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

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

[Out]

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

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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} \[ -\frac {2}{3} \sqrt {-x^3-1} \]

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*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 1.00 \[ -\frac {2}{3} \sqrt {-x^3-1} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/Sqrt[-1 - x^3],x]

[Out]

(-2*Sqrt[-1 - x^3])/3

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fricas [A]  time = 0.84, size = 11, normalized size = 0.73 \[ -\frac {2}{3} \, \sqrt {-x^{3} - 1} \]

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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giac [A]  time = 0.15, size = 11, normalized size = 0.73 \[ -\frac {2}{3} \, \sqrt {-x^{3} - 1} \]

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)

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maple [A]  time = 0.01, size = 23, normalized size = 1.53 \[ \frac {2 \left (x +1\right ) \left (x^{2}-x +1\right )}{3 \sqrt {-x^{3}-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.24, size = 11, normalized size = 0.73 \[ -\frac {2}{3} \, \sqrt {-x^{3} - 1} \]

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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mupad [B]  time = 0.02, size = 11, normalized size = 0.73 \[ -\frac {2\,\sqrt {-x^3-1}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(- x^3 - 1)^(1/2),x)

[Out]

-(2*(- x^3 - 1)^(1/2))/3

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sympy [A]  time = 0.21, size = 14, normalized size = 0.93 \[ - \frac {2 \sqrt {- x^{3} - 1}}{3} \]

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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