3.962 \(\int \frac{x^3}{\sqrt{16-x^4}} \, dx\)

Optimal. Leaf size=15 \[ -\frac{1}{2} \sqrt{16-x^4} \]

[Out]

-Sqrt[16 - x^4]/2

________________________________________________________________________________________

Rubi [A]  time = 0.0029264, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {261} \[ -\frac{1}{2} \sqrt{16-x^4} \]

Antiderivative was successfully verified.

[In]

Int[x^3/Sqrt[16 - x^4],x]

[Out]

-Sqrt[16 - x^4]/2

Rule 261

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

Rubi steps

\begin{align*} \int \frac{x^3}{\sqrt{16-x^4}} \, dx &=-\frac{1}{2} \sqrt{16-x^4}\\ \end{align*}

Mathematica [A]  time = 0.0021108, size = 15, normalized size = 1. \[ -\frac{1}{2} \sqrt{16-x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/Sqrt[16 - x^4],x]

[Out]

-Sqrt[16 - x^4]/2

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 23, normalized size = 1.5 \begin{align*}{\frac{ \left ( -2+x \right ) \left ( 2+x \right ) \left ({x}^{2}+4 \right ) }{2}{\frac{1}{\sqrt{-{x}^{4}+16}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(-x^4+16)^(1/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.980254, size = 15, normalized size = 1. \begin{align*} -\frac{1}{2} \, \sqrt{-x^{4} + 16} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(-x^4+16)^(1/2),x, algorithm="maxima")

[Out]

-1/2*sqrt(-x^4 + 16)

________________________________________________________________________________________

Fricas [A]  time = 1.46667, size = 30, normalized size = 2. \begin{align*} -\frac{1}{2} \, \sqrt{-x^{4} + 16} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(-x^4+16)^(1/2),x, algorithm="fricas")

[Out]

-1/2*sqrt(-x^4 + 16)

________________________________________________________________________________________

Sympy [A]  time = 0.214763, size = 10, normalized size = 0.67 \begin{align*} - \frac{\sqrt{16 - x^{4}}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(-x**4+16)**(1/2),x)

[Out]

-sqrt(16 - x**4)/2

________________________________________________________________________________________

Giac [A]  time = 1.20427, size = 15, normalized size = 1. \begin{align*} -\frac{1}{2} \, \sqrt{-x^{4} + 16} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(-x^4+16)^(1/2),x, algorithm="giac")

[Out]

-1/2*sqrt(-x^4 + 16)