3.5.95 \(\int \frac {x^2}{\sqrt {-1-x^3}} \, dx\) [495]

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

[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 = {267} \begin {gather*} -\frac {2}{3} \sqrt {-x^3-1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/Sqrt[-1 - x^3],x]

[Out]

(-2*Sqrt[-1 - x^3])/3

Rule 267

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 = 0.01, size = 15, normalized size = 1.00 \begin {gather*} -\frac {2}{3} \sqrt {-1-x^3} \end {gather*}

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 = 0.14, size = 12, normalized size = 0.80

method	result	size
derivativedivides	\(-\frac {2 \sqrt {-x^{3}-1}}{3}\)	\(12\)
default	\(-\frac {2 \sqrt {-x^{3}-1}}{3}\)	\(12\)
trager	\(-\frac {2 \sqrt {-x^{3}-1}}{3}\)	\(12\)
elliptic	\(-\frac {2 \sqrt {-x^{3}-1}}{3}\)	\(12\)
risch	\(\frac {\frac {2 x^{3}}{3}+\frac {2}{3}}{\sqrt {-x^{3}-1}}\)	\(17\)
gosper	\(\frac {2 \left (x +1\right ) \left (x^{2}-x +1\right )}{3 \sqrt {-x^{3}-1}}\)	\(23\)
meijerg	\(-\frac {i \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {x^{3}+1}\right )}{3 \sqrt {\pi }}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(-x^3-1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 11, normalized size = 0.73 \begin {gather*} -\frac {2}{3} \, \sqrt {-x^{3} - 1} \end {gather*}

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 = 0.34, size = 11, normalized size = 0.73 \begin {gather*} -\frac {2}{3} \, \sqrt {-x^{3} - 1} \end {gather*}

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 = 0.05, size = 14, normalized size = 0.93 \begin {gather*} - \frac {2 \sqrt {- x^{3} - 1}}{3} \end {gather*}

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 = 2.22, size = 11, normalized size = 0.73 \begin {gather*} -\frac {2}{3} \, \sqrt {-x^{3} - 1} \end {gather*}

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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Mupad [B]
time = 0.02, size = 11, normalized size = 0.73 \begin {gather*} -\frac {2\,\sqrt {-x^3-1}}{3} \end {gather*}

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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