3.367 \(\int \frac{\sqrt{x}}{1+x^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0061983, antiderivative size = 10, normalized size of antiderivative = 1., 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 = {329, 275, 203} \[ \frac{2}{3} \tan ^{-1}\left (x^{3/2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/(1 + x^3),x]

[Out]

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

Rule 329

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 275

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 203

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

Rubi steps

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

Mathematica [A]  time = 0.0024226, size = 10, normalized size = 1. \[ \frac{2}{3} \tan ^{-1}\left (x^{3/2}\right ) \]

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 = 0.005, size = 7, normalized size = 0.7 \begin{align*}{\frac{2}{3}\arctan \left ({x}^{{\frac{3}{2}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.51372, size = 8, normalized size = 0.8 \begin{align*} \frac{2}{3} \, \arctan \left (x^{\frac{3}{2}}\right ) \end{align*}

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 = 1.46513, size = 28, normalized size = 2.8 \begin{align*} \frac{2}{3} \, \arctan \left (x^{\frac{3}{2}}\right ) \end{align*}

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]  time = 1.80969, size = 42, normalized size = 4.2 \begin{align*} - \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{align*}

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 = 1.10189, size = 8, normalized size = 0.8 \begin{align*} \frac{2}{3} \, \arctan \left (x^{\frac{3}{2}}\right ) \end{align*}

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))