3.2.29 \(\int \frac {x^3}{\sqrt {4+x^2}} \, dx\) [129]

Optimal. Leaf size=25 \[ -4 \sqrt {4+x^2}+\frac {1}{3} \left (4+x^2\right )^{3/2} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \begin {gather*} \frac {1}{3} \left (x^2+4\right )^{3/2}-4 \sqrt {x^2+4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/Sqrt[4 + x^2],x]

[Out]

-4*Sqrt[4 + x^2] + (4 + x^2)^(3/2)/3

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

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Mathematica [A]
time = 0.01, size = 18, normalized size = 0.72 \begin {gather*} \frac {1}{3} \left (-8+x^2\right ) \sqrt {4+x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/Sqrt[4 + x^2],x]

[Out]

((-8 + x^2)*Sqrt[4 + x^2])/3

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Maple [A]
time = 0.04, size = 23, normalized size = 0.92

method	result	size
gosper	\(\frac {\sqrt {x^{2}+4}\, \left (x^{2}-8\right )}{3}\)	\(15\)
risch	\(\frac {\sqrt {x^{2}+4}\, \left (x^{2}-8\right )}{3}\)	\(15\)
trager	\(\sqrt {x^{2}+4}\, \left (\frac {x^{2}}{3}-\frac {8}{3}\right )\)	\(16\)
default	\(\frac {x^{2} \sqrt {x^{2}+4}}{3}-\frac {8 \sqrt {x^{2}+4}}{3}\)	\(23\)
meijerg	\(\frac {\frac {16 \sqrt {\pi }}{3}-\frac {2 \sqrt {\pi }\, \left (-x^{2}+8\right ) \sqrt {1+\frac {x^{2}}{4}}}{3}}{\sqrt {\pi }}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 1.88, size = 22, normalized size = 0.88 \begin {gather*} \frac {1}{3} \, \sqrt {x^{2} + 4} x^{2} - \frac {8}{3} \, \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="maxima")

[Out]

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

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Fricas [A]
time = 1.26, size = 14, normalized size = 0.56 \begin {gather*} \frac {1}{3} \, \sqrt {x^{2} + 4} {\left (x^{2} - 8\right )} \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]

1/3*sqrt(x^2 + 4)*(x^2 - 8)

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Sympy [A]
time = 0.10, size = 24, normalized size = 0.96 \begin {gather*} \frac {x^{2} \sqrt {x^{2} + 4}}{3} - \frac {8 \sqrt {x^{2} + 4}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2*sqrt(x**2 + 4)/3 - 8*sqrt(x**2 + 4)/3

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Giac [A]
time = 0.73, size = 19, normalized size = 0.76 \begin {gather*} \frac {1}{3} \, {\left (x^{2} + 4\right )}^{\frac {3}{2}} - 4 \, \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="giac")

[Out]

1/3*(x^2 + 4)^(3/2) - 4*sqrt(x^2 + 4)

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Mupad [B]
time = 0.02, size = 14, normalized size = 0.56 \begin {gather*} \frac {\sqrt {x^2+4}\,\left (x^2-8\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(x^2 + 4)^(1/2),x)

[Out]

((x^2 + 4)^(1/2)*(x^2 - 8))/3

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