3.1.42 \(\int \sqrt {\frac {x}{1-x^3}} \, dx\) [42]

Optimal. Leaf size=41 \[ \frac {2 \sqrt {\frac {x}{1-x^3}} \sqrt {1-x^3} \arcsin \left (x^{3/2}\right )}{3 \sqrt {x}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {6851, 335, 281, 222} \begin {gather*} \frac {2 \sqrt {\frac {x}{1-x^3}} \sqrt {1-x^3} \arcsin \left (x^{3/2}\right )}{3 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[x/(1 - x^3)],x]

[Out]

(2*Sqrt[x/(1 - x^3)]*Sqrt[1 - x^3]*ArcSin[x^(3/2)])/(3*Sqrt[x])

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 6851

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a*v^m*w^n)^FracPart[p]/(v^(m*Fr
acPart[p])*w^(n*FracPart[p]))), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !
FreeQ[v, x] &&  !FreeQ[w, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=\frac {\left (\sqrt {\frac {x}{1-x^3}} \sqrt {1-x^3}\right ) \int \frac {\sqrt {x}}{\sqrt {1-x^3}} \, dx}{\sqrt {x}}\\ &=\frac {\left (2 \sqrt {\frac {x}{1-x^3}} \sqrt {1-x^3}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-x^6}} \, dx,x,\sqrt {x}\right )}{\sqrt {x}}\\ &=\frac {\left (2 \sqrt {\frac {x}{1-x^3}} \sqrt {1-x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,x^{3/2}\right )}{3 \sqrt {x}}\\ &=\frac {2 \sqrt {\frac {x}{1-x^3}} \sqrt {1-x^3} \arcsin \left (x^{3/2}\right )}{3 \sqrt {x}}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.08, size = 48, normalized size = 1.17 \begin {gather*} \frac {2 \sqrt {-\frac {x}{-1+x^3}} \sqrt {-1+x^3} \log \left (x^{3/2}+\sqrt {-1+x^3}\right )}{3 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x/(1 - x^3)],x]

[Out]

(2*Sqrt[-(x/(-1 + x^3))]*Sqrt[-1 + x^3]*Log[x^(3/2) + Sqrt[-1 + x^3]])/(3*Sqrt[x])

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Maple [A]
time = 0.54, size = 43, normalized size = 1.05

method	result	size
default	\(\frac {2 \sqrt {-\frac {x}{x^{3}-1}}\, \left (x^{3}-1\right ) \arctanh \left (\frac {\sqrt {x^{4}-x}}{x^{2}}\right )}{3 \sqrt {\left (x^{3}-1\right ) x}}\)	\(43\)
trager	\(\frac {\mathit {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-2 \sqrt {-\frac {x}{x^{3}-1}}\, x^{4}-2 \mathit {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \sqrt {-\frac {x}{x^{3}-1}}\, x +\mathit {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )}{3}\)	\(60\)
elliptic	\(\frac {2 \sqrt {-\frac {x}{x^{3}-1}}\, \sqrt {\left (x^{3}-1\right ) x}\, \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -1\right )}}\, \left (x -1\right )^{2} \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x -1\right )}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -1\right )}}\, \left (F\left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -1\right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-\Pi \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -1\right )}}, \frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{x \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (x -1\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\)	\(312\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(-x/(x^3-1))^(1/2)*(x^3-1)/((x^3-1)*x)^(1/2)*arctanh((x^4-x)^(1/2)/x^2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-x/(x^3 - 1)), x)

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Fricas [A]
time = 0.66, size = 33, normalized size = 0.80 \begin {gather*} \frac {1}{3} \, \arctan \left (\frac {2 \, {\left (x^{4} - x\right )} \sqrt {-\frac {x}{x^{3} - 1}}}{2 \, x^{3} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*arctan(2*(x^4 - x)*sqrt(-x/(x^3 - 1))/(2*x^3 - 1))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {\frac {x}{1 - x^{3}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(x/(1 - x**3)), x)

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Giac [A]
time = 0.50, size = 16, normalized size = 0.39 \begin {gather*} \frac {2}{3} \, \arctan \left (\sqrt {\frac {1}{x^{3}} - 1}\right ) \mathrm {sgn}\left (x^{3} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*arctan(sqrt(1/x^3 - 1))*sgn(x^3 - 1)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \sqrt {-\frac {x}{x^3-1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x/(x^3 - 1))^(1/2),x)

[Out]

int((-x/(x^3 - 1))^(1/2), x)

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Chatgpt [F] Failed to verify
time = 1.00, size = 27, normalized size = 0.66 \begin {gather*} -\frac {2 \sqrt {-x^{3}+1}\, \sqrt {x}}{3}+\frac {2 \arcsin \left (\sqrt {-x^{3}+1}\right )}{3} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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