalized size = 0.74 \[ -\sqrt {4-x^2}\,\left (-\frac {x^4}{5}+\frac {4\,x^2}{15}+\frac {32}{15}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(4 - x^2)^(1/2),x)

[Out]

-(4 - x^2)^(1/2)*((4*x^2)/15 - x^4/5 + 32/15)

________________________________________________________________________________________

sympy [A]  time = 0.64, size = 39, normalized size = 1.26 \[ \frac {x^{4} \sqrt {4 - x^{2}}}{5} - \frac {4 x^{2} \sqrt {4 - x^{2}}}{15} - \frac {32 \sqrt {4 - x^{2}}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(-x**2+4)**(1/2),x)

[Out]

x**4*sqrt(4 - x**2)/5 - 4*x**2*sqrt(4 - x**2)/15 - 32*sqrt(4 - x**2)/15

________________________________________________________________________________________