3.495 \(\int \frac{x^2}{\sqrt{-1-x^3}} \, dx\)

Optimal. Leaf size=15 \[ -\frac{2}{3} \sqrt{-x^3-1} \]

[Out]

(-2*Sqrt[-1 - x^3])/3

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Rubi [A]  time = 0.0028506, 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} \[ -\frac{2}{3} \sqrt{-x^3-1} \]

Antiderivative was successfully verified.

[In]

Int[x^2/Sqrt[-1 - x^3],x]

[Out]

(-2*Sqrt[-1 - x^3])/3

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

Mathematica [A]  time = 0.002075, size = 15, normalized size = 1. \[ -\frac{2}{3} \sqrt{-x^3-1} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/Sqrt[-1 - x^3],x]

[Out]

(-2*Sqrt[-1 - x^3])/3

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Maple [A]  time = 0.004, size = 23, normalized size = 1.5 \begin{align*}{\frac{ \left ( 2+2\,x \right ) \left ({x}^{2}-x+1 \right ) }{3}{\frac{1}{\sqrt{-{x}^{3}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(-x^3-1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.04045, size = 15, normalized size = 1. \begin{align*} -\frac{2}{3} \, \sqrt{-x^{3} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*sqrt(-x^3 - 1)

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Fricas [A]  time = 1.7077, size = 28, normalized size = 1.87 \begin{align*} -\frac{2}{3} \, \sqrt{-x^{3} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*sqrt(-x^3 - 1)

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Sympy [A]  time = 0.136464, size = 14, normalized size = 0.93 \begin{align*} - \frac{2 \sqrt{- x^{3} - 1}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(-x**3-1)**(1/2),x)

[Out]

-2*sqrt(-x**3 - 1)/3

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Giac [A]  time = 1.11422, size = 15, normalized size = 1. \begin{align*} -\frac{2}{3} \, \sqrt{-x^{3} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*sqrt(-x^3 - 1)