3.4.67 \(\int \frac {\sqrt {x}}{1+x^3} \, dx\) [367]

Optimal. Leaf size=10 \[ \frac {2}{3} \tan ^{-1}\left (x^{3/2}\right ) \]

[Out]

2/3*arctan(x^(3/2))

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Rubi [A]
time = 0.00, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {335, 281, 209} \begin {gather*} \frac {2}{3} \text {ArcTan}\left (x^{3/2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/(1 + x^3),x]

[Out]

(2*ArcTan[x^(3/2)])/3

Rule 209

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

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]

Rubi steps

\begin {align*} \int \frac {\sqrt {x}}{1+x^3} \, dx &=2 \text {Subst}\left (\int \frac {x^2}{1+x^6} \, dx,x,\sqrt {x}\right )\\ &=\frac {2}{3} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,x^{3/2}\right )\\ &=\frac {2}{3} \tan ^{-1}\left (x^{3/2}\right )\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 10, normalized size = 1.00 \begin {gather*} \frac {2}{3} \tan ^{-1}\left (x^{3/2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/(1 + x^3),x]

[Out]

(2*ArcTan[x^(3/2)])/3

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Maple [A]
time = 0.23, size = 7, normalized size = 0.70

method	result	size
derivativedivides	\(\frac {2 \arctan \left (x^{\frac {3}{2}}\right )}{3}\)	\(7\)
default	\(\frac {2 \arctan \left (x^{\frac {3}{2}}\right )}{3}\)	\(7\)
meijerg	\(\frac {2 \arctan \left (x^{\frac {3}{2}}\right )}{3}\)	\(7\)
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {x^{3} \RootOf \left (\textit {\_Z}^{2}+1\right )+2 x^{\frac {3}{2}}-\RootOf \left (\textit {\_Z}^{2}+1\right )}{\left (x +1\right ) \left (x^{2}-x +1\right )}\right )}{3}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*arctan(x^(3/2))

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Maxima [A]
time = 0.51, size = 6, normalized size = 0.60 \begin {gather*} \frac {2}{3} \, \arctan \left (x^{\frac {3}{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*arctan(x^(3/2))

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Fricas [A]
time = 0.36, size = 6, normalized size = 0.60 \begin {gather*} \frac {2}{3} \, \arctan \left (x^{\frac {3}{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*arctan(x^(3/2))

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (8) = 16\).
time = 0.26, size = 42, normalized size = 4.20 \begin {gather*} - \frac {2 \operatorname {atan}{\left (\sqrt {x} \right )}}{3} + \frac {2 \operatorname {atan}{\left (2 \sqrt {x} - \sqrt {3} \right )}}{3} + \frac {2 \operatorname {atan}{\left (2 \sqrt {x} + \sqrt {3} \right )}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*atan(sqrt(x))/3 + 2*atan(2*sqrt(x) - sqrt(3))/3 + 2*atan(2*sqrt(x) + sqrt(3))/3

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Giac [A]
time = 1.40, size = 6, normalized size = 0.60 \begin {gather*} \frac {2}{3} \, \arctan \left (x^{\frac {3}{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*arctan(x^(3/2))

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Mupad [B]
time = 0.15, size = 6, normalized size = 0.60 \begin {gather*} \frac {2\,\mathrm {atan}\left (x^{3/2}\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)/(x^3 + 1),x)

[Out]

(2*atan(x^(3/2)))/3

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