3.2.25 \(\int x^3 \sqrt {4-x^2} \, dx\) [125]

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

[Out]

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

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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 = {272, 45} \begin {gather*} \frac {1}{5} \left (4-x^2\right )^{5/2}-\frac {4}{3} \left (4-x^2\right )^{3/2} \end {gather*}

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 45

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 272

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} \text {Subst}\left (\int \sqrt {4-x} x \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {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*}

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Mathematica [A]
time = 0.01, size = 27, normalized size = 0.87 \begin {gather*} \frac {1}{15} \sqrt {4-x^2} \left (-32-4 x^2+3 x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*Sqrt[4 - x^2],x]

[Out]

(Sqrt[4 - x^2]*(-32 - 4*x^2 + 3*x^4))/15

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Maple [A]
time = 0.04, size = 27, normalized size = 0.87

method	result	size
trager	\(\left (\frac {1}{5} x^{4}-\frac {4}{15} x^{2}-\frac {32}{15}\right ) \sqrt {-x^{2}+4}\)	\(23\)
gosper	\(\frac {\left (-2+x \right ) \left (2+x \right ) \left (3 x^{2}+8\right ) \sqrt {-x^{2}+4}}{15}\)	\(25\)
default	\(-\frac {x^{2} \left (-x^{2}+4\right )^{\frac {3}{2}}}{5}-\frac {8 \left (-x^{2}+4\right )^{\frac {3}{2}}}{15}\)	\(27\)
risch	\(-\frac {\left (3 x^{4}-4 x^{2}-32\right ) \left (x^{2}-4\right )}{15 \sqrt {-x^{2}+4}}\)	\(29\)
meijerg	\(-\frac {8 \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (1-\frac {x^{2}}{4}\right )^{\frac {3}{2}} \left (\frac {3 x^{2}}{4}+2\right )}{15}\right )}{\sqrt {\pi }}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 1.62, size = 26, normalized size = 0.84 \begin {gather*} -\frac {1}{5} \, {\left (-x^{2} + 4\right )}^{\frac {3}{2}} x^{2} - \frac {8}{15} \, {\left (-x^{2} + 4\right )}^{\frac {3}{2}} \end {gather*}

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)

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Fricas [A]
time = 0.68, size = 23, normalized size = 0.74 \begin {gather*} \frac {1}{15} \, {\left (3 \, x^{4} - 4 \, x^{2} - 32\right )} \sqrt {-x^{2} + 4} \end {gather*}

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)

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Sympy [A]
time = 0.15, size = 39, normalized size = 1.26 \begin {gather*} \frac {x^{4} \sqrt {4 - x^{2}}}{5} - \frac {4 x^{2} \sqrt {4 - x^{2}}}{15} - \frac {32 \sqrt {4 - x^{2}}}{15} \end {gather*}

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

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Giac [A]
time = 0.60, size = 30, normalized size = 0.97 \begin {gather*} \frac {1}{5} \, {\left (x^{2} - 4\right )}^{2} \sqrt {-x^{2} + 4} - \frac {4}{3} \, {\left (-x^{2} + 4\right )}^{\frac {3}{2}} \end {gather*}

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)

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

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)

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