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

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

[Out]

(-2*Sqrt[1 - x^3])/3

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Rubi [A]  time = 0.0028425, 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} \[ -\frac{2}{3} \sqrt{1-x^3} \]

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 = 0.0021221, size = 15, normalized size = 1. \[ -\frac{2}{3} \sqrt{1-x^3} \]

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 = 21, normalized size = 1.4 \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 = 0.95771, 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 = 1.45588, 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 = 0.188281, size = 12, normalized size = 0.8 \begin{align*} - \frac{2 \sqrt{1 - x^{3}}}{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(1 - x**3)/3

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Giac [A]  time = 1.16123, 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)