3.961 \(\int \frac{x^5}{\sqrt{16-x^4}} \, dx\)

Optimal. Leaf size=29 \[ 4 \sin ^{-1}\left (\frac{x^2}{4}\right )-\frac{1}{4} x^2 \sqrt{16-x^4} \]

[Out]

-(x^2*Sqrt[16 - x^4])/4 + 4*ArcSin[x^2/4]

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Rubi [A]  time = 0.0140243, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {275, 321, 216} \[ 4 \sin ^{-1}\left (\frac{x^2}{4}\right )-\frac{1}{4} x^2 \sqrt{16-x^4} \]

Antiderivative was successfully verified.

[In]

Int[x^5/Sqrt[16 - x^4],x]

[Out]

-(x^2*Sqrt[16 - x^4])/4 + 4*ArcSin[x^2/4]

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \frac{x^5}{\sqrt{16-x^4}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{16-x^2}} \, dx,x,x^2\right )\\ &=-\frac{1}{4} x^2 \sqrt{16-x^4}+4 \operatorname{Subst}\left (\int \frac{1}{\sqrt{16-x^2}} \, dx,x,x^2\right )\\ &=-\frac{1}{4} x^2 \sqrt{16-x^4}+4 \sin ^{-1}\left (\frac{x^2}{4}\right )\\ \end{align*}

Mathematica [A]  time = 0.0059448, size = 29, normalized size = 1. \[ 4 \sin ^{-1}\left (\frac{x^2}{4}\right )-\frac{1}{4} x^2 \sqrt{16-x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/Sqrt[16 - x^4],x]

[Out]

-(x^2*Sqrt[16 - x^4])/4 + 4*ArcSin[x^2/4]

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Maple [A]  time = 0.013, size = 24, normalized size = 0.8 \begin{align*} 4\,\arcsin \left ( 1/4\,{x}^{2} \right ) -{\frac{{x}^{2}}{4}\sqrt{-{x}^{4}+16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(-x^4+16)^(1/2),x)

[Out]

4*arcsin(1/4*x^2)-1/4*x^2*(-x^4+16)^(1/2)

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Maxima [A]  time = 1.45777, size = 59, normalized size = 2.03 \begin{align*} \frac{4 \, \sqrt{-x^{4} + 16}}{x^{2}{\left (\frac{x^{4} - 16}{x^{4}} - 1\right )}} - 4 \, \arctan \left (\frac{\sqrt{-x^{4} + 16}}{x^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(-x^4+16)^(1/2),x, algorithm="maxima")

[Out]

4*sqrt(-x^4 + 16)/(x^2*((x^4 - 16)/x^4 - 1)) - 4*arctan(sqrt(-x^4 + 16)/x^2)

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Fricas [A]  time = 1.44945, size = 86, normalized size = 2.97 \begin{align*} -\frac{1}{4} \, \sqrt{-x^{4} + 16} x^{2} - 8 \, \arctan \left (\frac{\sqrt{-x^{4} + 16} - 4}{x^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(-x^4+16)^(1/2),x, algorithm="fricas")

[Out]

-1/4*sqrt(-x^4 + 16)*x^2 - 8*arctan((sqrt(-x^4 + 16) - 4)/x^2)

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Sympy [A]  time = 2.44931, size = 80, normalized size = 2.76 \begin{align*} \begin{cases} - \frac{i x^{6}}{4 \sqrt{x^{4} - 16}} + \frac{4 i x^{2}}{\sqrt{x^{4} - 16}} - 4 i \operatorname{acosh}{\left (\frac{x^{2}}{4} \right )} & \text{for}\: \frac{\left |{x^{4}}\right |}{16} > 1 \\\frac{x^{6}}{4 \sqrt{16 - x^{4}}} - \frac{4 x^{2}}{\sqrt{16 - x^{4}}} + 4 \operatorname{asin}{\left (\frac{x^{2}}{4} \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(-x**4+16)**(1/2),x)

[Out]

Piecewise((-I*x**6/(4*sqrt(x**4 - 16)) + 4*I*x**2/sqrt(x**4 - 16) - 4*I*acosh(x**2/4), Abs(x**4)/16 > 1), (x**
6/(4*sqrt(16 - x**4)) - 4*x**2/sqrt(16 - x**4) + 4*asin(x**2/4), True))

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Giac [A]  time = 1.17697, size = 31, normalized size = 1.07 \begin{align*} -\frac{1}{4} \, \sqrt{-x^{4} + 16} x^{2} + 4 \, \arcsin \left (\frac{1}{4} \, x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(-x^4+16)^(1/2),x, algorithm="giac")

[Out]

-1/4*sqrt(-x^4 + 16)*x^2 + 4*arcsin(1/4*x^2)