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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, 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 = 0.00, size = 15, normalized size = 1.00 \[ -\frac {1}{2} \sqrt {16-x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/Sqrt[16 - x^4],x]

[Out]

-1/2*Sqrt[16 - x^4]

________________________________________________________________________________________

fricas [A]  time = 0.85, size = 11, normalized size = 0.73 \[ -\frac {1}{2} \, \sqrt {-x^{4} + 16} \]

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)

________________________________________________________________________________________

giac [A]  time = 0.16, size = 11, normalized size = 0.73 \[ -\frac {1}{2} \, \sqrt {-x^{4} + 16} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 23, normalized size = 1.53 \[ \frac {\left (x -2\right ) \left (x +2\right ) \left (x^{2}+4\right )}{2 \sqrt {-x^{4}+16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.30, size = 11, normalized size = 0.73 \[ -\frac {1}{2} \, \sqrt {-x^{4} + 16} \]

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)

________________________________________________________________________________________

mupad [B]  time = 1.23, size = 11, normalized size = 0.73 \[ -\frac {\sqrt {16-x^4}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(16 - x^4)^(1/2),x)

[Out]

-(16 - x^4)^(1/2)/2

________________________________________________________________________________________

sympy [A]  time = 0.32, size = 10, normalized size = 0.67 \[ - \frac {\sqrt {16 - x^{4}}}{2} \]

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

________________________________________________________________________________________