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

Optimal result

Integrand size = 15, antiderivative size = 15 \[ \int \frac {x^2}{\sqrt {-1-x^3}} \, dx=-\frac {2}{3} \sqrt {-1-x^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {267} \[ \int \frac {x^2}{\sqrt {-1-x^3}} \, dx=-\frac {2}{3} \sqrt {-x^3-1} \]

[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*} \text {integral}& = -\frac {2}{3} \sqrt {-1-x^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\sqrt {-1-x^3}} \, dx=-\frac {2}{3} \sqrt {-1-x^3} \]

[In]

Integrate[x^2/Sqrt[-1 - x^3],x]

[Out]

(-2*Sqrt[-1 - x^3])/3

Maple [A] (verified)

Time = 3.89 (sec) , antiderivative size = 12, normalized size of antiderivative = 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	\(-\frac {2 \sqrt {-x^{3}-1}}{3}\)	\(12\)
pseudoelliptic	\(-\frac {2 \sqrt {-x^{3}-1}}{3}\)	\(12\)
risch	\(\frac {\frac {2 x^{3}}{3}+\frac {2}{3}}{\sqrt {-x^{3}-1}}\)	\(17\)
gosper	\(\frac {2 \left (1+x \right ) \left (x^{2}-x +1\right )}{3 \sqrt {-x^{3}-1}}\)	\(23\)
meijerg	\(-\frac {i \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {x^{3}+1}\right )}{3 \sqrt {\pi }}\)	\(25\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \frac {x^2}{\sqrt {-1-x^3}} \, dx=-\frac {2}{3} \, \sqrt {-x^{3} - 1} \]

[In]

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

[Out]

-2/3*sqrt(-x^3 - 1)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {x^2}{\sqrt {-1-x^3}} \, dx=- \frac {2 \sqrt {- x^{3} - 1}}{3} \]

[In]

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

[Out]

-2*sqrt(-x**3 - 1)/3

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \frac {x^2}{\sqrt {-1-x^3}} \, dx=-\frac {2}{3} \, \sqrt {-x^{3} - 1} \]

[In]

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

[Out]

-2/3*sqrt(-x^3 - 1)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \frac {x^2}{\sqrt {-1-x^3}} \, dx=-\frac {2}{3} \, \sqrt {-x^{3} - 1} \]

[In]

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

[Out]

-2/3*sqrt(-x^3 - 1)

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \frac {x^2}{\sqrt {-1-x^3}} \, dx=-\frac {2\,\sqrt {-x^3-1}}{3} \]

[In]

int(x^2/(- x^3 - 1)^(1/2),x)

[Out]

-(2*(- x^3 - 1)^(1/2))/3